量子叠加态

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Quantum superposition of states and decoherence

thumb |直立=1.5 |量子态叠加和退相干 Quantum superposition of states and decoherence 量子态叠加与退相干

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Quantum superposition is a fundamental principle of quantum mechanics. It states that, much like waves in classical physics, any two (or more) quantum states can be added together ("superposed") and the result will be another valid quantum state; and conversely, that every quantum state can be represented as a sum of two or more other distinct states. Mathematically, it refers to a property of solutions to the Schrödinger equation; since the Schrödinger equation is linear, any linear combination of solutions will also be a solution.

“量子叠加”是量子力学的基本原理。它指出,就像经典物理中的波一样,任何两个(或更多)量子态可以加在一起(“叠加”),结果将是另一个有效的量子态;相反,每个量子态可以表示为两个或更多其他不同态的和。在数学上,它指的是薛定谔方程性质;由于Schrödinger方程是线性的,所以解的任何线性组合也将是一个解。

Quantum superposition is a fundamental principle of quantum mechanics. It states that, much like waves in classical physics, any two (or more) quantum states can be added together ("superposed") and the result will be another valid quantum state; and conversely, that every quantum state can be represented as a sum of two or more other distinct states. Mathematically, it refers to a property of solutions to the Schrödinger equation; since the Schrödinger equation is linear, any linear combination of solutions will also be a solution.

态叠加原理Quantum superposition是量子力学的基本原理。它指出,就像经典物理学中的波一样,任何两个(或更多)量子态可以叠加在一起(“叠加”) ,结果将是另一个有效的量子态; 反之,每个量子态可以表示为两个或更多其他不同状态的和。在数学上,它指的是 薛定谔方程Schrödinger equation的解的性质; 因为薛定谔方程是线性的,解的任何线性组合也是一个解。


An example of a physically observable manifestation of the wave nature of quantum systems is the interference peaks from an electron beam in a double-slit experiment. The pattern is very similar to the one obtained by diffraction of classical waves.

量子系统的波性质的物理可观察表现的一个例子是双缝实验电子束的干涉峰。这种模式与经典波的衍射获得的模式非常相似。

An example of a physically observable manifestation of the wave nature of quantum systems is the interference peaks from an electron beam in a double-slit experiment. The pattern is very similar to the one obtained by diffraction of classical waves.

量子系统波动本质的一个物理上可观察的表现就是双缝实验中电子束的干涉峰值。这种图样与经典波绕射得到的图样非常相似。


Another example is a quantum logical qubit state, as used in quantum information processing, which is a quantum superposition of the "basis states" [math]\displaystyle{ |0 \rangle }[/math] and [math]\displaystyle{ |1 \rangle }[/math].

另一个例子是量子逻辑 量子态,在量子信息处理中使用,它是“基态”[math]\displaystyle{ | 0\rangle }[/math][math]\displaystyle{ | 1\rangle }[/math]的量子叠加。

Another example is a quantum logical qubit state, as used in quantum information processing, which is a quantum superposition of the "basis states" [math]\displaystyle{ |0 \rangle }[/math] and [math]\displaystyle{ |1 \rangle }[/math].

另一个例子是量子逻辑量子比特状态,用于量子信息处理,是“基态” [math]\displaystyle{ |0 \rangle }[/math][math]\displaystyle{ |1 \rangle }[/math] 的态叠加。

Here [math]\displaystyle{ |0 \rangle }[/math] is the Dirac notation for the quantum state that will always give the result 0 when converted to classical logic by a measurement. Likewise [math]\displaystyle{ |1 \rangle }[/math] is the state that will always convert to 1. Contrary to a classical bit that can only be in the state corresponding to 0 or the state corresponding to 1, a qubit may be in a superposition of both states. This means that the probabilities of measuring 0 or 1 for a qubit are in general neither 0.0 nor 1.0, and multiple measurements made on qubits in identical states will not always give the same result.

这里[math]\displaystyle{ | 0\rangle }[/math]是量子态的狄拉克符号Dirac notation,当通过测量转换为经典逻辑时,它总是给出结果0。同样地,[math]\displaystyle{ | 1\rangle }[/math]是始终转换为1的状态。与经典的比特bit只能处于对应于0的状态或对应于1的状态相反,量子比特可以处于两种状态的叠加。这意味着对一个量子比特测量0或1的概率通常既不是0.0也不是1.0,对处于相同状态的量子比特进行多次测量并不总是得到相同的结果。

Here [math]\displaystyle{ |0 \rangle }[/math] is the Dirac notation for the quantum state that will always give the result 0 when converted to classical logic by a measurement. Likewise [math]\displaystyle{ |1 \rangle }[/math] is the state that will always convert to 1. Contrary to a classical bit that can only be in the state corresponding to 0 or the state corresponding to 1, a qubit may be in a superposition of both states. This means that the probabilities of measuring 0 or 1 for a qubit are in general neither 0.0 nor 1.0, and multiple measurements made on qubits in identical states will not always give the same result.

这里 < math > | | 0 rangle </math > 是量子态的 狄拉克Dirac 符号,当通过测量转换为经典逻辑时,它总是给出结果0。类似地,< math > | 1 rangle </math > 是总是转换为1的状态。与只能处于对应于0的状态或对应于1的状态的经典位相反,量子比特可以处于两种状态的叠加状态。这意味着一个量子比特测量0或1的概率通常既不是0.0也不是1.0,在相同状态下对量子比特进行多次测量并不总能得到相同的结果。


Concept概念

The principle of quantum superposition states that if a physical system may be in one of many configurations—arrangements of particles or fields—then the most general state is a combination of all of these possibilities, where the amount in each configuration is specified by a complex number.

The principle of quantum superposition states that if a physical system may be in one of many configurations—arrangements of particles or fields—then the most general state is a combination of all of these possibilities, where the amount in each configuration is specified by a complex number.

态叠加原理指出,如果一个物理系统可能处于多种构型中的一种---- 粒子或场的排列---- 那么最普遍的状态就是所有这些可能性的组合,其中每种构型的数量由一个复数来确定。


For example, if there are two configurations labelled by 0 and 1, the most general state would be

For example, if there are two configurations labelled by 0 and 1, the most general state would be

例如,如果有两个由0和1标记的配置,最一般的状态是


[math]\displaystyle{ c_0 {\mid} 0 \rangle + c_1 {\mid} 1 \rangle }[/math]

[math]\displaystyle{ c_0 {\mid} 0 \rangle + c_1 {\mid} 1 \rangle }[/math]



where the coefficients are complex numbers describing how much goes into each configuration.

where the coefficients are complex numbers describing how much goes into each configuration.

其中的系数是复数,用来描述每个配置中有多少。


The principle was described by Paul Dirac as follows:

The principle was described by Paul Dirac as follows:

保罗 · 狄拉克对这一原则的描述如下:


The general principle of superposition of quantum mechanics applies to the states [that are theoretically possible without mutual interference or contradiction] ... of any one dynamical system. It requires us to assume that between these states there exist peculiar relationships such that whenever the system is definitely in one state we can consider it as being partly in each of two or more other states. The original state must be regarded as the result of a kind of superposition of the two or more new states, in a way that cannot be conceived on classical ideas. Any state may be considered as the result of a superposition of two or more other states, and indeed in an infinite number of ways. Conversely, any two or more states may be superposed to give a new state...

The general principle of superposition of quantum mechanics applies to the states [that are theoretically possible without mutual interference or contradiction] ... of any one dynamical system. It requires us to assume that between these states there exist peculiar relationships such that whenever the system is definitely in one state we can consider it as being partly in each of two or more other states. The original state must be regarded as the result of a kind of superposition of the two or more new states, in a way that cannot be conceived on classical ideas. Any state may be considered as the result of a superposition of two or more other states, and indeed in an infinite number of ways. Conversely, any two or more states may be superposed to give a new state...

量子力学叠加的一般原理适用于任何一个动力系统的状态[理论上可能没有相互干涉或矛盾]。它要求我们假设这些状态之间存在着特殊的关系,比如每当系统明确处于一种状态时,我们可以认为它部分处于两种或两种以上的其他状态中的每一种。原始状态必须被看作是两个或两个以上新状态的一种叠加的结果,以一种不能用经典观点来构想的方式。任何状态都可以被认为是两个或两个以上其他状态叠加的结果,而且确实有无限多种方式。相反,任何两个或两个以上的状态可以叠加成一个新状态... ...


The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b, according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both a and b [i.e., either a or b]. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.[1]

The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b, according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both a and b [i.e., either a or b]. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.

如果我们考虑两个态A和B的叠加,那么叠加过程的非经典性质就清楚地显示出来了,这样就存在一个观测,当对处于A态的系统进行观测时,它肯定会导致一个特定的结果,比如说A,当在B状态下对系统进行观测时,肯定会导致一些不同的结果,比如说B。当对处于叠加状态的系统进行观测时,会有什么结果?答案是,根据概率定律,结果有时是a,有时是b,这取决于叠加过程中a和b的相对权重。它永远不会与a和b(即a或b)不同。因此,由叠加形成的状态的中间性质通过观察的特定结果介于原始状态的相应概率之间的概率来表示,而不是通过结果本身介于原始状态的相应结果之间来表示。


Anton Zeilinger, referring to the prototypical example of the double-slit experiment, has elaborated regarding the creation and destruction of quantum superposition:

Anton Zeilinger, referring to the prototypical example of the double-slit experiment, has elaborated regarding the creation and destruction of quantum superposition:

安东 · 泽林格,提到了双缝实验的原型例子,详细阐述了态叠加原理的创造和毁灭:


"[T]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is the essential criterion for quantum interference to appear.[2]

"[T]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is the essential criterion for quantum interference to appear.

“振幅叠加... ... 只有在没有办法知道,甚至在原则上,粒子走哪条路径的情况下才有效。重要的是要认识到,这并不意味着一个观察者实际上注意到发生了什么。如果路径信息原则上可以从实验中获得,或者即使它分散在环境中,超出了任何技术可能性可以恢复的范围,但原则上仍然“在那里” ,那么破坏干涉图样就足够了缺乏这样的信息是量子干涉出现的基本标准。

Theory理论

Examples案例

For an equation describing a physical phenomenon, the superposition principle states that a combination of solutions to a linear equation is also a solution of it. When this is true the equation is said to obey the superposition principle. Thus, if state vectors f1, f2 and f3 each solve the linear equation on ψ, then ψ = c1f1 + c2f2 + c3f3 would also be a solution, in which each c is a coefficient. The Schrödinger equation is linear, so quantum mechanics follows this.

对于描述物理现象的方程,叠加原理认为线性方程的解的组合也是它的解。当这点为真时,方程被认为服从叠加原理。因此,如果状态向量 f1, f2f3 每个都是线性方程 在 ψ的解,则 ψ = c1f1 + c2f2 + c3f3 也是一个解,其中每个 c是一个系数。薛定谔方程是线性的,所以量子力学服从这一原理。

For an equation describing a physical phenomenon, the superposition principle states that a combination of solutions to a linear equation is also a solution of it. When this is true the equation is said to obey the superposition principle. Thus, if state vectors , and each solve the linear equation on ψ, then would also be a solution, in which each is a coefficient. The Schrödinger equation is linear, so quantum mechanics follows this.

