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 free particle

ikxx Ae

Infinite space

sinx A kx

2mEk

2p mE

Finite box

Flat potential

ikxx Ae 2m E V

k

xx Ae

Scattering state E V

Bound state E V 2m V E

E

V

Varying potentialThe WKB approximation

V(x)

E

Classical region (E>V)

Turning points

The WKB approximation

V(x)

E

Classical region (E>V)

Locally constantor varying very slowly

In respect to wavelength

The WKB approximationClassical region

2 2

22

dV x E

m dx

2 2

2 2

d p

dx

2p x m E V x with

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

2p m E V x

The WKB approximationClassical region

0 0 a n

0

1' '

a

a p x dx n 0V

The WKB approximationat turning points

V(x)

E

Classical region (E>V)

Turning points

E V x 0p x

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)

The WKB approximationat turning points

2

1

1( )

2

x

x

p x dx n

Pb 8.7

Harmonic Oscillator

Pb 8.14

Hydrogenatom

2 21

2V x m x

2 2

20

1 ( 1)

4 2eff

e l lV r

r m r