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As the curvature increases, the amplitude of the wave alternates between positive and negative more rapidly, and also shortens the wavelength. So the inverse relation between momentum and wavelength is consistent with the energy the particle has, and so the energy of the particle has a connection to a wave, all in the same mathematical formulation.<ref name="Quanta 1974"/>

As the curvature increases, the amplitude of the wave alternates between positive and negative more rapidly, and also shortens the wavelength. So the inverse relation between momentum and wavelength is consistent with the energy the particle has, and so the energy of the particle has a connection to a wave, all in the same mathematical formulation.<ref name="Quanta 1974"/>

===Wave and particle motion===

===Wave and particle motion波与质点运动===

The Schrödinger equation can also be derived from a first order form similar to the manner in which the Klein–Gordon equation can be derived from the Dirac equation. In 1D the first order equation is given by

The Schrödinger equation can also be derived from a first order form similar to the manner in which the Klein–Gordon equation can be derived from the Dirac equation. In 1D the first order equation is given by

   | caption1  = Increasing levels of [[wavepacket]] localization, meaning the particle has a more localized position.

   | caption1  = Increasing levels of [[wavepacket]] localization, meaning the particle has a more localized position.

\end{align}

\end{align}

   | caption2  = In the limit ''ħ'' → 0, the particle's position and momentum become known exactly. This is equivalent to the classical particle.

   | caption2  = In the limit ''ħ'' → 0, the particle's position and momentum become known exactly. This is equivalent to the classical particle.

This equation allows for the inclusion of spin in nonrelativistic quantum mechanics. Squaring the above equation yields the Schrödinger equation in 1D. The matrices <math> \eta </math> obey the following properties

This equation allows for the inclusion of spin in nonrelativistic quantum mechanics. Squaring the above equation yields the Schrödinger equation in 1D. The matrices <math> \eta </math> obey the following properties

   | width2    = 150

   | width2    = 150

Schrödinger required that a [[wave packet]] solution near position <math>\mathbf{r}</math> with wave vector near <math>\mathbf{k}</math> will move along the trajectory determined by classical mechanics for times short enough for the spread in <math>\mathbf{k}</math> (and hence in velocity) not to substantially increase the spread in {{math|'''r'''}}. Since, for a given spread in {{math|'''k'''}}, the spread in velocity is proportional to Planck's constant <math>\hbar</math>, it is sometimes said that in the limit as <math>\hbar</math> approaches zero, the equations of classical mechanics are restored from quantum mechanics.<ref name="Analytical Mechanics 2008">''Analytical Mechanics'', L. N. Hand, J. D. Finch, Cambridge University Press, 2008, {{isbn|978-0-521-57572-0}}</ref> Great care is required in how that limit is taken, and in what cases.

Schrödinger required that a [[wave packet]] solution near position <math>\mathbf{r}</math> with wave vector near <math>\mathbf{k}</math> will move along the trajectory determined by classical mechanics for times short enough for the spread in <math>\mathbf{k}</math> (and hence in velocity) not to substantially increase the spread in {{math|'''r'''}}. Since, for a given spread in {{math|'''k'''}}, the spread in velocity is proportional to Planck's constant <math>\hbar</math>, it is sometimes said that in the limit as <math>\hbar</math> approaches zero, the equations of classical mechanics are restored from quantum mechanics.<ref name="Analytical Mechanics 2008">''Analytical Mechanics'', L. N. Hand, J. D. Finch, Cambridge University Press, 2008, {{isbn|978-0-521-57572-0}}</ref> Great care is required in how that limit is taken, and in what cases.

\eta^2=0  \\

\eta^2=0  \\

The limiting short-wavelength is equivalent to <math>\hbar</math> tending to zero because this is limiting case of increasing the wave packet localization to the definite position of the particle (see images right). Using the [[Heisenberg uncertainty principle]] for position and momentum, the products of uncertainty in position and momentum become zero as <math>\hbar \longrightarrow 0</math>:

The limiting short-wavelength is equivalent to <math>\hbar</math> tending to zero because this is limiting case of increasing the wave packet localization to the definite position of the particle (see images right). Using the [[Heisenberg uncertainty principle]] for position and momentum, the products of uncertainty in position and momentum become zero as <math>\hbar \longrightarrow 0</math>:

