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where <math>S</math> is the classical action and <math>H</math> is the Hamiltonian function (not operator). Here the generalized coordinates <math>q_i</math> for <math>i = 1, 2, 3</math> (used in the context of the HJE) can be set to the position in Cartesian coordinates as <math>\mathbf{r} = (q_1, q_2, q_3) = (x, y, z)</math>.

where <math>S</math> is the classical action and <math>H</math> is the Hamiltonian function (not operator). Here the generalized coordinates <math>q_i</math> for <math>i = 1, 2, 3</math> (used in the context of the HJE) can be set to the position in Cartesian coordinates as <math>\mathbf{r} = (q_1, q_2, q_3) = (x, y, z)</math>.

其中，s </math > 是经典动作，而 h </math > 是哈密顿函数(不是算符)。在这里，对于 < math > i = 1,2,3 </math > (在 HJE 上下文中使用) ，可以将广义坐标设置为笛卡尔坐标中的位置 < math > mathbf { r } = (q1，q2，q3) = (x，y，z) </math > 。

其中，<math>S</math> 是经典动作，而<math>H</math> 是哈密顿函数(不是算符)。在这里，对于 <math>i = 1, 2, 3</math> (在 HJE 上下文中使用) ，可以将广义坐标<math>q_i</math> 设置为笛卡尔坐标中的位置 <math>\mathbf{r} = (q_1, q_2, q_3) = (x, y, z)</math> 。

  |title=Hamilton's Analogy: Paths to the Schrödinger Equation

  |title=Hamilton's Analogy: Paths to the Schrödinger Equation

----> A modern version of his reasoning is reproduced below. The equation he found is:<ref name="verlagsgesellschaft1991">''Encyclopaedia of Physics'' (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) {{isbn|0-89573-752-3}}</ref>

----> A modern version of his reasoning is reproduced below. The equation he found is:

 

下面是他的推理的现代版本。他找到的方程式是：<ref name="verlagsgesellschaft1991">''Encyclopaedia of Physics'' (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) {{isbn|0-89573-752-3}}</ref>

where  is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and  is a function of time only.

where  is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and  is a function of time only.

However, by that time, [[Arnold Sommerfeld]] had [[Sommerfeld–Wilson quantization|refined the Bohr model]] with [[fine structure|relativistic corrections]].<ref>

However, by that time, [[Arnold Sommerfeld]] had [[Sommerfeld–Wilson quantization|refined the Bohr model]] with [[fine structure|relativistic corrections]].

 

然而，到那时，[[Arnold Sommerfeld]]已经[[Sommerfeld–Wilson量子化|改进了玻尔模型]]和[[fine structure |相对论修正]]。<ref>

<math> V(x_1,x_2,\ldots, x_N) = \sum_{n=1}^N V(x_n) \, .</math>

<math> V(x_1,x_2,\ldots, x_N) = \sum_{n=1}^N V(x_n) \, .</math>

  |isbn=978-3-87144-484-5

  |isbn=978-3-87144-484-5

}}</ref><ref>For an English source, see {{Cite journal

}}</ref><ref>For an English source, see 有关英文来源，请参阅{{Cite journal

  |last=Haar |first=T.

  |last=Haar |first=T.

For no potential,  0}}, so the particle is free and the equation reads: and atoms or ions in lattices, and approximating other potentials near equilibrium points. It is also the basis of perturbation methods in quantum mechanics.

For no potential,  0}}, so the particle is free and the equation reads: and atoms or ions in lattices, and approximating other potentials near equilibrium points. It is also the basis of perturbation methods in quantum mechanics.