对于一个描述物理现象的方程,叠加原理指出,一个线性方程的解的组合也是它的解。如果这是正确的,那么这个方程就是服从叠加原理的。因此,如果状态向量,每个解线性方程 ψ,那么也是一个解,其中每个都是一个系数。薛定谔方程是线性的,所以量子力学服从这一原理。


For example, consider an electron with two possible configurations, up and down. This describes the physical system of a qubit.

例如,考虑一个电子有两种可能的配置,上下。这描述了量子比特的物理系统。

For example, consider an electron with two possible configurations, up and down. This describes the physical system of a qubit.

例如,考虑一个有两种可能构型的电子,上下构型。这描述了量子位的物理系统。


[math]\displaystyle{ c_1 {\mid} {\uparrow} \rangle + c_2 {\mid} {\downarrow} \rangle }[/math]

[math]\displaystyle{ c_1 {\mid} {\uparrow} \rangle + c_2 {\mid} {\downarrow} \rangle }[/math]

1{ mid }{ uparrow } rangle + c2{ mid }{ downarrow } rangle </math >


is the most general state. But these coefficients dictate probabilities for the system to be in either configuration. The probability for a specified configuration is given by the square of the absolute value of the coefficient. So the probabilities should add up to 1. The electron is in one of those two states for sure.

is the most general state. But these coefficients dictate probabilities for the system to be in either configuration. The probability for a specified configuration is given by the square of the absolute value of the coefficient. So the probabilities should add up to 1. The electron is in one of those two states for sure.

是最普遍的状态。但是这些系数决定了系统处于任何一种配置状态的概率。一个特定构型的概率是系数的绝对值的平方。所以概率加起来应该等于1。电子肯定处于这两种状态之一。


[math]\displaystyle{ p_\text{up} = {\mid} c_1 {\mid}^2 }[/math]

[math]\displaystyle{ p_\text{up} = {\mid} c_1 {\mid}^2 }[/math]

1{ mid } ^ 2 </math >

[math]\displaystyle{ p_\text{down} = {\mid} c_2 \mid^2 }[/math]

[math]\displaystyle{ p_\text{down} = {\mid} c_2 \mid^2 }[/math]

2 mid ^ 2 </math >

[math]\displaystyle{ p_\text{up or down} = p_\text{up} + p_\text{down} = 1 }[/math]

[math]\displaystyle{ p_\text{up or down} = p_\text{up} + p_\text{down} = 1 }[/math]

1 </math > p text { up or down } = p text { up } + p text { down } = 1 </math >


Continuing with this example: If a particle can be in state  up and  down, it can also be in a state where it is an amount 3i/5 in up and an amount 4/5 in down.

Continuing with this example: If a particle can be in state  up and  down, it can also be in a state where it is an amount in up and an amount in down.

继续这个例子: 如果一个粒子可以处于向上 和向下的 状态,那么它也可以处于向上 3i/5 和向下4/5的状态。


[math]\displaystyle{ |\psi\rangle = {3\over 5} i {\mid}{\uparrow}\rangle + {4\over 5} {\mid}{\downarrow}\rangle. }[/math]

[math]\displaystyle{ |\psi\rangle = {3\over 5} i {\mid}{\uparrow}\rangle + {4\over 5} {\mid}{\downarrow}\rangle. }[/math]

< math > | psi rangle = {3 over 5} i { mid }{ uparrow } rangle + {4 over 5}{ mid }{ downarrow } rangle. </math >


In this, the probability for up is [math]\displaystyle{ \left|\frac{3i}{5}\right|^2=\frac{9}{25} }[/math]. The probability for down is [math]\displaystyle{ \left|\frac{4}{5}\right|^2=\frac{16}{25} }[/math]. Note that [math]\displaystyle{ \frac{9}{25}+\frac{16}{25}=1 }[/math].

In this, the probability for up is [math]\displaystyle{ \left|\frac{3i}{5}\right|^2=\frac{9}{25} }[/math]. The probability for down is [math]\displaystyle{ \left|\frac{4}{5}\right|^2=\frac{16}{25} }[/math]. Note that [math]\displaystyle{ \frac{9}{25}+\frac{16}{25}=1 }[/math].

在这里,向上的概率是 [math]\displaystyle{ \left|\frac{3i}{5}\right|^2=\frac{9}{25} }[/math]。向下的概率是 [math]\displaystyle{ \left|\frac{4}{5}\right|^2=\frac{16}{25} }[/math]。注意 [math]\displaystyle{ \frac{9}{25}+\frac{16}{25}=1 }[/math]


In the description, only the relative size of the different components matter, and their angle to each other on the complex plane. This is usually stated by declaring that two states which are a multiple of one another are the same as far as the description of the situation is concerned. Either of these describe the same state for any nonzero [math]\displaystyle{ \alpha }[/math]

In the description, only the relative size of the different components matter, and their angle to each other on the complex plane. This is usually stated by declaring that two states which are a multiple of one another are the same as far as the description of the situation is concerned. Either of these describe the same state for any nonzero [math]\displaystyle{ \alpha }[/math]

在描述中,只有不同部件的相对大小以及它们在复平面上的相对角度才是重要的。这种说法通常是宣布,两个国家是彼此的倍数就描述情况而言,是相同的。对于任何非零的 < math > alpha </math > ,它们都描述了相同的状态

[math]\displaystyle{ \lt math\gt 《数学》 |\psi \rangle \approx \alpha |\psi \rangle |\psi \rangle \approx \alpha |\psi \rangle | psi rangle approx alpha | psi rangle }[/math]

</math>

数学


The fundamental law of quantum mechanics is that the evolution is linear, meaning that if state A turns into A′ and B turns into B′ after 10 seconds, then after 10 seconds the superposition [math]\displaystyle{ \psi }[/math] turns into a mixture of A′ and B′ with the same coefficients as A and B.

The fundamental law of quantum mechanics is that the evolution is linear, meaning that if state A turns into A′ and B turns into B′ after 10 seconds, then after 10 seconds the superposition [math]\displaystyle{ \psi }[/math] turns into a mixture of A′ and B′ with the same coefficients as A and B.

量子力学的基本定律是: 进化是线性的,也就是说,如果状态 a 在10秒后变成 a ′ ,b 在10秒后变成 b ′ ,那么10秒后叠加态就会变成 a ′和 b ′的混合物,其系数与 a 和 b 的系数相同。


For example, if we have the following

For example, if we have the following

例如,如果我们有下面的


[math]\displaystyle{ {\mid} {\uparrow} \rangle \to {\mid} {\downarrow} \rangle }[/math]

[math]\displaystyle{ {\mid} {\uparrow} \rangle \to {\mid} {\downarrow} \rangle }[/math]

{ mid }{ uparrow } rangle to { mid }{ downarrow } rangle

[math]\displaystyle{ {\mid} {\downarrow} \rangle \to \frac{3i}{5} {\mid} {\uparrow} \rangle + \frac{4}{5} {\mid} {\downarrow} \rangle }[/math]

[math]\displaystyle{ {\mid} {\downarrow} \rangle \to \frac{3i}{5} {\mid} {\uparrow} \rangle + \frac{4}{5} {\mid} {\downarrow} \rangle }[/math]

{3 i }{5}{ mid }{ uparrow } rangle + frac {4}{ mid }{ downarrow } rangle </math >


Then after those 10 seconds our state will change to

Then after those 10 seconds our state will change to

10秒之后,我们的状态就会变成


[math]\displaystyle{ c_1 {\mid} {\uparrow} \rangle + c_2 {\mid} {\downarrow} \rangle \to c_1 \left( {\mid} {\downarrow} \rangle\right) + c_2 \left(\frac{3i}{5} {\mid} {\uparrow} \rangle + \frac{4}{5} {\mid} {\downarrow} \rangle \right) }[/math]

[math]\displaystyle{ c_1 {\mid} {\uparrow} \rangle + c_2 {\mid} {\downarrow} \rangle \to c_1 \left( {\mid} {\downarrow} \rangle\right) + c_2 \left(\frac{3i}{5} {\mid} {\uparrow} \rangle + \frac{4}{5} {\mid} {\downarrow} \rangle \right) }[/math]

< math > c _ 1{ mid }{ uparrow } rangle + c _ 2{ mid }{ downarrow } rangle to c _ 1 left ({ mid }{ downarrow } rangle right) + c _ 2 left (frac {3i }{ mid }{ uparrow } rangle + frac {4}{ mid }{ downarrow } right) </math >


So far there have just been 2 configurations, but there can be infinitely many.

So far there have just been 2 configurations, but there can be infinitely many.

到目前为止只发现两种构型,但是很可能有无穷多种。


In illustration, a particle can have any position, so that there are different configurations which have any value of the position x. These are written:

In illustration, a particle can have any position, so that there are different configurations which have any value of the position . These are written:

在例子中,一个粒子可以有任何位置,因此有不同的构型,它们具有任意的位置值。这些是写下来的:

[math]\displaystyle{ \lt math\gt 《数学》 |x\rangle |x\rangle 我们会找到他的 }[/math]

</math>

数学


The principle of superposition guarantees that there are states which are arbitrary superpositions of all the positions with complex coefficients:

The principle of superposition guarantees that there are states which are arbitrary superpositions of all the positions with complex coefficients:

叠加原理保证了所有复系数位置的任意叠加状态:


[math]\displaystyle{ \lt math\gt 《数学》 \sum_x \psi(x) |x\rangle \sum_x \psi(x) |x\rangle Sum _ x psi (x) | x rangle }[/math]

</math>

数学


This sum is defined only if the index x is discrete. If the index is over [math]\displaystyle{ \reals }[/math], then the sum is replaced by an integral. The quantity [math]\displaystyle{ \psi(x) }[/math] is called the wavefunction of the particle.

This sum is defined only if the index  is discrete. If the index is over [math]\displaystyle{ \reals }[/math], then the sum is replaced by an integral. The quantity [math]\displaystyle{ \psi(x) }[/math] is called the wavefunction of the particle.

仅当索引 x是离散的时候才定义这个和。如果索引大于[math]\displaystyle{ \reals }[/math] ,那么和将被替换为一个积分。数量 [math]\displaystyle{ \psi(x) }[/math] 被称为粒子的波函数。


If we consider a qubit with both position and spin, the state is a superposition of all possibilities for both:

If we consider a qubit with both position and spin, the state is a superposition of all possibilities for both:

如果我们考虑一个同时具有位置和自旋的量子位,那么这个状态就是两者的所有可能性的叠加:


[math]\displaystyle{ \lt math\gt 《数学》 \sum_x \psi_+(x)|x,{\uparrow}\rangle + \psi_-(x)|x,{\downarrow}\rangle \sum_x \psi_+(x)|x,{\uparrow}\rangle + \psi_-(x)|x,{\downarrow}\rangle Sum _ x psi _ + (x) | x,{ uparrow } rangle + psi _-(x) | x,{ downarrow } rangle \, }[/math]

\,</math>

,math


The configuration space of a quantum mechanical system cannot be worked out without some physical knowledge. The input is usually the allowed different classical configurations, but without the duplication of including both position and momentum.

