Techniques to find approximate solutions
to the Schrodinger equation
1. The perturbation theory
2. The variational principle
3. The WKB approximation
The WKB approximationWentzel- Kramers - Brillouin
Hendrik Kramers
Dutch1894- 1952
Leon Brillouin
French1889- 1969
GregorWentzel
German1898- 1978
The WKB approximation
The WKB approximation is based on the idea thatfor any given potential, the particle can be locally
seen as a free particle with a sinusoidal wave function,but whose wavelength varies very slowly in space.
ikxx Ae
The WKB approximation
V(x)
E
Classical region (E>V)
Locally constantor varying very slowly
In respect to wavelength
The WKB approximationClassical region
solution i xx A x e
2
2
2" '
pA x A x
2 'A x x cste
real part
imaginary part
The WKB approximationClassical region
solution i xx A x e
( )
CA x
p x
p xd
dx
2"
'A x
A xassumption
2
2
p
and
The WKB approximationClassical region
solution i xx A x e
( )
( )
ip x dxC
x ep x
where
2p x m E V x
22
( )C
xp x
Incidentally
Quiz 17a
In the WKB approximation,what can we say about the solution
for the wave function ?
A. The amplitude and the wavelength are fixed
B. The amplitude is fixed but the wavelength varies
C. The wavelength varies but the amplitude is fixed
D. Both the wavelength and the amplitude vary
E. There are multiple wavelengths for a given position
The WKB approximationClassical region
solution i xx A x e
0
( ') '
( )
xip x dxC
x ep x
Phase is a function of x
The WKB approximationClassical region
/if xx e
Another way to write the solution:
where f(x) is a complex function
Develop the function as power of f x
The WKB approximationClassical region
1( ') 'x p x dx
1( ') '
b
a
b a p x dx
When the phase is known at specific points:
Gives informationon the allowed
energies
i xx A x e
Quiz 17b
A. For any type of potential and any energy value
B. Only when
C. Only when
D. Only when the potential exhibits 1 turning point
E. Only when the potential exhibits 2 turning points
In which situation can we apply the formula
? 1( ') '
b
a
b a p x dx
E V x E V x
Example
Infinite Square well
sinx A x x
2p mE
The WKB approximationClassical region
0 0 a n
0
1' '
a
a p x dx n
The WKB approximationat turning points
V(x)
E
Classical region (E>V)
1x 2x
2
1
1( )
2
x
x
p x dx n
Connection formula (eq 8.51)