因为没有势，0} ，所以粒子是自由的，方程是: 和晶格中的原子或离子，并且近似平衡点附近的其他势。这也是量子力学微扰法的基础。

因为没有势，0}}，所以粒子是自由的，方程是: 和晶格中的原子或离子，并且近似平衡点附近的其他势。这也是量子力学微扰法的基础。

  |title=The Old Quantum Theory

  |title=The Old Quantum Theory

where <math>n \in  \{0, 1, 2, \ldots \}</math>, and the functions <math>  \mathcal{H}_n </math> are the Hermite polynomials of order <math>  n </math>. The solution set may be generated by

where <math>n \in  \{0, 1, 2, \ldots \}</math>, and the functions <math>  \mathcal{H}_n </math> are the Hermite polynomials of order <math>  n </math>. The solution set may be generated by

其中0,1,2，ldots </math > 中的 < math > n，而函数 < math > h } n </math > 是 < math > n </math > 的埃尔米特多项式。解决方案集可以由

其中<math>n \in  \{0, 1, 2, \ldots \}</math>，而函数 <math> \mathcal{H}_n </math> 是 <math> n </math> 的埃尔米特多项式。解决方案集可以由

<!-- <math>\frac{1}{c^2}\left(E + {e^2\over 4 \pi \varepsilon_0 r} \right)^2 \psi(x) = -\hbar^2 \nabla^2\psi(x) + \frac{m^2c^2}{\hbar^2} \psi(x).</math> ?-->

<!-- <math>\frac{1}{c^2}\left(E + {e^2\over 4 \pi \varepsilon_0 r} \right)^2 \psi(x) = -\hbar^2 \nabla^2\psi(x) + \frac{m^2c^2}{\hbar^2} \psi(x).</math> ?-->

He found the standing waves of this relativistic equation, but the relativistic corrections disagreed with Sommerfeld's formula. Discouraged, he put away his calculations and secluded himself with a mistress in a mountain cabin in December 1925.<ref>{{Cite news|last=Teresi|first=Dick|date=1990-01-07|title=THE LONE RANGER OF QUANTUM MECHANICS (Published 1990)|language=en-US|work=The New York Times|url=https://www.nytimes.com/1990/01/07/books/the-lone-ranger-of-quantum-mechanics.html|access-date=2020-10-13|issn=0362-4331}}</ref>

He found the standing waves of this relativistic equation, but the relativistic corrections disagreed with Sommerfeld's formula. Discouraged, he put away his calculations and secluded himself with a mistress in a mountain cabin in December 1925.<ref>{{Cite news|last=Teresi|first=Dick|date=1990-01-07|title=THE LONE RANGER OF QUANTUM MECHANICS (Published 1990)|language=en-US|work=The New York Times|url=https://www.nytimes.com/1990/01/07/books/the-lone-ranger-of-quantum-mechanics.html|access-date=2020-10-13|issn=0362-4331}}</ref>

他发现了这个相对论方程的驻波，但是相对论修正与索末菲公式不一致。1925年12月，他灰心丧气地放下算计，与一位情妇隐居在山间小屋里。<ref>{{Cite news|last=Teresi|first=Dick|date=1990-01-07|title=THE LONE RANGER OF QUANTUM MECHANICS (Published 1990)|language=en-US|work=The New York Times|url=https://www.nytimes.com/1990/01/07/books/the-lone-ranger-of-quantum-mechanics.html|access-date=2020-10-13|issn=0362-4331}}</ref>

他发现了这个相对论方程的驻波，但是相对论修正与索末菲公式不一致。1925年12月，他灰心丧气地放下计算，与一位情妇隐居在山间小屋里。<ref>{{Cite news|last=Teresi|first=Dick|date=1990-01-07|title=THE LONE RANGER OF QUANTUM MECHANICS (Published 1990)|language=en-US|work=The New York Times|url=https://www.nytimes.com/1990/01/07/books/the-lone-ranger-of-quantum-mechanics.html|access-date=2020-10-13|issn=0362-4331}}</ref>

The eigenvalues are

The eigenvalues are

The case <math>  n = 0 </math> is called the ground state, its energy is called the zero-point energy, and the wave function is a Gaussian.

The case <math>  n = 0 </math> is called the ground state, its energy is called the zero-point energy, and the wave function is a Gaussian.

