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第488行： + + − 在21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，有一个状态，一部分是这个，一部分是那个，具有正实数系数，概率，表示每个有多少。+ 第505行：
第507行： − 《数学》 第511行：
第512行： − Sum _ x rho (x) | x rangle 第517行：
第517行： − 数学 第528行：
第527行： + + − + + + − + 第571行：
第574行： − + 第609行：
第612行： − 《数学》 第621行：
第623行： − ，math 第633行：
第634行： − 《数学》 第645行：
第645行： − ，math. + +
→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.
在[[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.
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。
<math>
<math>
\sum_x \rho(x) |x\rangle
\sum_x \rho(x) |x\rangle
\sum_x \rho(x) |x\rangle
\sum_x \rho(x) |x\rangle
</math>
</math>
</math>
</math>
Where <math>\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>\rho</math> 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.
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.
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，从一个状态开始到 y 的概率是，从每个选项到 y 的概率的混合。这是概率的线性叠加原理。
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.
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.
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21量子力学，不同的可能性可以取消。
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。
where the probabilities <math>x,y,z</math> are positive numbers. Rescaling x,y,z so that
where the probabilities <math>x,y,z</math> are positive numbers. Rescaling x,y,z so that
其中 x，y，z 的概率是正数。重新标定 x，y，z，这样
其中 <math>x,y,z</math>的概率是正数。重新标定 x，y，z，则
:<math>
:<math>
<math>
<math>
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
<math>
<math>
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
\,</math>.
\,</math>.
A sphere has a large amount of symmetry, it can be viewed in different coordinate systems or [[linear algebra|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 [[linear algebra|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[[线性代数|基]]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.
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.