PlanetPhysics/Wave Equation of a Particle in a Scalar Potential

In order to form the wave equation of a particle in a potential $$V(\mathbf{r})$$, we operate at first under the conditions of the `geometrical optics approximation' and seek to form an equation of propagation for a wave packet $$\Psi(\mathbf{r},t)$$ moving in accordance with the de Broglie theory.

The center of the packet travels like a classical particle whose position, momentum, and energy we shall designate by $$\mathbf{r}_{cl.}$$, $$\mathbf{p}_{cl.}$$, and $$E_{cl.}$$, respectively. These quantities are connected by the relation $$ E_{cl.} = H(\mathbf{r}_{cl.},\mathbf{p}_{cl.}) = \frac{p^2_{cl.}}{2m} +V(\mathbf{r}_{cl.}) $$

$$ H(\mathbf{r}_{cl.}, \mathbf{p}_{cl.})$$ is the classical Hamiltonian. We suppose that $$V(\mathbf{r})$$ does not depend upon the time explicitly (conservative system), although this condition is not absolutely necessary for the present argument to hold. Consequently $$E_{cl.}$$ remains constant in time, while $$\mathbf{r}_{cl.}$$ and $$\mathbf{p}_{cl.}$$ are well-defined functions of $$t$$. Under the approximate conditions considered here, $$V(\mathbf{r})$$ remains practically constant over a region of the order of the size of the wave packet; therefore

$$ V(\mathbf{r}) \Psi(\mathbf{r},t) \approx V(\mathbf{r}_{cl.}) \Psi(\mathbf{r},t) $$

On the other hand, if we restrict ourselves to time intervals sufficiently short so that the relative variation of $$\mathbf{p}_{cl.}$$ remains negligible, $$\Psi(\mathbf{r},t)$$ may be considered as a superposition of plane waves of the type $$ \Psi(\mathbf{r},t) = \int F(\mathbf{p}) \exp^{i(\mathbf{p} \cdot \mathbf{r} - Et)/\hbar} d\mathbf{p} $$

whose frequencies are in the neighborhood of $$E_{cl.}/\hbar$$ and whose wave vectors lie close to $$\mathbf{p}_{cl.}/\hbar$$. Therefore

$$i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r},t) \approx E_{cl.} \Psi(\mathbf{r},t)$$ $$ \frac{\hbar}{i} \nabla \Psi(\mathbf{r},t) \approx \mathbf{p}_{cl.}(t) \Psi(\mathbf{r},t) $$

and taking the divergence of this last express ion, one obtains

$$ - \hbar^2 \nabla^2 \Psi(\mathbf{r},t) \approx p^2_{cl.} \Psi(\mathbf{r},t) $$

combining the relations (2),(3), and (4) and making use of equation (1), we obtain

$$ \mathrm{i} \hbar \frac{\partial}{\partial t} \Psi + \frac{\hbar^2}{2m} \nabla^2 \Psi - V \Psi \approx \left ( E_{cl.} - \frac{p^2_{cl.}}{2m} - V(\mathbf{r}_{cl.}) \right) \Psi \approx 0 $$

The wave packet $$\Psi(\mathbf{r},t)$$ satisfies - at least approximately - a wave equation of the type we are looking for. We are very naturally led to adopt this equation as the wave equation of a particle in a potential, and we postulate that in all generality, even when the conditions for the `geometrical optics' approximation are not fulfilled, the wave $$\Psi$$ satisfies the equation

$$ i \hbar \frac{\partial }{\partial t} \Psi(\mathbf{r},t) = \left( -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}) \right) \Psi(\mathbf{r},t) $$

It is the Schr\"odinger equation for a particle in a potential $$V(\mathbf{r})$$.

[1] Messiah, Albert. "Quantum mechanics: volume I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.

This entry is a derivative of the Public domain work [1].