这种情况下，n = 0 </math > 被称为基态，它的能量被称为零点能量，波函数是高斯函数。

这种情况下，<math>  n = 0 </math> 被称为基态，它的能量被称为零点能量，波函数是高斯函数。

  |url=http://gallica.bnf.fr/ark:/12148/bpt6k153811.image.langFR.f373.pagination

  |url=http://gallica.bnf.fr/ark:/12148/bpt6k153811.image.langFR.f373.pagination

|bibcode = 1926AnP...384..361S }}</ref> In the equation, Schrödinger computed the [[hydrogen spectral series]] by treating a [[hydrogen atom]]'s [[electron]] as a wave <math>\Psi (\mathbf{x}, t)</math>, moving in a [[potential well]] <math>V</math>, created by the [[proton]]. This computation accurately reproduced the energy levels of the [[Bohr model]]. In a paper, Schrödinger himself explained this equation as follows:

|bibcode = 1926AnP...384..361S }}</ref> In the equation, Schrödinger computed the [[hydrogen spectral series]] by treating a [[hydrogen atom]]'s [[electron]] as a wave <math>\Psi (\mathbf{x}, t)</math>, moving in a [[potential well]] <math>V</math>, created by the [[proton]]. This computation accurately reproduced the energy levels of the [[Bohr model]]. In a paper, Schrödinger himself explained this equation as follows:

在方程中，薛定谔通过将[[氢原子]]的[[电子]]视为波<math>\Psi（\mathbf{x}，t）</math>，在[[质子]]产生的[[势阱]]<math>V</math>中移动来计算[[氢光谱系列]]。这个计算准确地再现了[[玻尔模型]]的能级。在一篇论文中，薛定谔本人对这个等式的解释如下：

在方程中，薛定谔通过将[[氢原子]]的[[电子]]视为波<math>\Psi (\mathbf{x}, t)</math>，在[[质子]]产生的[[势阱]]<math>V</math>中移动来计算[[氢光谱系列]]。这个计算准确地再现了[[玻尔模型]]的能级。在一篇论文中，薛定谔本人对这个等式的解释如下：

The Hamiltonian for one particle in three dimensions is:

The Hamiltonian for one particle in three dimensions is:

<math>\nabla_n = \mathbf{e}_x \frac{\partial}{\partial x_n} + \mathbf{e}_y\frac{\partial}{\partial y_n} + \mathbf{e}_z\frac{\partial}{\partial z_n}\,,\quad \nabla_n^2 = \nabla_n\cdot\nabla_n = \frac{\partial^2}{{\partial x_n}^2} + \frac{\partial^2}{{\partial y_n}^2} + \frac{\partial^2}{{\partial z_n}^2}</math>

<math>\nabla_n = \mathbf{e}_x \frac{\partial}{\partial x_n} + \mathbf{e}_y\frac{\partial}{\partial y_n} + \mathbf{e}_z\frac{\partial}{\partial z_n}\,,\quad \nabla_n^2 = \nabla_n\cdot\nabla_n = \frac{\partial^2}{{\partial x_n}^2} + \frac{\partial^2}{{\partial y_n}^2} + \frac{\partial^2}{{\partial z_n}^2}</math>

数学部分，数学部分，数学部分，数学部分，数学部分,4 nabla n ^ 2 = nabla n cndot nabla n = frac { partial ^ 2}{ partial x _ n } ^ 2} + frac { partial y _ n } ^ 2}{ partial y _ n } ^ 2} + frac { partial ^ 2}{{ partial z _ n }{{ partial z _ n }}{{ partial z _ n } ^ 2} ^ 2} </math >  

,4 nabla n ^ 2 = nabla n cndot nabla n = frac { partial ^ 2}{ partial x _ n } ^ 2} + frac { partial y _ n } ^ 2}{ partial y _ n } ^ 2} + frac { partial ^ 2}{{ partial z _ n }{{ partial z _ n }}{{ partial z _ n } ^ 2} ^ 2} </math >  

Louis de Broglie in his later years proposed a real valued [[wave function]] connected to the complex wave function by a proportionality constant and developed the [[De Broglie–Bohm theory]].

Louis de Broglie in his later years proposed a real valued [[wave function]] connected to the complex wave function by a proportionality constant and developed the [[De Broglie–Bohm theory]].