Themes > Science > Physics > Quantum Physics > The Schrödinger equation

Reading: SB §41.5.

In 1926, Erwin Schrödinger suggested an equation to be satisfied by the matter waves. In classical physics the total energy E is conserved:
\begin{displaymath} K+V={p^2\over2m}+V=E=\hbox{constant}.\end{displaymath} (19)

His idea was that for matter waves, this equation would have to be rewritten to involve something waving', which he called the wave function $\psi$. He noticed that for a simple wave like $\sin(kx)$,one gets
\begin{displaymath} {{\rm d}^2\over{\rm d} x^2}\sin(kx)=-k^2\sin(kx).\end{displaymath} (20)
But according to de Broglie, $p=\hbar k$, and therefore the kinetic energy is
\begin{displaymath} K={\hbar^2k^2\over2m}.\end{displaymath} (21)

So Schrödinger proposed that the kinetic energy p2/2m should be `represented' (in one dimension) by
\begin{displaymath} -{\hbar^2\over2m}{{\rm d}^2\psi\over{\rm d} x^2}.\end{displaymath} (22)

So, he wrote
\begin{displaymath} -{\hbar^2\over2m}{{\rm d}^2\psi\over{\rm d} x^2}+V(x)\psi(x)=E\psi(x).\end{displaymath} (23)

(Note that E does not depend on x--it is a constant.) This is the one-dimensional time-independent Schrödinger equation. (There is also a time-dependent version, but we do not discuss it in this course.) It cannot be proved from classical physics--only justified by the predictions it makes.


Information provided by: http://www.cmmp.ucl.ac.uk