the wkb approximation

Introduction:

• Generally, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients.

• In Quantum Mechanics it is used to obtain approximate solutions to the time-independent equation in one dimension.

Applications in Quantum Mechanics

In quantum mechanics it is useful in 1. Calculating bound state energies (Whenever

the particle cannot move to infinity)2. Transmission probability through potential

barriers.These are given in next slides.

Main idea:• If potential is constant and energy of the particle is , then the particle

wave function has the form where (+) sign indicates : particle travelling to right (-) sign indicates : particle travelling to left

• General solution : Linear superposition of the two.

• The wave function is oscillatory, with fixed wavelength, .

• The amplitude (A) is fixed.

• If V (x) is not constant, but varies slow in comparison with the wavelength λ in a way that it is essentially constant over many λ, then the wave function is practically sinusoidal, but wavelength and amplitude slowly change with x. This is the inspiration behind WKB approximation. In effect, it identifies two different levels of x-dependence :- rapid oscillations, modulated by gradual variation in amplitude and wavelength.

• If E<V and V is constant, then wave function is where

If is not constant, but varies slowly in comparison with , the solution remain practically exponential, except that and are now slowly-varying function of .

Failure of this idea There is one place where this whole program is bound to fail, and that is in the immediate vicinity of a classical turning point, where . For here goes to infinity and can hardly be said to vary “slowly” in comparison. A proper handling of the turning points is the most difficult aspect of the WKB approximation, though the final results are simple to state and easy to implement. The diagram showing turning points is given in next slide.

The WKB approximation

V(x)

E

Turning points

The Classical Region• Let's now solve the Schrödinger equation using

WKB approximation can be rewritten in the following way: ; is the classical formula for the momentum

of a particle with total energy and potential energy . Let’s assume , so that is real. This is the classical region , as classically the particle is confined to this range of The classical and non-classical region is shown in the diagram on the next slide.


V(x)

E

Classical region (E>V)

Non-classical region (E<V)

Fig: Classically, the particle is confined to the region where

The function

In general, is some complex function; we can express it in terms of its amplitude , and its phase, – both of which are real :

Solving the Schrödinger equation

Using prime to denote the derivative with respect to we find:

and

Putting all these into (From Schrödinger equation ) , we get

Solving for real and imaginary parts we get,

The above equation cannot be solved in general, so we use WKB approximation: we assume amplitude A varies slowly, so that the A’’ term is negligible. We assume that << . Therefore, we drop that part and we get

And from second equation, we get

Where C is real constant.

Thus from the previous slides from the equations and making ‘C’ a complex constant, we get

And the general solution can be written as

where and are constants.

Alternate approachIn this approach, the wave function is expanded in powers of . Let, the wave function be:

Using this in: , we get --------(1)

Expanding S(x) in powers of :

; (neglecting higher powers of ).

where

For the above equation to be valid, the coefficient of each power of must vanish separately,

and, ; using the value of

or,

where A is a normalization

constant.

For the Schrodinger equation is ,

Proceeding in a similar fashion , the solution can be obtained as

; where B is a normalization constant.

Validity of WKB solutionThe zeroeth order WKB solution is:

; considering positive part only.

But we are interested in solving the following eqn.

Hence,

i.e. k(x) should not vary so rapidly This is the Validity condition for WKB

approximation.


V(x)

E

Classical region (E>V)



Turning points


E V x ( )

( )

i p x dx

WKBCx ep x

E V x

Excluding the turning points:

1 ( )

( )

p x dx

WKBCx ep x

Patching region


V(x)

E

Classical region (E>V)Non-classical region (E<V)

( )

( )

i p x dxCx ep x

1 ( )

( )

p x dxDx ep x

Connection Formulae𝑘2(𝑥)

𝑥

WKB soln not valid

a

Trigonometric WKB soln

Exponential WKB soln Turning point

Barrier to the right of turning point

Barrier to the right of turning point

And,

Barrier to the left of turning point

WKB soln not valid

𝑘2(𝑥)

𝑥

Trigonometric WKB soln

Exponential WKB soln

b

Turning point

Barrier to the left of turning point

And,

WKB Examples

Example 1

Potential Square well with a Bumpy Surface

Potential Square well with a Bumpy Surface

Suppose we have an infinite square well with a bumpy bottom as shown in figure: and

𝑽 (𝒙 )

𝒙𝟎

Inside the well, , we have

or, must go to zero at and . So, putting the values we get respectively,

and ,this quantization condition determines the allowed energies.

𝜓 (𝑥)≅ 1√𝑝(𝑥 )

[𝐶1𝑒𝑖ℏ0

𝑥

𝑝 (𝑥 )𝑑𝑥+𝐶2𝑒

−𝑖ℏ

0

𝑥

𝑝 (𝑥 )𝑑𝑥 ]

Special Case:

If the well has a flat bottom i.e. (), then and from quantization equation, we get

Solving these, we get value of :

which is the formula for the discrete energy

levels of the infinite square well.

Example 2

Tunneling

In the non-classical region (), ; is complexLet us consider the following example : problem of scattering from a rectangular barrier also called tunneling.

To the left of the barrier (),

where is the incident amplitude and is the reflected amplitude, and .To the right of the barrier ,

where is the transmitted amplitude.In the tunneling region () WKB approximation gives,

The transmission probability is

where T is transmission probability.

Example 3

Eigen value equation for Bound State

Eigen value equation for Bound StateHere, a and b are the classical turning points.

= ;

Using connection formula at

Now using connection formula we get that goes to exponentially increasing solution in region III which is not a condition for bound state. Hence, the wave function to be well behaved

This is the Eigen value equation for bound state using WKB approximation.

Now, using this equation the energy Eigen values for Linear Harmonic Oscillator (LHO) can be calculated as shown in next slide :

LHO energy Eigen values

For LHO potential is

where, is the classical amplitude or turning points.

Using this value of in Eigen value equation for bound state,

On solving the above equation, or, ;

- which are the energy Eigen values for a Linear Harmonic Oscillator.

the wkb approximation

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