The configuration space of a quantum mechanical system cannot be worked out without some physical knowledge. The input is usually the allowed different classical configurations, but without the duplication of including both position and momentum.

没有一些物理知识,量子力学系统的位形空间是无法计算出来的。输入通常是允许的不同的经典结构,但没有包括位置和动量的重复。


A pair of particles can be in any combination of pairs of positions. A state where one particle is at position x and the other is at position y is written [math]\displaystyle{ |x,y\rangle }[/math]. The most general state is a superposition of the possibilities:

A pair of particles can be in any combination of pairs of positions. A state where one particle is at position x and the other is at position y is written [math]\displaystyle{ |x,y\rangle }[/math]. The most general state is a superposition of the possibilities:

一对粒子可以处于任意对位置的组合中。一个粒子位于 x 位置,另一个粒子位于 y 位置的状态写成了[math]\displaystyle{ |x,y\rangle }[/math]。最普遍的状态是各种可能性的叠加:


[math]\displaystyle{ \lt math\gt 《数学》 \sum_{xy} A(x,y) |x,y\rangle \sum_{xy} A(x,y) |x,y\rangle Sum _ { xy } a (x,y) | x,y rangle \, }[/math]

\,</math>

,math


The description of the two particles is much larger than the description of one particle—it is a function in twice the number of dimensions. This is also true in probability, when the statistics of two random variables are correlated. If two particles are uncorrelated, the probability distribution for their joint position P(x, y) is a product of the probability of finding one at one position and the other at the other position:

两个粒子的描述比一个粒子的描述要大得多,它是两倍于维数的函数。当两个随机变量的统计是相关时,概率也是如此。如果两个粒子不相关,则它们的联合位置{math | P(x',y')}的概率分布是在一个位置找到一个粒子和在另一个位置找到另一个粒子的概率的乘积:

The description of the two particles is much larger than the description of one particle—it is a function in twice the number of dimensions. This is also true in probability, when the statistics of two random variables are correlated. If two particles are uncorrelated, the probability distribution for their joint position is a product of the probability of finding one at one position and the other at the other position:

对两个粒子的描述比对一个粒子的描述要大得多ーー这是维数的两倍的函数。当两个随机变量的统计量相关时,概率也是如此。如果两个粒子是不相关的,那么它们联合位置的概率分布就是在一个位置发现一个粒子而在另一个位置发现另一个粒子的概率的乘积:

[math]\displaystyle{ \lt math\gt 《数学》 P(x,y) = P_x (x) P_y(y) P(x,y) = P_x (x) P_y(y) P (x,y) = p_x (x) p_y (y) \, }[/math]

\,</math>

,math


In quantum mechanics, two particles can be in special states where the amplitudes of their position are uncorrelated. For quantum amplitudes, the word entanglement replaces[citation needed] the word correlation, but the analogy模板:Which is exact. A disentangled wave function has the form:

In quantum mechanics, two particles can be in special states where the amplitudes of their position are uncorrelated. For quantum amplitudes, the word entanglement replaces the word correlation, but the analogy is exact. A disentangled wave function has the form:

在量子力学中,两个粒子可以处于特殊的状态,其位置振幅是不相关的。对于量子振幅,纠缠这个词取代了关联这个词,但这个类比是准确的。一个解除纠缠的波函数有这样的形式:

[math]\displaystyle{ \lt math\gt 《数学》 A(x,y) = \psi_x(x)\psi_y(y) A(x,y) = \psi_x(x)\psi_y(y) A (x,y) = psi _ x (x) psi _ y (y) \, }[/math]

\,</math>

,math


while an entangled wavefunction does not have this form.

while an entangled wavefunction does not have this form.

而纠缠波函数没有这种形式。

模板:Expand section

Analogy with probability概率推论

In probability theory there is a similar principle. If a system has a probabilistic description, this description gives the probability of any configuration, and given any two different configurations, there is a state which is partly this and partly that, with positive real number coefficients, the probabilities, which say how much of each there is.

概率论中也有一个类似的原理。如果一个系统有一个概率描述,这个描述给出了任何配置的概率,并且给定了任何两个不同的配置,则存在一种状态部分为这个部分为那个,有正实数系数,其概率表示每个部分配置有多少。

In probability theory there is a similar principle. If a system has a probabilistic description, this description gives the probability of any configuration, and given any two different configurations, there is a state which is partly this and partly that, with positive real number coefficients, the probabilities, which say how much of each there is.

在概率论也有类似的原则。如果一个系统有一个概率描述,这个描述给出了任何配置的概率,给定任何两个不同的配置,则存在一种状态部分为这个部分为那个,有正实数系数,其概率表示每个部分配置有多少。


For example, if we have a probability distribution for where a particle is, it is described by the "state"

For example, if we have a probability distribution for where a particle is, it is described by the "state"

例如,如果我们有一个粒子在哪里的概率分布,它被描述为状态

[math]\displaystyle{ \lt math\gt \sum_x \rho(x) |x\rangle \sum_x \rho(x) |x\rangle }[/math]

</math>


Where [math]\displaystyle{ \rho }[/math] is the probability density function, a positive number that measures the probability that the particle will be found at a certain location.

Where [math]\displaystyle{ \rho }[/math] is the probability density function, a positive number that measures the probability that the particle will be found at a certain location.

其中,[math]\displaystyle{ \rho }[/math]是概率密度函数,一个正数,用来测量粒子在某个位置被发现的概率。


The evolution equation is also linear in probability, for fundamental reasons. If the particle has some probability for going from position x to y, and from z to y, the probability of going to y starting from a state which is half-x and half-z is a half-and-half mixture of the probability of going to y from each of the options. This is the principle of linear superposition in probability.

出于根本原因,进化方程在概率上也是线性的。如果粒子有一定概率从位置“x”到“y”以及从“z”到“y”,则从状态一半x和一半z开始到“y”的概率是从每个选项到“y”的概率的一半和一半的混合。这就是概率线性叠加原理。

The evolution equation is also linear in probability, for fundamental reasons. If the particle has some probability for going from position x to y, and from z to y, the probability of going to y starting from a state which is half-x and half-z is a half-and-half mixture of the probability of going to y from each of the options. This is the principle of linear superposition in probability.

由于基本原因,发展方程在概率上也是线性的。如果粒子有一定的概率从 x 到 y,从 z 到 y,则从状态一半x和一半z开始到“y”的概率是从每个选项到“y”的概率的一半和一半的混合。这是概率的线性叠加原理。


Quantum mechanics is different, because the numbers can be positive or negative. While the complex nature of the numbers is just a doubling, if you consider the real and imaginary parts separately, the sign of the coefficients is important. In probability, two different possible outcomes always add together, so that if there are more options to get to a point z, the probability always goes up. In quantum mechanics, different possibilities can cancel.

量子力学是不同的,因为数字可以是正的,也可以是负的。虽然数字的复数性质只是一个倍增,但如果你分开考虑实部和虚部,系数的符号就很重要。在概率论中,两种不同的可能结果总是加在一起,所以如果有更多的选择到达“z”点,概率总是上升。而在量子力学中,不同的可能性可以抵消。

Quantum mechanics is different, because the numbers can be positive or negative. While the complex nature of the numbers is just a doubling, if you consider the real and imaginary parts separately, the sign of the coefficients is important. In probability, two different possible outcomes always add together, so that if there are more options to get to a point z, the probability always goes up. In quantum mechanics, different possibilities can cancel.

量子力学是不同的,因为这些数字可以是正的也可以是负的。虽然这些数字的复杂本质只是一个双倍,但是如果你分别考虑实部和虚部,系数的符号就很重要了。在概率上,两个不同的可能结果总是相加,所以如果有更多的选项达到一个点 z,概率总是上升的。在量子力学,不同的可能性可以取消。


In probability theory with a finite number of states, the probabilities can always be multiplied by a positive number to make their sum equal to one. For example, if there is a three state probability system:

In probability theory with a finite number of states, the probabilities can always be multiplied by a positive number to make their sum equal to one. For example, if there is a three state probability system:

在概率论有限的情况下,概率总是可以乘以一个正数使得它们的和等于1。例如,如果有一个三态概率系统:

[math]\displaystyle{ \lt math\gt 《数学》 x |1\rangle + y |2\rangle + z |3\rangle x |1\rangle + y |2\rangle + z |3\rangle X | 1 rangle + y | 2 rangle + z | 3 rangle \, }[/math]

\,</math>

,math

where the probabilities [math]\displaystyle{ x,y,z }[/math] are positive numbers. Rescaling x,y,z so that

where the probabilities [math]\displaystyle{ x,y,z }[/math] are positive numbers. Rescaling x,y,z so that

其中 [math]\displaystyle{ x,y,z }[/math]的概率是正数。重新标定 x,y,z,则

[math]\displaystyle{ \lt math\gt 《数学》 x+y+z=1 x+y+z=1 X + y + z = 1 \, }[/math]

\,</math>

,math

The geometry of the state space is a revealed to be a triangle. In general it is a simplex. There are special points in a triangle or simplex corresponding to the corners, and these points are those where one of the probabilities is equal to 1 and the others are zero. These are the unique locations where the position is known with certainty.

The geometry of the state space is a revealed to be a triangle. In general it is a simplex. There are special points in a triangle or simplex corresponding to the corners, and these points are those where one of the probabilities is equal to 1 and the others are zero. These are the unique locations where the position is known with certainty.

状态空间的几何形状被揭示为一个三角形。一般来说,它是一个单形。在一个三角形或单纯形中,有一些特殊的点对应于角点,这些点是其中一个概率等于1,其他的概率等于零的点。这些都是独特的位置,其中的位置是确定的。


In a quantum mechanical system with three states, the quantum mechanical wavefunction is a superposition of states again, but this time twice as many quantities with no restriction on the sign:

In a quantum mechanical system with three states, the quantum mechanical wavefunction is a superposition of states again, but this time twice as many quantities with no restriction on the sign:

在一个具有三个态的量子力学系统中,量子力学波函数又是一个态的叠加,但是这次是两倍的量,不受符号的限制:

[math]\displaystyle{ \lt math\gt A|1\rangle + B|2\rangle + C|3\rangle = (A_r + iA_i) |1\rangle + (B_r + i B_i) |2\rangle + (C_r + iC_i) |3\rangle A|1\rangle + B|2\rangle + C|3\rangle = (A_r + iA_i) |1\rangle + (B_r + i B_i) |2\rangle + (C_r + iC_i) |3\rangle A | 1 rangle + b | 2 rangle + c | 3 rangle = (a _ r + iA _ i) | 1 rangle + (b _ r + i b _ i) | 2 rangle + (c _ r + iC _ i) | 3 rangle \, }[/math]

\,</math>


rescaling the variables so that the sum of the squares is 1, the geometry of the space is revealed to be a high-dimensional sphere

rescaling the variables so that the sum of the squares is 1, the geometry of the space is revealed to be a high-dimensional sphere

重新调整变量的大小,使得平方和为1,空间的几何形状被揭示为一个高维球体

[math]\displaystyle{ \lt math\gt A_r^2 + A_i^2 + B_r^2 + B_i^2 + C_r^2 + C_i^2 = 1 A_r^2 + A_i^2 + B_r^2 + B_i^2 + C_r^2 + C_i^2 = 1 2 + a i ^ 2 + b r ^ 2 + b i ^ 2 + c r ^ 2 + c i ^ 2 = 1 \, }[/math].

