the wkb approximation
TRANSCRIPT
![Page 1: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/1.jpg)
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.
![Page 2: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/2.jpg)
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.
![Page 3: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/3.jpg)
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.
![Page 4: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/4.jpg)
• 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.
![Page 5: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/5.jpg)
• 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 .
![Page 6: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/6.jpg)
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.
![Page 7: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/7.jpg)
The WKB approximation
V(x)
E
Turning points
![Page 8: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/8.jpg)
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.
![Page 9: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/9.jpg)
The WKB approximation
V(x)
E
Classical region (E>V)
Non-classical region (E<V)
Fig: Classically, the particle is confined to the region where
![Page 10: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/10.jpg)
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 :
![Page 11: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/11.jpg)
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
![Page 12: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/12.jpg)
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
![Page 13: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/13.jpg)
And from second equation, we get
Where C is real constant.
![Page 14: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/14.jpg)
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.
![Page 15: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/15.jpg)
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 :
![Page 16: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/16.jpg)
; (neglecting higher powers of ).
where
For the above equation to be valid, the coefficient of each power of must vanish separately,
![Page 17: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/17.jpg)
and, ; using the value of
or,
where A is a normalization
constant.
![Page 18: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/18.jpg)
For the Schrodinger equation is ,
Proceeding in a similar fashion , the solution can be obtained as
; where B is a normalization constant.
![Page 19: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/19.jpg)
Validity of WKB solutionThe zeroeth order WKB solution is:
; considering positive part only.
![Page 20: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/20.jpg)
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.
![Page 21: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/21.jpg)
The WKB approximation
V(x)
E
Classical region (E>V)
Non-classical region (E<V)
Non-classical region (E<V)
Turning points
![Page 22: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/22.jpg)
The WKB approximation
E V x ( )
( )
i p x dx
WKBCx ep x
E V x
Excluding the turning points:
1 ( )
( )
p x dx
WKBCx ep x
![Page 23: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/23.jpg)
Patching region
The WKB approximation
V(x)
E
Classical region (E>V)Non-classical region (E<V)
( )
( )
i p x dxCx ep x
1 ( )
( )
p x dxDx ep x
![Page 24: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/24.jpg)
Connection Formulae𝑘2(𝑥)
𝑥
WKB soln not valid
a
Trigonometric WKB soln
Exponential WKB soln Turning point
Barrier to the right of turning point
![Page 25: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/25.jpg)
Barrier to the right of turning point
And,
![Page 26: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/26.jpg)
Barrier to the left of turning point
WKB soln not valid
𝑘2(𝑥)
𝑥
Trigonometric WKB soln
Exponential WKB soln
b
Turning point
![Page 27: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/27.jpg)
Barrier to the left of turning point
And,
![Page 28: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/28.jpg)
WKB Examples
![Page 29: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/29.jpg)
Example 1
Potential Square well with a Bumpy Surface
![Page 30: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/30.jpg)
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
![Page 31: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/31.jpg)
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 ]
![Page 32: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/32.jpg)
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.
![Page 33: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/33.jpg)
Example 2
Tunneling
![Page 34: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/34.jpg)
In the non-classical region (), ; is complexLet us consider the following example : problem of scattering from a rectangular barrier also called tunneling.
![Page 35: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/35.jpg)
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,
![Page 36: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/36.jpg)
The transmission probability is
where T is transmission probability.
![Page 37: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/37.jpg)
Example 3
Eigen value equation for Bound State
![Page 38: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/38.jpg)
Eigen value equation for Bound StateHere, a and b are the classical turning points.
= ;
Using connection formula at
![Page 39: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/39.jpg)
For :
![Page 40: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/40.jpg)
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
![Page 41: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/41.jpg)
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 :
![Page 42: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/42.jpg)
LHO energy Eigen values
For LHO potential is
where, is the classical amplitude or turning points.
![Page 43: The wkb approximation](https://reader035.vdocuments.us/reader035/viewer/2022081507/587d3a781a28ab2a448b6d47/html5/thumbnails/43.jpg)
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.