Editing Schrödinger equation (section)

== Semiclassical limit ==
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.<ref name=":0" />{{rp|302}} 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
<math display="block">m\frac{d}{dt}\langle x\rangle = \langle p\rangle;\quad \frac{d}{dt}\langle p\rangle =  -\left\langle V'(X)\right\rangle.</math>
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
<math display="block">-V'\left(\left\langle X\right\rangle\right)</math>
which is typically not the same as <math>-\left\langle V'(X)\right\rangle</math>. For a general <math>V'</math>, therefore, quantum mechanics can lead to predictions where expectation values do not mimic the classical behavior. 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.

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.

The Schrödinger equation in its general form
<math display="block"> 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)
<math display="block"> -\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).<ref name=":0" />{{rp|308}} 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>.

Substituting
<math display="block"> \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 \to 0</math> in the resulting equation yield the [[Hamilton–Jacobi equation]].