\left\lbrace \eta, \eta^\dagger \right\rbrace= 2 I

\left\lbrace \eta, \eta^\dagger \right\rbrace= 2 I

\end{align}

\end{align}

:<math> \sigma(x) \sigma(p_x) \geqslant \frac{\hbar}{2} \quad \rightarrow \quad \sigma(x) \sigma(p_x) \geqslant 0 \,\!</math>

:<math> \sigma(x) \sigma(p_x) \geqslant \frac{\hbar}{2} \quad \rightarrow \quad \sigma(x) \sigma(p_x) \geqslant 0 \,\!</math>

</math>

</math>

where {{math|''σ''}} denotes the (root mean square) [[measurement uncertainty]] in {{math|''x''}} and {{math|''p_x''}} (and similarly for the {{math|''y''}} and {{math|''z''}} directions) which implies the position and momentum can only be known to arbitrary precision in this limit.

where {{math|''σ''}} denotes the (root mean square) [[measurement uncertainty]] in {{math|''x''}} and {{math|''p_x''}} (and similarly for the {{math|''y''}} and {{math|''z''}} directions) which implies the position and momentum can only be known to arbitrary precision in this limit.

The 3 dimensional version of the equation is given by

The 3 dimensional version of the equation is given by

One simple way to compare classical to quantum mechanics is to consider the time-evolution of the ''expected'' position and ''expected'' momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics. The quantum expectation values satisfy the [[Ehrenfest theorem]]. For a one-dimensional quantum particle moving in a potential <math>V</math>, the Ehrenfest theorem says<ref>{{harvnb|Hall|2013}} Section 3.7.5</ref>

One simple way to compare classical to quantum mechanics is to consider the time-evolution of the ''expected'' position and ''expected'' momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics. The quantum expectation values satisfy the [[Ehrenfest theorem]]. For a one-dimensional quantum particle moving in a potential <math>V</math>, the Ehrenfest theorem says<ref>{{harvnb|Hall|2013}} Section 3.7.5</ref>

<math display="block">

<math display="block">

\begin{align}

\begin{align}

Although the first of these equations is consistent with the classical behavior, the second is not: If the pair <math>(\langle X\rangle,\langle P\rangle)</math> were to satisfy Newton's second law, the right-hand side of the second equation would have to be

Although the first of these equations is consistent with the classical behavior, the second is not: If the pair <math>(\langle X\rangle,\langle P\rangle)</math> were to satisfy Newton's second law, the right-hand side of the second equation would have to be

-i \gamma_i \partial_i \psi = (i  \eta \partial_t  + \eta^\dagger m) \psi  

-i \gamma_i \partial_i \psi = (i  \eta \partial_t  + \eta^\dagger m) \psi  

\end{align}

\end{align}

which is typically not the same as <math>-\left\langle V'(X)\right\rangle</math>. In the case of the quantum harmonic oscillator, however, <math>V'</math> is linear and this distinction disappears, so that in this very special case, the expected position and expected momentum do exactly follow the classical trajectories.

which is typically not the same as <math>-\left\langle V'(X)\right\rangle</math>. In the case of the quantum harmonic oscillator, however, <math>V'</math> is linear and this distinction disappears, so that in this very special case, the expected position and expected momentum do exactly follow the classical trajectories.

</math>

</math>

For general systems, the best we can hope for is that the expected position and momentum will ''approximately'' follow the classical trajectories. If the wave function is highly concentrated around a point <math>x_0</math>, then <math>V'\left(\left\langle X\right\rangle\right)</math> and <math>\left\langle V'(X)\right\rangle</math> will be ''almost'' the same, since both will be approximately equal to <math>V'(x_0)</math>. In that case, the expected position and expected momentum will remain very close to the classical trajectories, at least for as long as the wave function remains highly localized in position.<ref>{{harvnb|Hall|2013}} p. 78</ref> When Planck's constant is small, it is possible to have a state that is well localized in ''both'' position and momentum. The small uncertainty in momentum ensures that the particle ''remains'' well localized in position for a long time, so that expected position and momentum continue to closely track the classical trajectories.

