# 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

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