Propagators

We briefly describe the solution of Schrodinger's equation for a time independent potential $V(\vec{r})$. Suppose the energy eigenfunctions are $\psi_{E}(\vec{r})$ with amplitudes $c_{E}(t)$. The general solution of the Schrodinger equation

(1)
\begin{align} -\frac{\hbar^2}{2m}\nabla^2\Psi(\vec{r},t) + V(\vec{r})\Psi(\vec{r},t) = E\Psi(\vec{r},t) \end{align}

is given by

(2)
\begin{align} \Psi(\vec{r},t) = \sum_{E} c_{E}(t)\psi_{E}(\vec{r}) \end{align}

Here,

(3)
\begin{align} c_{E}(t) = \int_{\mathbb{R}^3} \psi_{E}^{*}(\vec{r})\Psi(\vec{r},t) d^{3}\vec{r} \end{align}

Using the orthonormality of the eigenfunctions $\psi_{E}$, we get

(4)
\begin{align} i\hbar\frac{d}{dt}c_{E}(t) = Ec_{E}(t) \end{align}

the solution to which is

(5)
\begin{align} c_{E}(t) = c_{E}(t_0)e^{\left[-\frac{i}{\hbar}E(t-t_{0})\right]} \end{align}

Let $P_{E}$ denote the probability of obtaining energy $E$. From the QM Postulate we have,

(6)
\begin{equation} P_{E} = |c_{E}(t)|^2 = |c_{E}(t_0)|^2 \end{equation}

Hence $P_{E}$ is constant. We can write

(7)
\begin{align} \Psi(\vec{r},t) = \sum_{E}d_{E}(t_0)\psi_{E}(\vec{r})e^{-\frac{i}{\hbar}Et} \end{align}

where $d_{E}(t_0) = c_{E}(t_0)e^{\frac{i}{\hbar}Et_0$

Thus

(8)
\begin{align} \Psi(\vec{r},t) = \sum_{E}\left[\int_{\mathbb{R}^3}\psi_{E}^{*}(\vec{r'})\Psi(\vec{r'},t_0)d^{3}\vec{r'}\right]e^{-\frac{i}{\hbar}E(t-t_0)}\psi_{E}(\vec{r}) = \int_{\mathbb{R}^3}K(\vec{r},t;\vec{r'},t_0)\Psi(\vec{r'},t_0)d\vec{r'} \end{align}

where the propagator $K$ is given by

(9)
\begin{align} K(\vec{r},t;\vec{r'},t_0) = \sum_{E}\psi_{E}^{*}(\vec{r'})\psi_{E}(\vec{r})e^{-\frac{i}{\hbar}E(t-t_0)} \end{align}

The propagator can be thought of as the probability amplitude that a particle at $\vec{r'}$ will 'propagate' to $\vec{r}$ in a time interval $(t-t_0)$.

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