# 量子叠加态

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

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.

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.

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.

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

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

Here $\displaystyle{ |0 \rangle }$ is the Dirac notation for the quantum state that will always give the result 0 when converted to classical logic by a measurement. Likewise $\displaystyle{ |1 \rangle }$ 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.

Here $\displaystyle{ |0 \rangle }$ is the Dirac notation for the quantum state that will always give the result 0 when converted to classical logic by a measurement. Likewise $\displaystyle{ |1 \rangle }$ 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.

## 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.

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

$\displaystyle{ c_0 {\mid} 0 \rangle + c_1 {\mid} 1 \rangle }$

$\displaystyle{ c_0 {\mid} 0 \rangle + c_1 {\mid} 1 \rangle }$

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.

The principle was described by Paul Dirac as follows:

The principle was described by Paul Dirac as follows:

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...

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.

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.

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:

"[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.

"[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.

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.

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

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

$\displaystyle{ c_1 {\mid} {\uparrow} \rangle + c_2 {\mid} {\downarrow} \rangle }$

$\displaystyle{ c_1 {\mid} {\uparrow} \rangle + c_2 {\mid} {\downarrow} \rangle }$

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.

$\displaystyle{ p_\text{up} = {\mid} c_1 {\mid}^2 }$

$\displaystyle{ p_\text{up} = {\mid} c_1 {\mid}^2 }$

1{ mid } ^ 2 </math >

$\displaystyle{ p_\text{down} = {\mid} c_2 \mid^2 }$

$\displaystyle{ p_\text{down} = {\mid} c_2 \mid^2 }$

2 mid ^ 2 </math >

$\displaystyle{ p_\text{up or down} = p_\text{up} + p_\text{down} = 1 }$

$\displaystyle{ p_\text{up or down} = p_\text{up} + p_\text{down} = 1 }$

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.

$\displaystyle{ |\psi\rangle = {3\over 5} i {\mid}{\uparrow}\rangle + {4\over 5} {\mid}{\downarrow}\rangle. }$

$\displaystyle{ |\psi\rangle = {3\over 5} i {\mid}{\uparrow}\rangle + {4\over 5} {\mid}{\downarrow}\rangle. }$

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

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

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

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 $\displaystyle{ \alpha }$

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 $\displaystyle{ \alpha }$

$\displaystyle{ \lt math\gt 《数学》 |\psi \rangle \approx \alpha |\psi \rangle |\psi \rangle \approx \alpha |\psi \rangle | psi rangle approx alpha | psi rangle }$

[/itex]

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 $\displaystyle{ \psi }$ 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 $\displaystyle{ \psi }$ turns into a mixture of A′ and B′ with the same coefficients as A and B.

For example, if we have the following

For example, if we have the following

$\displaystyle{ {\mid} {\uparrow} \rangle \to {\mid} {\downarrow} \rangle }$

$\displaystyle{ {\mid} {\uparrow} \rangle \to {\mid} {\downarrow} \rangle }$

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

$\displaystyle{ {\mid} {\downarrow} \rangle \to \frac{3i}{5} {\mid} {\uparrow} \rangle + \frac{4}{5} {\mid} {\downarrow} \rangle }$

$\displaystyle{ {\mid} {\downarrow} \rangle \to \frac{3i}{5} {\mid} {\uparrow} \rangle + \frac{4}{5} {\mid} {\downarrow} \rangle }$

{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

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

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

< math > 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.

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:

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

[/itex]

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:

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

[/itex]

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

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

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:

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

\,[/itex]

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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.

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 $\displaystyle{ |x,y\rangle }$. 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 $\displaystyle{ |x,y\rangle }$. The most general state is a superposition of the possibilities:

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

\,[/itex]

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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:

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:

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

\,[/itex]

，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:

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

\,[/itex]

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while an entangled wavefunction does not have this form.

while an entangled wavefunction does not have this form.

### 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.

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.

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"

$\displaystyle{ \lt math\gt \sum_x \rho(x) |x\rangle \sum_x \rho(x) |x\rangle }$

[/itex]

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

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

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.

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.

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.

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.

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:

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

\,[/itex]

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where the probabilities $\displaystyle{ x,y,z }$ are positive numbers. Rescaling x,y,z so that

where the probabilities $\displaystyle{ x,y,z }$ are positive numbers. Rescaling x,y,z so that

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

\,[/itex]

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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.

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:

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

\,[/itex]

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

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

\,[/itex].

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.

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.

### 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:

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

\,[/itex]

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So that the infinite list of amplitudes $\displaystyle{ (\ldots, \psi_{-2}, \psi_{-1}, \psi_0, \psi_1, \psi_2, \ldots) }$ 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 $\displaystyle{ (\ldots, \psi_{-2}, \psi_{-1}, \psi_0, \psi_1, \psi_2, \ldots) }$ 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:

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

[/itex]

For a particle described by probability theory random walking on a line, the analogous thing is the list of probabilities $\displaystyle{ (\ldots,P_{-2},P_{-1},P_0,P_1,P_2,\ldots) }$, which give the probability of any position. The quantities that describe how they change in time are the transition probabilities $\displaystyle{ K_{x\rightarrow y}(t) }$, 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 $\displaystyle{ (\ldots,P_{-2},P_{-1},P_0,P_1,P_2,\ldots) }$, which give the probability of any position. The quantities that describe how they change in time are the transition probabilities $\displaystyle{ K_{x\rightarrow y}(t) }$, 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

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

\,[/itex]

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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:

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

\,[/itex]

，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.