\,</math>.



A sphere has a large amount of symmetry, it can be viewed in different coordinate systems or bases. So unlike a probability theory, a quantum theory has a large number of different bases in which it can be equally well described. The geometry of the phase space can be viewed as a hint that the quantity in quantum mechanics which corresponds to the probability is the absolute square of the coefficient of the superposition.

球体具有大量的对称性,可以在不同的坐标系或中查看。因此,与概率论不同的是,量子理论有大量不同的基础,在这些基础上,量子理论同样可以得到很好的描述。相空间的几何结构可以看作是一个暗示,量子力学中对应概率的量是叠加系数的“绝对平方”。

A sphere has a large amount of symmetry, it can be viewed in different coordinate systems or bases. So unlike a probability theory, a quantum theory has a large number of different bases in which it can be equally well described. The geometry of the phase space can be viewed as a hint that the quantity in quantum mechanics which corresponds to the probability is the absolute square of the coefficient of the superposition.

一个球体具有大量的对称性,它可以在不同的坐标系或基础上观察。因此,与概率论不同,量子理论有大量不同的基,在这些基中,量子理论可以得到同样好的描述。相空间的几何形状可以看作是一个暗示,即量子力学中与概率相对应的量是叠加系数的绝对平方。

Hamiltonian evolution哈密顿演化

The numbers that describe the amplitudes for different possibilities define the kinematics, the space of different states. The dynamics describes how these numbers change with time. For a particle that can be in any one of infinitely many discrete positions, a particle on a lattice, the superposition principle tells you how to make a state:

The numbers that describe the amplitudes for different possibilities define the kinematics, the space of different states. The dynamics describes how these numbers change with time. For a particle that can be in any one of infinitely many discrete positions, a particle on a lattice, the superposition principle tells you how to make a state:

描述不同可能性振幅的数字定义了运动学,不同状态的空间。动态描述了这些数字是如何随着时间变化的。对于一个可以处于无限多个离散位置中任意一个的粒子,一个点阵上的粒子,叠加原理能告诉你如何创造一个状态:


[math]\displaystyle{ \lt math\gt 《数学》 \sum_n \psi_n |n\rangle \sum_n \psi_n |n\rangle [咒语] \, }[/math]

\,</math>

,math


So that the infinite list of amplitudes [math]\displaystyle{ (\ldots, \psi_{-2}, \psi_{-1}, \psi_0, \psi_1, \psi_2, \ldots) }[/math] completely describes the quantum state of the particle. This list is called the state vector, and formally it is an element of a Hilbert space, an infinite-dimensional complex vector space. It is usual to represent the state so that the sum of the absolute squares of the amplitudes is one:

So that the infinite list of amplitudes [math]\displaystyle{ (\ldots, \psi_{-2}, \psi_{-1}, \psi_0, \psi_1, \psi_2, \ldots) }[/math] completely describes the quantum state of the particle. This list is called the state vector, and formally it is an element of a Hilbert space, an infinite-dimensional complex vector space. It is usual to represent the state so that the sum of the absolute squares of the amplitudes is one:

所以无限的振幅列表完全描述了这个粒子的量子态。这个列表被称为状态向量,形式上它是希尔伯特空间的一个元素,一个无限维的复向量空间。通常用振幅的绝对平方和来表示状态:

[math]\displaystyle{ \lt math\gt 《数学》 \sum \psi_n^*\psi_n = 1 \sum \psi_n^*\psi_n = 1 总和 psi n ^ * psi n = 1 }[/math]

</math>

数学


For a particle described by probability theory random walking on a line, the analogous thing is the list of probabilities [math]\displaystyle{ (\ldots,P_{-2},P_{-1},P_0,P_1,P_2,\ldots) }[/math], which give the probability of any position. The quantities that describe how they change in time are the transition probabilities [math]\displaystyle{ \scriptstyle K_{x\rightarrow y}(t) }[/math], which gives the probability that, starting at x, the particle ends up at y time t later. The total probability of ending up at y is given by the sum over all the possibilities

For a particle described by probability theory random walking on a line, the analogous thing is the list of probabilities [math]\displaystyle{ (\ldots,P_{-2},P_{-1},P_0,P_1,P_2,\ldots) }[/math], which give the probability of any position. The quantities that describe how they change in time are the transition probabilities [math]\displaystyle{ \scriptstyle K_{x\rightarrow y}(t) }[/math], which gives the probability that, starting at x, the particle ends up at y time t later. The total probability of ending up at y is given by the sum over all the possibilities

对于一个由概率论随机行走描述的粒子,类似的事情是概率列表 [math]\displaystyle{ (\ldots,P_{-2},P_{-1},P_0,P_1,P_2,\ldots) }[/math],它给出了任何位置的概率。描述粒子在时间上如何变化的量是转移概率[math]\displaystyle{ \scriptstyle K_{x\rightarrow y}(t) }[/math] ,它给出了粒子从 x 开始到 y 时间 t 结束的概率。总概率以 y 收尾,是所有可能性的和


[math]\displaystyle{ \lt math\gt 《数学》 P_y(t_0+t) = \sum_x P_x(t_0) K_{x\rightarrow y}(t) P_y(t_0+t) = \sum_x P_x(t_0) K_{x\rightarrow y}(t) P _ y (t _ 0 + t) = sum _ x p _ x (t _ 0) k _ { x right tarrow y }(t) \, }[/math]

\,</math>

,math


The condition of conservation of probability states that starting at any x, the total probability to end up somewhere must add up to 1:

The condition of conservation of probability states that starting at any x, the total probability to end up somewhere must add up to 1:

概率守恒的条件是,从任意 x 开始,到达某处的总概率必须等于1:


[math]\displaystyle{ \lt math\gt 《数学》 \sum_y K_{x\rightarrow y} = 1 \sum_y K_{x\rightarrow y} = 1 1 = 1 \, }[/math]

\,</math>

,math


So that the total probability will be preserved, K is what is called a stochastic matrix.

So that the total probability will be preserved, K is what is called a stochastic matrix.

所以总概率保持不变,k 就是所谓的转移矩阵。


When no time passes, nothing changes: for 0 elapsed time [math]\displaystyle{ \scriptstyle K{x\rightarrow y}(0) = \delta_{xy} }[/math], the K matrix is zero except from a state to itself. So in the case that the time is short, it is better to talk about the rate of change of the probability instead of the absolute change in the probability.

When no time passes, nothing changes: for 0 elapsed time [math]\displaystyle{ \scriptstyle K{x\rightarrow y}(0) = \delta_{xy} }[/math], the K matrix is zero except from a state to itself. So in the case that the time is short, it is better to talk about the rate of change of the probability instead of the absolute change in the probability.

当没有时间流逝时,什么也不会改变: 对于0经过的时间 < math > scriptstyle k { x right tarrow y }(0) = delta _ { xy } </math > ,k 矩阵除了从一个状态到它本身之外是零。所以在时间很短的情况下,最好讨论概率的变化率,而不是概率的绝对变化率。


[math]\displaystyle{ \lt math\gt 《数学》 P_y(t+dt) = P_y(t) + dt \, \sum_x P_x R_{x\rightarrow y} P_y(t+dt) = P_y(t) + dt \, \sum_x P_x R_{x\rightarrow y} P _ y (t + dt) = p _ y (t) + dt,sum _ x p _ x r _ { x right tarrow y } \, }[/math]

\,</math>

,math


where [math]\displaystyle{ \scriptstyle R_{x\rightarrow y} }[/math] is the time derivative of the K matrix:

where [math]\displaystyle{ \scriptstyle R_{x\rightarrow y} }[/math] is the time derivative of the K matrix:

其中 < math > scriptstyle r _ { x right tarrow y } </math > 是 k 矩阵的时间导数:


[math]\displaystyle{ \lt math\gt 《数学》 R_{x\rightarrow y} = {K_{x\rightarrow y} \, dt - \delta_{xy} \over dt}. R_{x\rightarrow y} = {K_{x\rightarrow y} \, dt - \delta_{xy} \over dt}. R _ { x right tarrow y } = { k _ { x right tarrow y } ,dt-delta _ { xy }/dt }. \, }[/math]

\,</math>

,math


The equation for the probabilities is a differential equation that is sometimes called the master equation:

The equation for the probabilities is a differential equation that is sometimes called the master equation:

概率的等式是一个微分方程,有时也被称为主方程:


[math]\displaystyle{ \lt math\gt 《数学》 {dP_y \over dt} = \sum_x P_x R_{x\rightarrow y} {dP_y \over dt} = \sum_x P_x R_{x\rightarrow y} { dP _ y over dt } = sum _ x p _ x r _ { x right tarrow y } \, }[/math]

\,</math>

,math


The R matrix is the probability per unit time for the particle to make a transition from x to y. The condition that the K matrix elements add up to one becomes the condition that the R matrix elements add up to zero:

The R matrix is the probability per unit time for the particle to make a transition from x to y. The condition that the K matrix elements add up to one becomes the condition that the R matrix elements add up to zero:

R矩阵是粒子从 x 到 y 转变为单位时间的概率。K矩阵元素加起来等于1的条件成为 R 矩阵元素加起来等于零的条件:


[math]\displaystyle{ \lt math\gt \sum_y R_{x\rightarrow y} = 0 \sum_y R_{x\rightarrow y} = 0 \, }[/math]

\,</math>



One simple case to study is when the R matrix has an equal probability to go one unit to the left or to the right, describing a particle that has a constant rate of random walking. In this case [math]\displaystyle{ \scriptstyle R_{x\rightarrow y} }[/math] is zero unless y is either x + 1, x, or x − 1, when y is x + 1 or x − 1, the R matrix has value c, and in order for the sum of the R matrix coefficients to equal zero, the value of [math]\displaystyle{ R_{x\rightarrow x} }[/math] must be −2c. So the probabilities obey the discretized diffusion equation:

一个简单的例子是当R矩阵有相等的概率向左或向右移动一个单位时,描述了一个具有恒定随机游动速率的粒子。在这种情况下,[math]\displaystyle{ \scriptstyle R{x\rightarrow y} }[/math]为零,除非y是x + 1, x, 或 x − 1,当“y”为x + 1, x, 或 x − 1,时,“R”矩阵具有值“c”,并且为了使“R”矩阵系数之和等于零,[math]\displaystyle{ R{x\rightarrow x} }[/math]的值必须是−2c'。所以概率服从“离散扩散方程”:

One simple case to study is when the R matrix has an equal probability to go one unit to the left or to the right, describing a particle that has a constant rate of random walking. In this case [math]\displaystyle{ \scriptstyle R_{x\rightarrow y} }[/math] is zero unless y is either x + 1, x, or x − 1, when y is x + 1 or x − 1, the R matrix has value c, and in order for the sum of the R matrix coefficients to equal zero, the value of [math]\displaystyle{ R_{x\rightarrow x} }[/math] must be −2c. So the probabilities obey the discretized diffusion equation:

要研究的一个简单的例子是,当 R 矩阵向左或向右移动一个单位的概率相等时,描述的是一个具有恒定随机游动速率的粒子。在这种情况下 < math > scriptstyle r _ { x right tarrow y } </math > 是0,除非 y 是 x + 1,x,或 x-1,当 y 是 x + 1或 x-1时,r 矩阵具有值 c,为了使 r 矩阵系数之和等于零,< math > r { x right tarrow x } </math > 的值必须是-2c。所以概率服从离散化的扩散方程:


[math]\displaystyle{ \lt math\gt 《数学》 {dP_x \over dt } = c(P_{x+1} - 2P_x + P_{x-1}) {dP_x \over dt } = c(P_{x+1} - 2P_x + P_{x-1}) { dP _ x over dt } = c (p _ { x + 1}-2P _ x + p _ { x-1}) \, }[/math]

\,</math>

,math


which, when c is scaled appropriately and the P distribution is smooth enough to think of the system in a continuum limit becomes:

which, when c is scaled appropriately and the P distribution is smooth enough to think of the system in a continuum limit becomes:

当 c 适当缩放,p 分布足够平滑,可以认为系统是一个连续的极限时,它变成:


[math]\displaystyle{ \lt math\gt {\partial P(x,t) \over \partial t} = c {\partial^2 P \over \partial x^2 } {\partial P(x,t) \over \partial t} = c {\partial^2 P \over \partial x^2 } { partial p (x,t) over partial t } = c { partial ^ 2p over partial x ^ 2} \, }[/math]

\,</math>



Which is the diffusion equation.

Which is the diffusion equation.

也就是扩散方程。


Quantum amplitudes give the rate at which amplitudes change in time, and they are mathematically exactly the same except that they are complex numbers. The analog of the finite time K matrix is called the U matrix:

Quantum amplitudes give the rate at which amplitudes change in time, and they are mathematically exactly the same except that they are complex numbers. The analog of the finite time K matrix is called the U matrix:

量子振幅给出了时间振幅变化的速率,它们在数学上完全相同,除了它们是复数。有限时间 K 矩阵的类似物称为 U 矩阵:


[math]\displaystyle{ \lt math\gt 《数学》 \psi_n(t) = \sum_m U_{nm}(t) \psi_m \psi_n(t) = \sum_m U_{nm}(t) \psi_m Psi _ n (t) = sum _ m u _ { nm }(t) psi _ m \, }[/math]

\,</math>

,math


Since the sum of the absolute squares of the amplitudes must be constant, [math]\displaystyle{ U }[/math] must be unitary:

Since the sum of the absolute squares of the amplitudes must be constant, [math]\displaystyle{ U }[/math] must be unitary:

因为振幅的绝对平方和必须是常数,所以 < math > u </math > 必须归一:


[math]\displaystyle{ \lt math\gt \sum_n U^*_{nm} U_{np} = \delta_{mp} \sum_n U^*_{nm} U_{np} = \delta_{mp} \, }[/math]

\,</math>


or, in matrix notation,

or, in matrix notation,

或者用矩阵表示法,

[math]\displaystyle{ \lt math\gt U^\dagger U = I U^\dagger U = I U ^ dagger u = i \, }[/math]

\,</math>



The rate of change of U is called the Hamiltonian H, up to a traditional factor of i:

The rate of change of U is called the Hamiltonian H, up to a traditional factor of i:

U 的变化率称为哈密顿量 H,最高可达传统因子i:


[math]\displaystyle{ \lt math\gt H_{mn} = i{d \over dt} U_{mn} H_{mn} = i{d \over dt} U_{mn} H { mn } = i { d over dt } u { mn } }[/math]

</math>



The Hamiltonian gives the rate at which the particle has an amplitude to go from m to n. The reason it is multiplied by i is that the condition that U is unitary translates to the condition:

The Hamiltonian gives the rate at which the particle has an amplitude to go from m to n. The reason it is multiplied by i is that the condition that U is unitary translates to the condition:

哈密顿函数给出了粒子振幅从 m 到 n 的速率。它被乘以 i 的原因是条件 “U 是幺正的”转换为条件:


[math]\displaystyle{ \lt math\gt (I + i H^\dagger \, dt )(I - i H \, dt ) = I (I + i H^\dagger \, dt )(I - i H \, dt ) = I (i + i h ^ dagger,dt)(i-i h,dt) = i }[/math]

</math>


[math]\displaystyle{ \lt math\gt H^\dagger - H = 0 H^\dagger - H = 0 H ^ dagger-h = 0 \, }[/math]

\,</math>


which says that H is Hermitian. The eigenvalues of the Hermitian matrix H are real quantities, which have a physical interpretation as energy levels. If the factor i were absent, the H matrix would be antihermitian and would have purely imaginary eigenvalues, which is not the traditional way quantum mechanics represents observable quantities like the energy.

也就是说H是厄米特Hermitian的。厄米矩阵“H”的本征值是实数,它的物理解释是能级。如果因子“i”不存在,H矩阵将是反厄米矩阵,并且将具有纯粹的虚本征值,这不是量子力学表示能量等可观测量的传统方法。

which says that H is Hermitian. The eigenvalues of the Hermitian matrix H are real quantities, which have a physical interpretation as energy levels. If the factor i were absent, the H matrix would be antihermitian and would have purely imaginary eigenvalues, which is not the traditional way quantum mechanics represents observable quantities like the energy.

也就是说H是Hermitian。厄米矩阵“H”的本征值是实数,它的物理解释是能级。如果因子“i”不存在,H矩阵将是反厄米矩阵,并且将具有纯粹的虚本征值,这不是量子力学表示能量等可观测量的传统方法。


For a particle that has equal amplitude to move left and right, the Hermitian matrix H is zero except for nearest neighbors, where it has the value c. If the coefficient is everywhere constant, the condition that H is Hermitian demands that the amplitude to move to the left is the complex conjugate of the amplitude to move to the right. The equation of motion for [math]\displaystyle{ \psi }[/math] is the time differential equation:

For a particle that has equal amplitude to move left and right, the Hermitian matrix H is zero except for nearest neighbors, where it has the value c. If the coefficient is everywhere constant, the condition that H is Hermitian demands that the amplitude to move to the left is the complex conjugate of the amplitude to move to the right. The equation of motion for [math]\displaystyle{ \psi }[/math] is the time differential equation:

对于一个振幅相等的粒子,它的左右运动,埃尔米特矩阵 h 是零,除了最近的邻居,它的值是 c。如果系数处处不变,条件是 h 是厄米特的,要求振幅向左移动的条件是振幅向右移动的共轭复数。运动方程式表示的是时间微分方程:


[math]\displaystyle{ \lt math\gt i{d \psi_n \over dt} = c^* \psi_{n+1} + c \psi_{n-1} i{d \psi_n \over dt} = c^* \psi_{n+1} + c \psi_{n-1} I { d psi _ n over dt } = c ^ * psi { n + 1} + c psi _ { n-1} }[/math]

</math>



In the case in which left and right are symmetric, c is real. By redefining the phase of the wavefunction in time, [math]\displaystyle{ \psi\rightarrow \psi e^{i2ct} }[/math], the amplitudes for being at different locations are only rescaled, so that the physical situation is unchanged. But this phase rotation introduces a linear term.

In the case in which left and right are symmetric, c is real. By redefining the phase of the wavefunction in time, [math]\displaystyle{ \psi\rightarrow \psi e^{i2ct} }[/math], the amplitudes for being at different locations are only rescaled, so that the physical situation is unchanged. But this phase rotation introduces a linear term.

在左右对称的情况下,c 是实数。通过重新定义波函数在时间上的相位,不同位置的振幅只是重新调整了,这样物理情况就不会改变。但是这个相位旋转引入了一个线性项。

[math]\displaystyle{ \lt math\gt i{d \psi_n \over dt} = c \psi_{n+1} - 2c\psi_n + c\psi_{n-1}, i{d \psi_n \over dt} = c \psi_{n+1} - 2c\psi_n + c\psi_{n-1}, }[/math]

</math>


which is the right choice of phase to take the continuum limit. When [math]\displaystyle{ c }[/math] is very large and [math]\displaystyle{ \psi }[/math] is slowly varying so that the lattice can be thought of as a line, this becomes the free Schrödinger equation:

which is the right choice of phase to take the continuum limit. When [math]\displaystyle{ c }[/math] is very large and [math]\displaystyle{ \psi }[/math] is slowly varying so that the lattice can be thought of as a line, this becomes the free Schrödinger equation:

这是正确的选择阶段,以连续的极限。当 < math > c </math > 非常大,< math > psi </math > 缓慢变化,以至于格子可以看作一条线时,这就成了自由薛定谔方程:

[math]\displaystyle{ \lt math\gt i{ \partial \psi \over \partial t } = - {\partial^2 \psi \over \partial x^2} i{ \partial \psi \over \partial t } = - {\partial^2 \psi \over \partial x^2} I { partial psi over partial t } =-{ partial ^ 2 psi over partial x ^ 2} }[/math]

</math>



If there is an additional term in the H matrix that is an extra phase rotation that varies from point to point, the continuum limit is the Schrödinger equation with a potential energy:

If there is an additional term in the H matrix that is an extra phase rotation that varies from point to point, the continuum limit is the Schrödinger equation with a potential energy:

如果在 H 矩阵中有一个额外的项是一个额外的相位旋转,这个相位旋转点与点之间变化,连续极限就是具有势能的薛定谔方程:

[math]\displaystyle{ \lt math\gt i{ \partial \psi \over \partial t} = - {\partial^2 \psi \over \partial x^2} + V(x) \psi i{ \partial \psi \over \partial t} = - {\partial^2 \psi \over \partial x^2} + V(x) \psi I { partial psi over partial t } =-{ partial ^ 2 psi over partial x ^ 2} + v (x) psi }[/math]

</math>



These equations describe the motion of a single particle in non-relativistic quantum mechanics.

These equations describe the motion of a single particle in non-relativistic quantum mechanics.

