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wave equation of a particle in a scalar potential (Topic)

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

$\displaystyle E_{cl.} = H(\mathbf{r}_{cl.},\mathbf{p}_{cl.}) = \frac{p^2_{cl.}}{2m} +V(\mathbf{r}_{cl.})$ (1)

$H(\mathbf{r}_{cl.}, \mathbf{p}_{cl.})$ 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

$\displaystyle V(\mathbf{r}) \Psi(\mathbf{r},t) \approx V(\mathbf{r}_{cl.}) \Psi(\mathbf{r},t)$ (2)

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

$\displaystyle \Psi(\mathbf{r},t) = \int F(\mathbf{p}) \exp^{i(\mathbf{p} \cdot \mathbf{r} - Et)/\hbar} d\mathbf{p}$ (3)

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

$\displaystyle i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r},t) \approx E_{cl.} \Psi(\mathbf{r},t)$
$\displaystyle \frac{\hbar}{i} \nabla \Psi(\mathbf{r},t) \approx \mathbf{p}_{cl.}(t) \Psi(\mathbf{r},t)$ (4)

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

$\displaystyle - \hbar^2 \nabla^2 \Psi(\mathbf{r},t) \approx p^2_{cl.} \Psi(\mathbf{r},t)$ (5)

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

$\displaystyle \i \hbar \frac{\partial}{\partial t} \Psi + \frac{\hbar^2}{2m} \n... ... ( 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

$\displaystyle i \hbar \frac{\partial }{\partial t} \Psi(\mathbf{r},t) = \left( -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}) \right) \Psi(\mathbf{r},t)$ (6)

It is the Schrödinger 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].



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See Also: wave equation of a free particle, wave equation of a charged a particle in an electromagnetic field


Cross-references: work, domain, volume, quantum mechanics, divergence, vectors, type, functions, system, Hamiltonian, relation, energy, momentum, position, wave, wave equation
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This is version 2 of wave equation of a particle in a scalar potential, born on 2009-04-05, modified 2009-04-05.
Object id is 629, canonical name is WaveEquationOfAParticleInAScalarPotential.
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Physics Classification: 03.65.-w (Quantum mechanics )
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