WKB The Approximation
WKB: Wentzel, Kramer, Brittain So far we have worked a lot on calculating ground states. Now let's look at excited states. For that we will use the WKB approximation.
Basic idea: We want to obtain approximate solutions of the Schrodinger Equation in one dimension for the cases when the wave function oscillates fast but the potential is a slowly varying function.
A 1) If the potential V is a constant and the energy of the particle E>V, then the wave function has the form:
signs: particle is travelling (to the right +)) to the left The general solution is a superposition of the two. The wave function is oscillatory with a fixed wavelength and fixed amplitude. 2) If V(x) is not a constant but varies slow in comparison with the wavelength, in a way that is essentially constant over many
Then the wave function is practically sinusoidal, but wavelengths and amplitude slowly change with x.
That is: Rapid oscillations are modulated by gradual changes in amplitude and wavelength.
B) 1) If E<V and V is a constant, then the wave function is
2) If V is not constant but varies slowly in comparison with 1/k, then the wave function is practically and exponential A and k are slowly-varying functions of x.
C) Problem: Turning points: when V=E. Close to them V(x) is not slowly varying in comparison with or I/k since or (1/k)
The "classical" region Let's solve the Schrodinger Equation using WKB
rewrite
or
We assume for now E>V(x) and k real. is some complex function an can be expressed as
We can plug (2) back into (1) and separate real an imaginary parts in two equations:
so
We solve first (4) which is easer:
To solve equation (3) is more complicated since it is a non-linear second order differential equation. To solve it we are going to make the approximation that A varies slowly with x. So the second derivat...