这些方程描述了非相对论量子力学中单个粒子的运动。

Quantum mechanics in imaginary time虚时间量子力学

The analogy between quantum mechanics and probability is very strong, so that there are many mathematical links between them. In a statistical system in discrete time, t=1,2,3, described by a transition matrix for one time step [math]\displaystyle{ \scriptstyle K_{m\rightarrow n} }[/math], the probability to go between two points after a finite number of time steps can be represented as a sum over all paths of the probability of taking each path:

量子力学和概率论之间的类比性很强,因此它们之间有许多数学联系。在离散时间的统计系统中,t=1,2,3,由一个时间步的转移矩阵[math]\displaystyle{ \scriptstyle K_{m\rightarrow n} }[/math]描述,在有限个时间步之后,两点之间经过的概率可以表示为走每条路径的概率在所有路径上的和:

The analogy between quantum mechanics and probability is very strong, so that there are many mathematical links between them. In a statistical system in discrete time, t=1,2,3, described by a transition matrix for one time step [math]\displaystyle{ \scriptstyle K_{m\rightarrow n} }[/math], the probability to go between two points after a finite number of time steps can be represented as a sum over all paths of the probability of taking each path:

量子力学和概率论之间的类比性很强,因此它们之间有许多数学联系。在离散时间的统计系统中,t=1,2,3,由一个时间步的转移矩阵[math]\displaystyle{ \scriptstyle K{m\rightarrow n} }[/math]描述,在有限个时间步之后,两点之间经过的概率可以表示为走每条路径的概率在所有路径上的和:

[math]\displaystyle{ \lt math\gt K_{x\rightarrow y}(T) = \sum_{x(t)} \prod_t K_{x(t)x(t+1)} K_{x\rightarrow y}(T) = \sum_{x(t)} \prod_t K_{x(t)x(t+1)} K _ { x right tarrow y }(t) = sum _ { x (t)} prod _ t k _ { x (t) x (t + 1)} \, }[/math]

\,</math>


where the sum extends over all paths [math]\displaystyle{ x(t) }[/math] with the property that [math]\displaystyle{ x(0)=0 }[/math] and [math]\displaystyle{ x(T)=y }[/math]. The analogous expression in quantum mechanics is the path integral.

其中,和扩展到所有路径[math]\displaystyle{ x(t) }[/math],其属性为[math]\displaystyle{ x(0)=0 }[/math][math]\displaystyle{ x(T)=y }[/math]。量子力学中的类似表达式是路径积分

where the sum extends over all paths [math]\displaystyle{ x(t) }[/math] with the property that [math]\displaystyle{ x(0)=0 }[/math] and [math]\displaystyle{ x(T)=y }[/math]. The analogous expression in quantum mechanics is the path integral.

其中和扩展到所有路径 < math > x (t) </math > ,其属性是 < math > x (0) = 0 </math > 和 < math > x (t) = y </math > 。量子力学中类似的表达式是路径积分。


A generic transition matrix in probability has a stationary distribution, which is the eventual probability to be found at any point no matter what the starting point. If there is a nonzero probability for any two paths to reach the same point at the same time, this stationary distribution does not depend on the initial conditions. In probability theory, the probability m for the stochastic matrix obeys detailed balance when the stationary distribution [math]\displaystyle{ \rho_n }[/math] has the property:

概率论中的一般转移矩阵具有平稳分布,即无论起点是什么,在任何一点上找到的最终概率。如果任意两条路径同时到达同一点的概率为非零,则此平稳分布不依赖于初始条件。在概率论中,当平稳分布具有以下性质时,随机矩阵的概率m服从精细平衡

A generic transition matrix in probability has a stationary distribution, which is the eventual probability to be found at any point no matter what the starting point. If there is a nonzero probability for any two paths to reach the same point at the same time, this stationary distribution does not depend on the initial conditions. In probability theory, the probability m for the stochastic matrix obeys detailed balance when the stationary distribution [math]\displaystyle{ \rho_n }[/math] has the property:

一般的概率转移矩阵有一个平稳分布,这是在任何一点上,无论起点是什么,最终被发现的概率。如果任意两条路径同时到达同一点的概率是非零的,则这种稳态分布不依赖于初始条件。在概率论中,当稳态分布 < math > rho </math > 具有以下性质时,转移矩阵的概率 m 服从精细平衡:


[math]\displaystyle{ \lt math\gt \rho_n K_{n\rightarrow m} = \rho_m K_{m\rightarrow n} \rho_n K_{n\rightarrow m} = \rho_m K_{m\rightarrow n} 右旋糖胺 = 右旋糖胺 = 右旋糖胺 \, }[/math]

\,</math>

,math


Detailed balance says that the total probability of going from m to n in the stationary distribution, which is the probability of starting at m [math]\displaystyle{ \rho_m }[/math] times the probability of hopping from m to n, is equal to the probability of going from n to m, so that the total back-and-forth flow of probability in equilibrium is zero along any hop. The condition is automatically satisfied when n=m, so it has the same form when written as a condition for the transition-probability R matrix.

精细平衡表明,平稳分布中从m到n的总概率,即从m开始的概率乘以从m跳到n的概率,等于从n到m的概率,所以,在平衡状态下,沿着任何一个跳跃,总的来回概率为零。当n=m时,该条件自动满足,因此它在作为转移概率R矩阵的条件编写时具有相同的形式。

Detailed balance says that the total probability of going from m to n in the stationary distribution, which is the probability of starting at m [math]\displaystyle{ \rho_m }[/math] times the probability of hopping from m to n, is equal to the probability of going from n to m, so that the total back-and-forth flow of probability in equilibrium is zero along any hop. The condition is automatically satisfied when n=m, so it has the same form when written as a condition for the transition-probability R matrix.

精细平衡表明,平稳分布中从m到n的总概率,即从m开始的概率乘以从m跳到n的概率,等于从n到m的概率,所以,在平衡状态下,沿着任何一个跳跃,总的来回概率为零。当n=m时,该条件自动满足,因此它在作为转移概率R矩阵的条件编写时具有相同的形式。


[math]\displaystyle{ \lt math\gt \rho_n R_{n\rightarrow m} = \rho_m R_{m\rightarrow n} \rho_n R_{n\rightarrow m} = \rho_m R_{m\rightarrow n} 右旋糖胺等于右旋糖胺等于右旋糖胺 \, }[/math]

\,</math>



When the R matrix obeys detailed balance, the scale of the probabilities can be redefined using the stationary distribution so that they no longer sum to 1:

When the R matrix obeys detailed balance, the scale of the probabilities can be redefined using the stationary distribution so that they no longer sum to 1:

当 R 矩阵满足精细平衡时,可以使用平稳分布重新定义概率的大小,使它们不再和为1:


[math]\displaystyle{ \lt math\gt p'_n = \sqrt{\rho_n}\;p_n p'_n = \sqrt{\rho_n}\;p_n 2. p’ n = sqrt { rho n } ; p _ n \, }[/math]

\,</math>



In the new coordinates, the R matrix is rescaled as follows:

In the new coordinates, the R matrix is rescaled as follows:

在新的坐标系中,R 矩阵重新调整如下:


[math]\displaystyle{ \lt math\gt \sqrt{\rho_n} R_{n\rightarrow m} {1\over \sqrt{\rho_m}} = H_{nm} \sqrt{\rho_n} R_{n\rightarrow m} {1\over \sqrt{\rho_m}} = H_{nm} 1 over sqrt { rho _ m } = h _ { nm } \, }[/math]

\,</math>



and H is symmetric

and H is symmetric

H 是对称的

[math]\displaystyle{ \lt math\gt H_{nm} = H_{mn} H_{nm} = H_{mn} H _ { nm } = h _ { mn } \, }[/math]

\,</math>



This matrix H defines a quantum mechanical system:

This matrix H defines a quantum mechanical system:

这个矩阵 H 定义了一个量子力学系统:


[math]\displaystyle{ \lt math\gt i{d \over dt} \psi_n = \sum H_{nm} \psi_m i{d \over dt} \psi_n = \sum H_{nm} \psi_m I { d over dt } psi _ n = sum h { nm } psi _ m \, }[/math]

\,</math>

whose Hamiltonian has the same eigenvalues as those of the R matrix of the statistical system. The eigenvectors are the same too, except expressed in the rescaled basis. The stationary distribution of the statistical system is the ground state of the Hamiltonian and it has energy exactly zero, while all the other energies are positive. If H is exponentiated to find the U matrix:

其哈密顿量与统计系统的R矩阵的特征值相同。特征向量也是相同的,除了在重新缩放的基础上表示。统计系统的平稳分布是哈密顿量的“基态”,它的能量正好为零,而所有其他能量都为正。如果H被指数化以找到U矩阵:

whose Hamiltonian has the same eigenvalues as those of the R matrix of the statistical system. The eigenvectors are the same too, except expressed in the rescaled basis. The stationary distribution of the statistical system is the ground state of the Hamiltonian and it has energy exactly zero, while all the other energies are positive. If H is exponentiated to find the U matrix:

其哈密顿量与统计系统的 R 矩阵具有相同的特征值。特征向量也是相同的,除了用重标度基表示。统计系统的稳态分布是哈密顿量的基态,它的能量精确为零,而其它所有能量都是正的。如果 H 是求 U 矩阵的幂:


[math]\displaystyle{ \lt math\gt U(t) = e^{-iHt} U(t) = e^{-iHt} U (t) = e ^ {-iHt } \, }[/math]

\,</math>



and t is allowed to take on complex values, the K' matrix is found by taking time imaginary.

and t is allowed to take on complex values, the K' matrix is found by taking time imaginary.

T 可以取复数值,k’矩阵可以通过取时间虚数来求。


[math]\displaystyle{ \lt math\gt K'(t) = e^{-Ht} K'(t) = e^{-Ht} K’(t) = e ^ {-Ht } \, }[/math]

\,</math>



For quantum systems which are invariant under time reversal the Hamiltonian can be made real and symmetric, so that the action of time-reversal on the wave-function is just complex conjugation. If such a Hamiltonian has a unique lowest energy state with a positive real wave-function, as it often does for physical reasons, it is connected to a stochastic system in imaginary time. This relationship between stochastic systems and quantum systems sheds much light on supersymmetry.

对于在时间反转下不变的量子系统,哈密顿量可以是实的和对称的,因此时间反转对波函数的作用仅仅是复共轭。如果这样的哈密顿量有一个唯一的具有正实波函数的最低能量态,就像它经常由于物理原因所做的那样,它在虚时间内与一个随机系统相连。随机系统和量子系统之间的这种关系为超对称性提供了很多线索。

For quantum systems which are invariant under time reversal the Hamiltonian can be made real and symmetric, so that the action of time-reversal on the wave-function is just complex conjugation. If such a Hamiltonian has a unique lowest energy state with a positive real wave-function, as it often does for physical reasons, it is connected to a stochastic system in imaginary time. This relationship between stochastic systems and quantum systems sheds much light on supersymmetry.

对于具有时间反转不变性的量子系统,哈密顿量可以实现对称性,因此时间反转对波函数的作用仅仅是复共轭。如果这样一个哈密顿量有一个唯一的最低能态和一个正的实波函数,就像它经常由于物理原因所做的那样,它在虚时间内连接到一个随机系统。随机系统和量子系统之间的这种关系揭示了超对称性。

Experiments and applications实验与应用

Successful experiments involving superpositions of relatively large (by the standards of quantum physics) objects have been performed.[3]

Successful experiments involving superpositions of relatively large (by the standards of quantum physics) objects have been performed.