对于一般系统，我们所能期望的最好结果是，预期的位置和动量将“近似”地遵循经典轨迹。如果波函数高度集中在一个点上，那么<math>V'\left（\left\langle x\right\rangle\right）</math>和<math>\left\langle V'（x）\right\rangle</math>将“几乎”相同，因为两者都将近似等于<math>V'（x\u 0）</math>。在这种情况下，预期位置和预期动量将保持非常接近经典轨迹，至少只要波函数在位置上保持高度局部化。<ref>{harvnb | Hall | 2013}p.78</ref>当普朗克常数很小时，有可能有一个状态在“位置和动量”都很好地局部化。动量的小不确定性确保了粒子在很长一段时间内“保持”良好的局部化位置，因此预期的位置和动量继续与经典轨迹密切相关。

Here <math> \eta=(\gamma_0+i \gamma_5)/\sqrt{2} </math> is a <math>4 \times 4</math> nilpotent matrix and <math> \gamma_i </math> are the Dirac gamma matrices (<math>i=1,2,3</math>). The Schrödinger equation in 3D can be obtained by squaring the above equation.  

Here <math> \eta=(\gamma_0+i \gamma_5)/\sqrt{2} </math> is a <math>4 \times 4</math> nilpotent matrix and <math> \gamma_i </math> are the Dirac gamma matrices (<math>i=1,2,3</math>). The Schrödinger equation in 3D can be obtained by squaring the above equation.  

The Schrödinger equation in its general form

The Schrödinger equation in its general form

 

:<math> i\hbar \frac{\partial}{\partial t} \Psi\left(\mathbf{r},t\right) = \hat{H} \Psi\left(\mathbf{r},t\right) \,\!</math>

:<math> i\hbar \frac{\partial}{\partial t} \Psi\left(\mathbf{r},t\right) = \hat{H} \Psi\left(\mathbf{r},t\right) \,\!</math>

is closely related to the [[Hamilton–Jacobi equation]] (HJE)

is closely related to the [[Hamilton–Jacobi equation]] (HJE)

 

:<math> -\frac{\partial}{\partial t} S(q_i,t) = H\left(q_i,\frac{\partial S}{\partial q_i},t \right) \,\!</math>

:<math> -\frac{\partial}{\partial t} S(q_i,t) = H\left(q_i,\frac{\partial S}{\partial q_i},t \right) \,\!</math>

where <math>S</math> is the classical [[action (physics)|action]] and <math>H</math> is the [[Hamiltonian mechanics|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>.<ref name="Analytical Mechanics 2008"/>

where <math>S</math> is the classical [[action (physics)|action]] and <math>H</math> is the [[Hamiltonian mechanics|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>.<ref name="Analytical Mechanics 2008"/>

Substituting

Substituting

 

:<math> \Psi = \sqrt{\rho(\mathbf{r},t)} e^{iS(\mathbf{r},t)/\hbar}\,\!</math>

:<math> \Psi = \sqrt{\rho(\mathbf{r},t)} e^{iS(\mathbf{r},t)/\hbar}\,\!</math>

where <math>\rho</math> is the probability density, into the Schrödinger equation and then taking the limit <math>\hbar \longrightarrow 0</math> in the resulting equation yield the Hamilton–Jacobi equation.

where <math>\rho</math> is the probability density, into the Schrödinger equation and then taking the limit <math>\hbar \longrightarrow 0</math> in the resulting equation yield the Hamilton–Jacobi equation.

The implications are as follows:

The implications are as follows:

* The motion of a particle, described by a (short-wavelength) wave packet solution to the Schrödinger equation, is also described by the Hamilton–Jacobi equation of motion.

*粒子的运动由薛定谔方程的（短波长）波包解描述，也由哈密顿-雅可比运动方程描述。

* The Schrödinger equation includes the wave function, so its wave packet solution implies the position of a (quantum) particle is fuzzily spread out in wave fronts. On the contrary, the Hamilton–Jacobi equation applies to a (classical) particle of definite position and momentum, instead the position and momentum at all times (the trajectory) are deterministic and can be simultaneously known.

* The Schrödinger equation includes the wave function, so its wave packet solution implies the position of a (quantum) particle is fuzzily spread out in wave fronts. On the contrary, the Hamilton–Jacobi equation applies to a (classical) particle of definite position and momentum, instead the position and momentum at all times (the trajectory) are deterministic and can be simultaneously known.

 

==Nonrelativistic quantum mechanics==

==Nonrelativistic quantum mechanics==