The WKB Approximation

Consider the 1-dimensional, quantum mechanical system consisting of a single structureless particle.
How well do the classical equations of motion describe the motion of this particle?

We have Ehrenfest’s theorem, .

When the potential varies slowly with distance, then the classical equations of motion work best.

Assume the potential is a slowly varying function of distance, V=V(x), independent of time.  Then solutions of the form can be found. fE(x) is a solution of

,

where

 .

Let us try a solution of the form f(x)=Aexp((i/h)S(x)).  [Any complex function of x can be written as f(x)=A(x)exp((i/h)S(x)).]  Substituting this solution into the time-independent Schroedinger equation we obtain

.

(For a potential well S(x)=±hkx inside the well and ±ihrx outside the well.)

Assume that h can, in some sense, be regarded as a small quantity and that S(x) can be expanded in powers of h, S(x)=S0(x)+hS1(x)+L.

Then

 ,  (E>V(x)).

We assume that and collect terms with equal powers of h.  [Note: h2k2 is zeroth order in h, since Image2342a.gif (1220 bytes).

.

.

We have used 

Therefore

In the classically allowed region counts the oscillations of the wave function.  An increase of 2ph corresponds to an additional phase of 2p.

Similarly, in regions where E<V(x) we have

This is the WKB (Wentzel, Kramers, Brillouin) approximation. When is it valid?

For our first order expansion to be accurate we need that the magnitude of higher order terms decreases rapidly.  We need or .

The local deBroglie wavelength is l=2p/k.  Therefore , i.e. the change in l over a distance l /4p is small compared to l . This holds when the potential varies slowly and the momentum of the particle is nearly constant over several wavelength.

Near the classical turning points the WKB solutions become invalid, because k goes to zero here.  We have to find a way to connect an oscillating solution to an exponential solution across a turning point if we want to solve barrier penetration problems or find bound states.

The WKB approximation for bound states

We want to find the wave function of a particle in a given potential well.  Assume the energy of the particle is E and that the classical turning points are x1 and x2, x1<x2, i.e. we have a potential well with two sloping sides.

For x<x1 the wave function is of the form 

For x>x2 the wave function is of the form 

In the region between x1 and x2 it is of the form   .

At x=x1 and x=x2 the wave function f and its derivative have to be continuous.  How do we apply these boundary conditions?

Near x1 and x2 we expand the potential in a Taylor series expansion in x and neglect all terms of order higher than 1.

Near x1 we have , and near x2 we have .

In the neighborhood of x1 the time-independent Schroedinger equation then becomes ,

and in the neighborhood of x2 the time-independent Schroedinger equation becomes .

Let us define  .

Then we obtain near x1.

The solutions of this equation which vanish asymptotically as  z® ¥ or x® -¥ are the Airy functions. They are defined through 

, 

which for large |z| has the asymptotic form

.

If the energy is high enough the linear approximation to the potential remains valid over many wavelength. The Airy functions can therefore be the connecting wave functions through the turning point at x1.

If we define then we find near x=x2 and the Airy functions can also be the connecting wave functions through the turning point at x2.  Here z® ¥ as x® ¥ .

In the neighborhood of x1 we have .

Therefore .

Similarly .

By comparing this with the asymptotic forms of the Airy functions we note that  (x<x1) must continue on the right side as  (x>x1).

In the neighborhood of x2 we similarly find that  (x>x2) must continue in region 2 as   (x<x2).  Both expressions for f2(x) are approximations to the same eigenfunction. We therefore need

.

Writing , we require  .

This condition is only satisfied if 

.

This can be rewritten as

or .

Here denote an integral over one complete cycle of the classical motion.  The WKB method for bound states therefore leads to the Wilson- Sommerfeld quantization rule except that n is replaced by .  It leads to a quantization of the classical action .

If our potential well has one or two vertical walls, the results of the WKB approximation differ only in the number that is subtracted from n (-1/4 or 0, respectively).  Since the WKB approximation works best for large n in the semiclassical regime, this distinction is more in appearance than in substance.

Write your own program

This link shows you how to find  a numerical solution for  and presents you with an example program.

Penetration of a potential barrier

The WKB method leads to the following expression for the transmission coefficient:

.

To the same approximation R=1-T.  Details can be found in Powell and Crasemann, Quantum Mechanics.

Problem:

Use the WKBJ approximation to derive the energy levels of a particle confined to the one-dimensional potential V(x)=F|x|.
Solution:

For stationary bound states we want . In our problem . We need

.

. .

. For n=1 we have .

Problem:

A particle of a given energy E>0 is confined by a potential energy function which is given by V(x)=c|x|.

(a) Describe in detail, quantitatively and qualitatively as best as you can, what an excited state wavefunction looks like for region E>V(x) and region E<V(x).

(b) Introduce the parity operator P and prove that each eigenstate with an eigenenergy E is also an eigenstate of the parity operator P.

(c) With the energy diagram ordered accordingly to the magnitude of the eigenenergy E, show that the ground state (lowest eigenenergy state) is an even parity state, followed by the next excited state being an odd parity state.  All subsequent excited states possess even or odd parity alternatively.

Solution:
(a) The time-independent Schroedinger equation is , with  .

Let us concentrate in the region E>V(x).  For excited states the WKB approximation can be used. except near the turning points. counts the oscillations of the wavefunction.  Near x=0 the wavefunction oscillates rapidly since k is large.  Near the turning points it oscillates less rapidly since k is much smaller.  The amplitude increases as we approach the turning points.

Now concentrate on the region where E<V(x).  In this region the wavefunction decreases rapidly as we move towards larger |x| away from the turning points.

(b) The parity operator is defined through Py(x)=y(-x). [P,H]=PH-HP.  If V(x)=V(-x) then [P,H]=0 since

 

.

Since P and H commute, a basis of common eigenfunctions can be found.  The eigenstates of H are not degenerate.  They are found from matching the solutions for E<V(x) to the solutions for E>V(x) at the turning points.  Each successive match requires an additional half oscillation and therefore a different k(x) (or E).  Since the eigenvalues of H are not degenerate a unique common eigenbasis exists, all eigenfunctions of H are eigenfunctions of P.

(c) The ground state is the state with the smallest number of oscillations which satisfies the matching condition at the turning points.  We need only ½ oscillation, the state has even parity.  The state with the next lowest energy must have an additional ½ oscillation to satisfy the boundary conditions, it has odd parity.  (Proceed in the same fashion up the energy ladder.)