Correspondence Principle

Starting with Plane Waves

Free particles are described in quantum mechanics by plane wave solutions

$$
\psi(\vec{r},t) \sim e^{i (\vec{k} \cdot \vec{r} – \omega t)}
$$

and by the de-Broglie wave particle dualism a relation between particle momentum \(\vec{p}\) and wave vector \(\vec{k}\) is established

$$
E = \hbar \omega, \quad p = \hbar k, \quad \omega = c k
$$

One can see that

$$
i \hbar \partial_t \psi = i \hbar (-i \omega) \psi = \hbar \omega \psi = E \psi
$$

$$
-i\hbar \nabla \psi = -i \hbar (i \vec{k}) = \hbar \vec{k} \psi = \vec{p} \psi
$$

which motivates the correspondence principles of quantum mechanics, that is replacing classical quantities by quantum mechanical operators

$$
E \to i \hbar \partial_t, \quad \vec{p} \to – i \hbar \nabla, \quad \vec{r} \to \vec{r}
$$

Motivating Schrödinger Equation

The correspondence principle can be used to motivate the form of Schrödinger equation. Starting with the classical Hamilton function

$$
H(q,p,t) = \frac{p^2}{2m} + V(q,t) = E
$$

Schrödinger equation is obtained by substituting energy \(E\) by the Hamilton operator \(\hat{H} = i \hbar \partial_t \), momentum \(\vec{p}\) by the momentum operator \(\hat{\vec{p}} = -i \hbar \nabla\) and position \(\vec{r}\) by the position operator \(\hat{\vec{r}}\), yielding

$$
\hat{H} \psi(\vec{r},t) = \left( -\frac{\hbar^2}{2m} \nabla^2 \psi(\vec{r},t) + V(\vec{r},t) \right) \psi(\vec{r},t) = i \hbar \partial_t \psi(\vec{r},t)
$$