When no time passes, nothing changes: for 0 elapsed time $\displaystyle{ K{x\rightarrow y}(0) = \delta_{xy} }$, 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 $\displaystyle{ K{x\rightarrow y}(0) = \delta_{xy} }$, 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.

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

\,[/itex]

，math

where $\displaystyle{ R_{x\rightarrow y} }$ is the time derivative of the K matrix:

where $\displaystyle{ R_{x\rightarrow y} }$ is the time derivative of the K matrix:

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

\,[/itex]

，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:

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

\,[/itex]

，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:

$\displaystyle{ \lt math\gt \sum_y R_{x\rightarrow y} = 0 \sum_y R_{x\rightarrow y} = 0 \, }$

\,[/itex]

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 $\displaystyle{ R_{x\rightarrow y} }$ 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 $\displaystyle{ R_{x\rightarrow x} }$ must be −2c. So the probabilities obey the discretized diffusion equation:

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 $\displaystyle{ R_{x\rightarrow y} }$ 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 $\displaystyle{ R_{x\rightarrow x} }$ must be −2c. So the probabilities obey the discretized diffusion equation:

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

\,[/itex]

，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:

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

\,[/itex]

Which is the diffusion equation.

Which is the diffusion equation.

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:

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

\,[/itex]

，math

Since the sum of the absolute squares of the amplitudes must be constant, $\displaystyle{ U }$ must be unitary:

Since the sum of the absolute squares of the amplitudes must be constant, $\displaystyle{ U }$ must be unitary:

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

\,[/itex]

or, in matrix notation,

or, in matrix notation,

$\displaystyle{ \lt math\gt U^\dagger U = I U^\dagger U = I U ^ dagger u = i \, }$

\,[/itex]

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:

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

[/itex]

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:

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

[/itex]

$\displaystyle{ \lt math\gt H^\dagger - H = 0 H^\dagger - H = 0 H ^ dagger-h = 0 \, }$

\,[/itex]

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.

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.

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 $\displaystyle{ \psi }$ 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 $\displaystyle{ \psi }$ is the time differential equation:

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

[/itex]

In the case in which left and right are symmetric, c is real. By redefining the phase of the wavefunction in time, $\displaystyle{ \psi\rightarrow \psi e^{i2ct} }$, 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, $\displaystyle{ \psi\rightarrow \psi e^{i2ct} }$, 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.

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

[/itex]

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

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

[/itex]

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:

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

[/itex]

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.

### 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 $\displaystyle{ K_{m\rightarrow n} }$, 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:

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 $\displaystyle{ K_{m\rightarrow n} }$, 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:

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

\,[/itex]

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

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

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 $\displaystyle{ \rho_n }$ has the property:

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 $\displaystyle{ \rho_n }$ has the property:

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

\,[/itex]

，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 $\displaystyle{ \rho_m }$ 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.

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 $\displaystyle{ \rho_m }$ 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.

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

\,[/itex]

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:

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

\,[/itex]

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

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

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

\,[/itex]

and H is symmetric

and H is symmetric

H 是对称的

$\displaystyle{ \lt math\gt H_{nm} = H_{mn} H_{nm} = H_{mn} H _ { nm } = h _ { mn } \, }$

\,[/itex]

This matrix H defines a quantum mechanical system:

This matrix H defines a quantum mechanical system:

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

\,[/itex]

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:

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:

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

\,[/itex]

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。

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

\,[/itex]

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.

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.

## 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.

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

• 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.
• 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。
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。 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.

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.

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.

• 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.
• 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。
• 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.
• 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。
• An experiment has been proposed, with a bacterial cell cooled to 10 mK, using an electromechanical oscillator. 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。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.,

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

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

(| 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.

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.

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.

Write each state with the value q of the physical quantity as a vector in some basis $\displaystyle{ \psi^q_n }$, 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 $\displaystyle{ \psi^q_n }$, 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

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

[/itex]

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.

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.

## 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. 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.

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.

## References参考文献

1. P.A.M. Dirac (1947). The Principles of Quantum Mechanics (2nd ed.). Clarendon Press. p. 12.
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11. 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. 
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. 
13. 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, .
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15. Dirac, P. A. M. (1930/1958), p. 9.
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17. Friedman, J. R., Patel, V., Chen, W., Tolpygo, S. K., Lukens, J. E. (2000)."Quantum superposition of distinct macroscopic states", Nature 406: 43–46.
18. 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)
19. Moyer, Michael (September 2009). "Quantum Entanglement, Photosynthesis and Better Solar Cells". Scientific American. Retrieved 12 May 2010.
20. Moyer, Michael (September 2009). "Quantum Entanglement, Photosynthesis and Better Solar Cells". Scientific American. Retrieved 12 May 2010.
21. Feynman, R. P., Leighton, R. B., Sands, M. (1965), § 1-1.
22. 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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