The Time Independent Schrödinger Equation

Second order differential equations, like the Schrödinger Equation, can be solved by separation of variables. These separated solutions can then be used to solve the problem in general.

Assume that we can factorize the solution between time and space.

\begin{displaymath}\bgroup\color{black} \psi(x,t)=u(x)T(t) \egroup\end{displaymath}

Plug this into the Schrödinger Equation.

\begin{displaymath}\bgroup\color{black} \left({-\hbar^2\over 2m}{\partial^2u(x)\...
i\hbar u(x){\partial T(t)\over\partial t} \egroup\end{displaymath}

Put everything that depends on \bgroup\color{black}$x$\egroup on the left and everything that depends on \bgroup\color{black}$t$\egroup on the right.

\begin{displaymath}\bgroup\color{black} {\left({-\hbar^2\over 2m}{\partial^2u(x)...
...\hbar{\partial T(t)\over\partial t}\over T(t)}=const.=E \egroup\end{displaymath}

Since we have a function of only \bgroup\color{black}$x$\egroup set equal to a function of only \bgroup\color{black}$t$\egroup, they both must equal a constant. In the equation above, we call the constant \bgroup\color{black}$E$\egroup, (with some knowledge of the outcome). We now have an equation in \bgroup\color{black}$t$\egroup set equal to a constant

\begin{displaymath}\bgroup\color{black}i\hbar{\partial T(t)\over\partial t}=E\; T(t)\egroup\end{displaymath}

which has a simple general solution,

\begin{displaymath}\bgroup\color{black} T(t)=C e^{-iEt/\hbar} \egroup\end{displaymath}

and an equation in \bgroup\color{black}$x$\egroup set equal to a constant

\begin{displaymath}\bgroup\color{black} {-\hbar^2\over 2m}{\partial^2u(x)\over \partial x^2}+V(x)u(x)=E\; u(x) \egroup\end{displaymath}

which depends on the problem to be solved (through \bgroup\color{black}$V(x)$\egroup).

The \bgroup\color{black}$x$\egroup equation is often called the Time Independent Schrödinger Equation.

\bgroup\color{black}$\displaystyle {-\hbar^2\over 2m}{\partial^2u(x)\over \partial x^2}+V(x)u(x)=E\; u(x) $\egroup
Here, \bgroup\color{black}$E$\egroup is a constant. The full time dependent solution is.
\bgroup\color{black}$\displaystyle \psi(x,t)=u(x)e^{-iEt/\hbar}$\egroup

* Example: Solve the Schrödinger equation for a constant potential $V_0$.*

Jim Branson 2013-04-22