成功的实验涉及到相对较大(以量子物理学的标准)物体的叠加。


  • A 2013 experiment superposed molecules containing 15,000 each of protons, neutrons and electrons. The molecules were of compounds selected for their good thermal stability, and were evaporated into a beam at a temperature of 600 K. The beam was prepared from highly purified chemical substances, but still contained a mixture of different molecular species. Each species of molecule interfered only with itself, as verified by mass spectrometry.[12]
  • 2013年的一项实验将含有15000个质子、中子和电子的分子叠加在一起。这些分子是由于其良好的热稳定性而选择的化合物,并在600 K的温度下蒸发成束。束是由高度纯化的化学物质制备的,但仍然包含不同分子种类的混合物。每种分子都只与自身发生干扰,这一点已被质谱学证实。[13]
By use of very low temperatures, very fine experimental arrangements were made to protect in near isolation and preserve the coherence of intermediate states, for a duration of time, between preparation and detection, of SQUID currents. Such a SQUID current is a coherent physical assembly of perhaps billions of electrons. Because of its coherence, such an assembly may be regarded as exhibiting "collective states" of a macroscopic quantal entity. For the principle of superposition, after it is prepared but before it is detected, it may be regarded as exhibiting an intermediate state. It is not a single-particle state such as is often considered in discussions of interference, for example by Dirac in his famous dictum stated above.[16] Moreover, though the 'intermediate' state may be loosely regarded as such, it has not been produced as an output of a secondary quantum analyser that was fed a pure state from a primary analyser, and so this is not an example of superposition as strictly and narrowly defined.

:通过使用非常低的温度,进行了非常精细的实验安排,以在制备和检测鱿鱼电流之间的一段时间内,近乎隔离地保护和保持中间状态的一致性。这样的鱿鱼电流是一个可能由数十亿个电子组成的相干物理集合。由于它的一致性,这样一个集合可以被认为表现出宏观量子实体的“集体状态”。对于叠加原理,在制备之后但在检测之前,可以认为它表现出中间状态。它不是一个单一的粒子状态,就像在讨论干涉时经常考虑的那样,例如狄拉克在他著名的格言中所说的那样。[16] 此外,尽管“中间”态可以粗略地视为这样,但它并不是作为次级量子分析器的输出而产生的,次级量子分析器从初级分析器馈送纯态,因此这不是严格和狭义定义的叠加示例。

By use of very low temperatures, very fine experimental arrangements were made to protect in near isolation and preserve the coherence of intermediate states, for a duration of time, between preparation and detection, of SQUID currents. Such a SQUID current is a coherent physical assembly of perhaps billions of electrons. Because of its coherence, such an assembly may be regarded as exhibiting "collective states" of a macroscopic quantal entity. For the principle of superposition, after it is prepared but before it is detected, it may be regarded as exhibiting an intermediate state. It is not a single-particle state such as is often considered in discussions of interference, for example by Dirac in his famous dictum stated above. Moreover, though the 'intermediate' state may be loosely regarded as such, it has not been produced as an output of a secondary quantum analyser that was fed a pure state from a primary analyser, and so this is not an example of superposition as strictly and narrowly defined.

通过使用极低的温度,作出了非常精细的实验安排,以便在准备和检测超导量子干涉仪电流之间的一段时间内,保持近乎隔离的保护和中间状态的一致性。这样的超导量子干涉仪电流是一个可能由数十亿个电子组成的相干物理组装。由于它的连贯性,这样的组合可以看作是宏观量子实体的“集体状态”。对于叠加原理来说,在准备好之后但是在被检测到之前,它可以被看作是展示了一个居间态。它不是一个单粒子态,就像在讨论干涉时经常考虑的那样,例如狄拉克在他的著名名言中所说的那样。此外,虽然中间态可以被宽泛地认为是这样的,但它并不是作为从一个初级分析器输入一个纯态的次级量子分析器的输出而产生的,因此这不是一个严格和狭义定义的叠加的例子。


Nevertheless, after preparation, but before measurement, such a SQUID state may be regarded in a manner of speaking as a "pure" state that is a superposition of a clockwise and an anti-clockwise current state. In a SQUID, collective electron states can be physically prepared in near isolation, at very low temperatures, so as to result in protected coherent intermediate states. What is remarkable here is that there are two well-separated self-coherent collective states that exhibit such metastability. The crowd of electrons tunnels back and forth between the clockwise and the anti-clockwise states, as opposed to forming a single intermediate state in which there is no definite collective sense of current flow.[17][18]

Nevertheless, after preparation, but before measurement, such a SQUID state may be regarded in a manner of speaking as a "pure" state that is a superposition of a clockwise and an anti-clockwise current state. In a SQUID, collective electron states can be physically prepared in near isolation, at very low temperatures, so as to result in protected coherent intermediate states. What is remarkable here is that there are two well-separated self-coherent collective states that exhibit such metastability. The crowd of electrons tunnels back and forth between the clockwise and the anti-clockwise states, as opposed to forming a single intermediate state in which there is no definite collective sense of current flow.

然而,在准备之后,但在测量之前,这样的 SQUID 态可以被认为是一种“纯”态,是顺时针和逆时针电流态的叠加态。在超导量子干涉仪中,可以在非常低的温度下以接近孤立的物理方式制备集体电子态,从而产生受保护的相干中间态。这里值得注意的是,有两个完全分离的自相干集体态展示了这样的亚稳态。电子群在顺时针和逆时针状态之间来回穿梭,而不是形成一个没有明确的电流集体感的单一居间态。


  • A piezoelectric "tuning fork" has been constructed, which can be placed into a superposition of vibrating and non-vibrating states. The resonator comprises about 10 trillion atoms.[21]
  • 一个压电音叉”已经建成,它可以放置到一个振动和非振动状态的叠加。谐振器由大约10万亿个原子组成。[22]
  • Recent research indicates that chlorophyll within plants appears to exploit the feature of quantum superposition to achieve greater efficiency in transporting energy, allowing pigment proteins to be spaced further apart than would otherwise be possible.[23][24]
  • 最近的研究表明,植物内的叶绿素似乎利用量子叠加的特性来实现更高的能量传输效率,使得色素蛋白质的间隔比其他可能的要远。[23][25]
  • An experiment has been proposed, with a bacterial cell cooled to 10 mK, using an electromechanical oscillator.[26] At that temperature, all metabolism would be stopped, and the cell might behave virtually as a definite chemical species. For detection of interference, it would be necessary that the cells be supplied in large numbers as pure samples of identical and detectably recognizable virtual chemical species. It is not known whether this requirement can be met by bacterial cells. They would be in a state of suspended animation during the experiment.
  • 已经提出了一个实验,使用机电振荡器将细菌细胞冷却到10 mK。[27]在这个温度下,所有的新陈代谢都会停止,细胞实际上可能表现为一种特定的化学物质。为了检测干扰,有必要大量提供细胞作为相同和可检测可识别的虚拟化学物种的纯样品。目前尚不清楚细菌细胞能否满足这一要求。在实验过程中,它们会处于一种暂停活动的状态。

In quantum computing the phrase "cat state" often refers to the GHZ state, the special entanglement of qubits wherein the qubits are in an equal superposition of all being 0 and all being 1; i.e.,

In quantum computing the phrase "cat state" often refers to the GHZ state, the special entanglement of qubits wherein the qubits are in an equal superposition of all being 0 and all being 1; i.e.,

在量子计算中,短语“猫态”通常指的是 GHZ 态,即量子比特之间的特殊纠缠,其中量子比特处于全部为0且全部为1的等量叠加态中; 即,


[math]\displaystyle{ | \psi \rangle = \frac{1}{\sqrt{2}} \bigg( | 00\ldots0 \rangle + |11\ldots1 \rangle \bigg). }[/math]

[math]\displaystyle{ | \psi \rangle = \frac{1}{\sqrt{2}} \bigg( | 00\ldots0 \rangle + |11\ldots1 \rangle \bigg). }[/math]

(| 00 ldots0 rangle + | 11 ldots1 rangle bigg).数学

Formal interpretation形式解释

Applying the superposition principle to a quantum mechanical particle, the configurations of the particle are all positions, so the superpositions make a complex wave in space. The coefficients of the linear superposition are a wave which describes the particle as best as is possible, and whose amplitude interferes according to the Huygens principle.

叠加原理应用于量子力学粒子,粒子的构型都是位置,因此叠加会在空间中产生一个复杂的波。线性叠加的系数是一个尽可能最好地描述粒子的波,它的振幅根据惠更斯原理干涉

Applying the superposition principle to a quantum mechanical particle, the configurations of the particle are all positions, so the superpositions make a complex wave in space. The coefficients of the linear superposition are a wave which describes the particle as best as is possible, and whose amplitude interferes according to the Huygens principle.

将叠加原理应用于量子力学粒子,粒子的构型都是位置,所以叠加在一起会在空间产生复杂的波动。线性叠加系数是尽可能最好地描述粒子的波,其振幅按惠更斯原理干扰。


For any physical property in quantum mechanics, there is a list of all the states where that property has some value. These states are necessarily perpendicular to each other using the Euclidean notion of perpendicularity which comes from sums-of-squares length, except that they also must not be i multiples of each other. This list of perpendicular states has an associated value which is the value of the physical property. The superposition principle guarantees that any state can be written as a combination of states of this form with complex coefficients.模板:Clarify

For any physical property in quantum mechanics, there is a list of all the states where that property has some value. These states are necessarily perpendicular to each other using the Euclidean notion of perpendicularity which comes from sums-of-squares length, except that they also must not be i multiples of each other. This list of perpendicular states has an associated value which is the value of the physical property. The superposition principle guarantees that any state can be written as a combination of states of this form with complex coefficients.

对于量子力学中的任何物理属性,都有一个列表,上面列出了该属性具有某种价值的所有状态。这些状态必然是垂直于彼此使用欧氏垂直的概念,它来自于平方和的长度,除非它们也不能是彼此的倍数。这个垂直状态列表有一个关联值,该值是物理属性的值。叠加原理保证任何状态都可以写成这种形式的状态与复系数的组合。


Write each state with the value q of the physical quantity as a vector in some basis [math]\displaystyle{ \psi^q_n }[/math], a list of numbers at each value of n for the vector which has value q for the physical quantity. Now form the outer product of the vectors by multiplying all the vector components and add them with coefficients to make the matrix

Write each state with the value q of the physical quantity as a vector in some basis [math]\displaystyle{ \psi^q_n }[/math], a list of numbers at each value of n for the vector which has value q for the physical quantity. Now form the outer product of the vectors by multiplying all the vector components and add them with coefficients to make the matrix

用物理量的值 q 作为向量写入每个状态,在某个基础上 < math > psi ^ q _ n </math > ,一个数字列表,每个数值 n 表示物理量值 q 的向量。现在通过将所有矢量分量相乘,然后将它们与系数相加得到矩阵,从而得到矢量的外积

[math]\displaystyle{ \lt math\gt 《数学》 A_{nm} = \sum_q q \psi^{*q}_n \psi^q_m A_{nm} = \sum_q q \psi^{*q}_n \psi^q_m A _ { nm } = sum _ q q psi ^ { * q } n psi ^ q _ m }[/math]

</math>

数学

where the sum extends over all possible values of q. This matrix is necessarily symmetric because it is formed from the orthogonal states, and has eigenvalues q. The matrix A is called the observable associated to the physical quantity. It has the property that the eigenvalues and eigenvectors determine the physical quantity and the states which have definite values for this quantity.

where the sum extends over all possible values of q. This matrix is necessarily symmetric because it is formed from the orthogonal states, and has eigenvalues q. The matrix A is called the observable associated to the physical quantity. It has the property that the eigenvalues and eigenvectors determine the physical quantity and the states which have definite values for this quantity.

