[NOTES/QM-11001] Time Dependent Schrodinger Equation :Solution for Wave function at time \(t\)

We assume that the Hamiltonian of the particle is independent of time and that it can be written in the form \begin{equation}\label{EQ01} H = \frac{p^2}{2m} + V(\vec{r}) \end{equation} The Schrodinger equation for the particle moving in potential $V(\vec{r})$ can be written as \begin{equation} -i \hbar\frac{\partial\psi(t)}{\partial t} = {1\over 2M} \nabla\,^2 \psi(t) + V(\vec{r}) \psi (t). \label{E24} \end{equation} The possible states of the particle at a time $t$ will be represented by square integrable wave function wave function $\psi(\vec{r},t)$. When time does not appear in the Hamiltonian of a system, the equation of motion can be solved by the method of separation of variables. Thus by substituting \begin{equation} \psi(\vec{r},t) = u(\vec{r}) T(t) \label{E25} \end{equation} in \eqref{E24} we get \begin{equation} i\hbar {1\over T(t)}\frac{d T(t)}{dt} = \hat{H} u(\vec{r}) \label{E26} \end{equation} Equating each side to a constant, say $E$ we get two equations for $u(\vec{r})$, and for $T(t)$, as follows. \begin{eqnarray} i \hbar \frac{d}{dt} T(t) & =& E T(t) \label{E27} \\ \hat{H} u(\vec{r}) & = & E u(\vec{r}) \label{E28} \end{eqnarray} Let $u_1(\vec{r}), u_2(\vec{r}), ...$ denote the eigenvectors of the Hamiltonian $\hat{H}$ with the eigenvalues $E_1,E_2, ... $, respectively. \begin{equation}\label{EQ301} \hat{H} u_k(\vec{r}) = E_k u_k(\vec{r}) ; \qquad k=1,2,3,... \end{equation} \eqref{E27} and \eqref{E28} have solutions given by \begin{equation} T_k(t) = \text{const} \times \exp(-iE_kt/\hbar), \qquad H u_k(\vec{r}) = E_ku_k(\vec{r}) \label{E29} \end{equation} and \eqref{E24} has {\it an infinite number of solutions}, one for each real \(k\), given by \begin{equation} \phi_k(\vec{r},t) = u_k(\vec{r},t) \exp(-iE_kt/\hbar) \label{E30} \end{equation} and the most general solution is a linear combination of solutions $\phi_k(\vec{r},t)$ in \eqref{E30} and is given by \begin{equation} \psi(\vec{r},t) = \sum_{k=1}^{\infty} c_k u_k(\vec{r}) \exp(-iE_kt)/\hbar) . \label{E31} \end{equation} If the wave function at time $t_0$, is $\psi(\vec{r},t_0)\equiv \psi_o(\vec{r})$, the expression \eqref{E31} evaluated at $t=t_0$ gives \begin{equation} \psi_0(\vec{r}) = \sum_{k=1}^{\infty} c_k u_k(\vec{r}) \exp(-iE_kt_0)/\hbar) \end{equation} Using the orthogonality of energy eigen functions we can find the coefficients $\alpha_k$ and are given by. \begin{equation} (u_k,\psi_0) = c_k \exp(-iE_kt_0)/\hbar) \end{equation} and the knowledge of the coefficients $c_k$ allows us to compute the wave function at time $t$ from \eqref{E31}. The final answer for the wave function at time $t$ is given by \begin{equation} \label{EQ12} \psi(\vec{r},t) = \sum_{k=1}^{\infty} c_k u_k(\vec{r}) \exp(-iE_kt)/\hbar)\psi(\vec{r},t) = \sum_{k=1}^{\infty} (u_k, \psi_0)\exp(-iE_k(t-t_0)/\hbar) \,u_k(\vec{r}) \end{equation} $\psi_0$ is the wave function of the system at time $t_0$.