其中和超过了 q 的所有可能值。这个矩阵必然是对称的,因为它是由正交态构成的,并且具有特征值 q。矩阵 a 称为与物理量相关联的可观测量。它的性质是特征矢量决定物理量和对这个量有确定值的状态。


Every physical quantity has a Hermitian linear operator associated to it, and the states where the value of this physical quantity is definite are the eigenstates of this linear operator. The linear combination of two or more eigenstates results in quantum superposition of two or more values of the quantity. If the quantity is measured, the value of the physical quantity will be random, with a probability equal to the square of the coefficient of the superposition in the linear combination. Immediately after the measurement, the state will be given by the eigenvector corresponding to the measured eigenvalue.

Every physical quantity has a Hermitian linear operator associated to it, and the states where the value of this physical quantity is definite are the eigenstates of this linear operator. The linear combination of two or more eigenstates results in quantum superposition of two or more values of the quantity. If the quantity is measured, the value of the physical quantity will be random, with a probability equal to the square of the coefficient of the superposition in the linear combination. Immediately after the measurement, the state will be given by the eigenvector corresponding to the measured eigenvalue.

每个物理量都有一个与其相关的厄米线性算子,这个物理量值为确定的状态就是这个线性算子的本征态。两个或两个以上本征态的线性组合导致两个或两个以上本征态的态叠加原理。如果量被测量,物理量的值将是随机的,其概率等于线性组合的叠加系数的平方。在测量之后,状态将由与被测量特征值相对应的特征向量给出。

Physical interpretation物理解释

It is natural to ask why ordinary everyday objects and events do not seem to display quantum mechanical features such as superposition. Indeed, this is sometimes regarded as "mysterious", for instance by Richard Feynman.[28] In 1935, Erwin Schrödinger devised a well-known thought experiment, now known as Schrödinger's cat, which highlighted this dissonance between quantum mechanics and classical physics. One modern view is that this mystery is explained by quantum decoherence.[citation needed] A macroscopic system (such as a cat) may evolve over time into a superposition of classically distinct quantum states (such as "alive" and "dead"). The mechanism that achieves this is a subject of significant research, one mechanism suggests that the state of the cat is entangled with the state of its environment (for instance, the molecules in the atmosphere surrounding it), when averaged over the possible quantum states of the environment (a physically reasonable procedure unless the quantum state of the environment can be controlled or measured precisely) the resulting mixed quantum state for the cat is very close to a classical probabilistic state where the cat has some definite probability to be dead or alive, just as a classical observer would expect in this situation. Another proposed class of theories is that the fundamental time evolution equation is incomplete, and requires the addition of some type of fundamental Lindbladian, the reason for this addition and the form of the additional term varies from theory to theory. A popular theory is Continuous spontaneous localization, where the lindblad term is proportional to the spatial separation of the states, this too results in a quasi-classical probabilistic state.

人们很自然地会问,为什么普通的日常物体和事件似乎没有表现出叠加等量子力学特征。事实上,这有时被认为是“神秘的”,例如理查德·费曼。[29]1935年,Erwin Schrödinger设计了一个著名的思维实验,现在被称为Schrödinger's cat,这突出了量子力学和经典物理学之间的矛盾。一种现代观点认为,这个谜可以用量子退相干来解释。.[citation needed] 宏观系统(如猫)可能会随着时间的推移演化成经典不同量子态(如“活”和“死”)的叠加。实现这一点的机制是一个重要的研究课题,一种机制表明猫的状态与其周围环境的状态(例如,周围大气中的分子)纠缠在一起,当对环境的可能量子态进行平均时(除非环境的量子态可以精确地控制或测量,否则这是一个物理上合理的过程),cat的结果混合量子态非常接近经典概率态,其中cat具有一些确定的概率不管是死是活,就像一个经典的观察者在这种情况下所期望的那样。提出的另一类理论是,基本时间演化方程是不完备的,需要添加某种类型的基本项林德布拉德项Lindbladian,添加的原因和附加项的形式因理论而异。一个流行的理论是连续自发局域化,其中林得布拉德lindblad项与状态的空间分离成正比,这也导致了准经典概率状态。

It is natural to ask why ordinary everyday objects and events do not seem to display quantum mechanical features such as superposition. Indeed, this is sometimes regarded as "mysterious", for instance by Richard Feynman. In 1935, Erwin Schrödinger devised a well-known thought experiment, now known as Schrödinger's cat, which highlighted this dissonance between quantum mechanics and classical physics. One modern view is that this mystery is explained by quantum decoherence. A macroscopic system (such as a cat) may evolve over time into a superposition of classically distinct quantum states (such as "alive" and "dead"). The mechanism that achieves this is a subject of significant research, one mechanism suggests that the state of the cat is entangled with the state of its environment (for instance, the molecules in the atmosphere surrounding it), when averaged over the possible quantum states of the environment (a physically reasonable procedure unless the quantum state of the environment can be controlled or measured precisely) the resulting mixed quantum state for the cat is very close to a classical probabilistic state where the cat has some definite probability to be dead or alive, just as a classical observer would expect in this situation. Another proposed class of theories is that the fundamental time evolution equation is incomplete, and requires the addition of some type of fundamental Lindbladian, the reason for this addition and the form of the additional term varies from theory to theory. A popular theory is Continuous spontaneous localization, where the lindblad term is proportional to the spatial separation of the states, this too results in a quasi-classical probabilistic state.

人们自然会问,为什么日常生活中的普通物体和事件似乎没有显示出叠加等量子力学特征。事实上,这有时被认为是“神秘的” ,例如理查德 · 费曼。1935年,埃尔温·薛定谔设计了一个著名的思想实验,现在被称为薛定谔猫,它强调了量子力学和经典物理学之间的不协调。一种现代的观点认为这个谜团可以用量子退相干来解释。一个宏观系统(例如一只猫)可能随着时间的推移演变成一个经典的不同量子态的叠加态(例如“活的”和“死的”)。实现这一点的机制是一个重要的研究课题,一种机制表明,猫的状态与其环境的状态(例如,大气中的分子)纠缠在一起,当超过环境可能的量子状态(这是一个物理上合理的程序,除非环境的量子状态可以被精确控制或测量)平均后,猫的混合量子状态非常接近于经典概率状态,在这种状态下,猫有一定的可能是死的或活的,正如经典观察者在这种情况下所期待的那样。另一类被提出的理论是,基本的时间/发展方程是不完整的,并且需要增加某种类型的基本的林德布拉迪安---- 这种增加的原因和附加项的形式因理论而异。一个流行的理论是连续自发局部化,其中林德布拉德项是成正比的空间分离的状态,这也导致了一个准经典的概率状态。

See also请参阅

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References参考文献

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  5. "Schrödinger's Cat Now Made Of Light". 27 August 2014.
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  9. Nairz, Olaf. "standinglightwave".
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  11. Nairz, Olaf. "standinglightwave".
  12. Eibenberger, S., Gerlich, S., Arndt, M., Mayor, M., Tüxen, J. (2013). "Matter-wave interference with particles selected from a molecular library with masses exceeding 10 000 amu", Physical Chemistry Chemical Physics, 15: 14696-14700. [1]
  13. Eibenberger, S., Gerlich, S., Arndt, M., Mayor, M., Tüxen, J. (2013). "Matter-wave interference with particles selected from a molecular library with masses exceeding 10 000 amu", Physical Chemistry Chemical Physics, 15: 14696-14700. [2]
  14. Leggett, A. J. (1986). "The superposition principle in macroscopic systems", pp. 28–40 in Quantum Concepts of Space and Time, edited by R. Penrose and C.J. Isham, .
  15. Leggett, A. J. (1986). "The superposition principle in macroscopic systems", pp. 28–40 in Quantum Concepts of Space and Time, edited by R. Penrose and C.J. Isham, .
  16. 16.0 16.1 Dirac, P. A. M. (1930/1958), p. 9.
  17. Physics World: Schrodinger's cat comes into view
  18. Friedman, J. R., Patel, V., Chen, W., Tolpygo, S. K., Lukens, J. E. (2000)."Quantum superposition of distinct macroscopic states", Nature 406: 43–46.
  19. "How to Create Quantum Superpositions of Living Things">
  20. "How to Create Quantum Superpositions of Living Things">
  21. Scientific American: Macro-Weirdness: "Quantum Microphone" Puts Naked-Eye Object in 2 Places at Once: A new device tests the limits of Schrödinger's cat
  22. Scientific American: Macro-Weirdness: "Quantum Microphone" Puts Naked-Eye Object in 2 Places at Once: A new device tests the limits of Schrödinger's cat
  23. 23.0 23.1 Scholes, Gregory; Elisabetta Collini; Cathy Y. Wong; Krystyna E. Wilk; Paul M. G. Curmi; Paul Brumer; Gregory D. Scholes (4 February 2010). "Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature". Nature. 463 (7281): 644–647. Bibcode:2010Natur.463..644C. doi:10.1038/nature08811. PMID 20130647. Unknown parameter |s2cid= ignored (help)
  24. Moyer, Michael (September 2009). "Quantum Entanglement, Photosynthesis and Better Solar Cells". Scientific American. Retrieved 12 May 2010.
  25. Moyer, Michael (September 2009). "Quantum Entanglement, Photosynthesis and Better Solar Cells". Scientific American. Retrieved 12 May 2010.
  26. "Could 'Schrödinger's bacterium' be placed in a quantum superposition?">
  27. "Could 'Schrödinger's bacterium' be placed in a quantum superposition?">
  28. Feynman, R. P., Leighton, R. B., Sands, M. (1965), § 1-1.
  29. Feynman,R.P.,莱顿,R.B.,桑兹,M.(1965),§1-1。


Bibliography of cited references引用文献目录

  • Cohen-Tannoudji, C., Diu, B., Laloë, F. (1973/1977). Quantum Mechanics, translated from the French by S. R. Hemley, N. Ostrowsky, D. Ostrowsky, second edition, volume 1, Wiley, New York, .
  • Dirac, P. A. M. (1930/1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press.
  • Feynman, R. P., Leighton, R.B., Sands, M. (1965). The Feynman Lectures on Physics, volume 3, Addison-Wesley, Reading, MA.
  • Merzbacher, E. (1961/1970). Quantum Mechanics, second edition, Wiley, New York.
  • Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam.
  • Wheeler, J. A.; Zurek, W.H. (1983). Quantum Theory and Measurement. Princeton NJ: Princeton University Press. 


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