WKB Report

7/28/2019 WKB Report

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Cody Arceneaux

Quantum Mechanics II

The Wentzel-Kramers-Brillouin (WKB) approximation is a method of finding

approximate solutions of linear, second order differential equations. It was originally developed

by Liouville and Green, and is sometimes referred to as the Liouville-Green (LG) approximation.

Also, because of the work by Jeffereys before Wentzel, Kramers, and Brillouin, it is sometimes

referred to as the WKBJ or the JWKB approximation.

1

In quantum mechanics, the WKB approximation is used to find a semi-classical solution

to the Schrodinger equation. The time-dependent version of the Schrodinger equation is

).4 (1)This is one of the more general problems that would need to be solved. A more simplified

explanation can be used to get the idea before moving on to a more generalized case.

The explanation of the WKB approximation in Griffiths (2005)2

and Shankar (1994)3

begins with a particle in one-dimension of energy E in a constant potential V. The Schrodinger

equation is used in the form of

* ( ) + . (2)For simplicity, letp(x) = . Then, use as a wavefunction

(3)

and insert this into Eq. 2 to get

. (4)(x) needs to be expanded into a power series for

(x) = 0 + 1+ 2 + . (5)For the WKB approximation, terms of order2or higher will be discarded. The expansion is

placed into the above equation, and (after taking derivatives and performing some algebra) it

ends up as 0= p(x). Taking the integral of this and placing it back into wavefunction resultsin


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(6)which is the WKB approximation for the one-dimensional time-independent Schrodinger

equation.

2,3

Sakurai (1994)4, though, goes through a derivation that is time-dependent and draws

heavily on Hamiltonian dynamics. In this derivation, let

(7)where | |2. Putting this through the time-dependent Schrodinger equation resultsin

[() || ] (8)

For the WKB approximation, Eq. 8 and has all terms of order or higher removed because it isso small. The real part of Eq. 8 is then

||

, (9)and this equivalent to the Hamilton-Jacobi theorem with S(x,t) is Hamiltons principle function.

This can be easily recognized from almost any book on classical mechanics, such as Fetter and

Walecka (2005, pp. 185-186, 189)5. With a constant Hamiltonian, S(x,t) can be separated as

(10)where W(x) is Hamiltons characteristic function. The solution of the Hamilton-Jacobi equation

is

= . (11)Looking at the imaginary part of Eq. 8 and using the fact that for a stationary state

it isfound that

() . (12)


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This means that is a constant. Taking that fact and rearranging terms leads to

. (13)

with C a constant. Combing all of this together results in

(14)which is a time-dependent WKB approximation. This will only work in the case of

| | | |(15)

Eq. 15 used with the de Broglie wavelength leads to

| | (16)which shows that the WKB approximation only works for small wavelengths.

This explanation works for the classically allowed region where E>V, but it does not

work in the classically (but not quantum) forbidden region where V>E. This can be found easily

enough by replacingE-V(x) in Eq. 14 and ends up being

(17)Neither the V>Ecase or theE>Vcase are able to explain the turning points whenE=V.

4

Figure 1 illustrates where the turning points are.


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Fig. 1: V, E, and the turning points x1 and x2

Brack and Bhaduri (2003)6

specifically point out that for Figure 1, the approximated

wavefunction in the central region is

(18)

in the region on the right is

(19)

and in the region on the left is

(20)To solve this, the solutions need to be combined at the turning points, and a patch

wavefunction will be used at the turning points. This patching wavefunction ends up being an

Airy function.6

The integration constants for the two WKB approximation wavefunctions need

to be matched to the Airy function. When this is done, Eq. 19 (without the time dependence)goes to

(21)and Eq. 20 (without the time dependence) goes to


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()

(22)

Looking at Eq. 21 and Eq. 22, it is apparent that they differ by an integer multiple of. It can bediscerned that ()

4,6 (23)

which is often used to approximate the quantized energy of various systems.

The WKB approximation has a more general use to solve non-quantum problems. If the

highest order derivative of a differential equation is multiplied by a small number (in thesolution of the Schrodinger equation, this is ), following steps similar to the derivationspresented here will find the WKB approximation of that problem. In that sense, it is a useful tool

across numerous fields.7


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References

1Spigler, Renato and Vianello, Marco (1998). "A Survey on the LiouvilleGreen (WKB)

approximation for linear difference equations of the second order". In Elaydi, Saber;

Gyri, I.; and Ladas, G. E. Advances in difference equations: proceedings of the SecondInternational Conference on Difference Equations : Veszprm, Hungary, August 711,

1995. CRC Press.

2Griffiths, David J. (2005). Introduction to Quantum Mechanics, Second Edition. Upper Saddle

River: Pearson Education, Inc.

3Shankar, R. (1994). Principles of Quantum Mechanics, Second Edition. New York: Plenum

Press.

4Sakurai, J.J. and Tuan, San Fu, ed. (1994). Modern Quantum Mechanics, Revised Edition.Reading: Addison-Wesley Publishing Company, Inc.

5Fetter, Alexander L. and Walecka, John Dirk. (2003). Theoretical Mechanics of Particles and

Continua. Mineola: Dover Publications, Inc.

6Brack, Matthias and Bhaduri, Rajat K. (2003). Semiclassical Physics. Boulder: Westview

Press.

7Bender, Carl M. and Orszag, Steven A. (1999). Advanced Mathematical Methods for Scientists

and Engineers. New York: Springer-Verlag New York, Inc.
http://books.google.com/?id=a36iXw5_VzcC&pg=PA567&dq=Liouville-Green+WKBJ+WKB+LGhttp://books.google.com/?id=a36iXw5_VzcC&pg=PA567&dq=Liouville-Green+WKBJ+WKB+LGhttp://books.google.com/?id=a36iXw5_VzcC&pg=PA567&dq=Liouville-Green+WKBJ+WKB+LGhttp://books.google.com/?id=a36iXw5_VzcC&pg=PA567&dq=Liouville-Green+WKBJ+WKB+LGhttp://books.google.com/?id=a36iXw5_VzcC&pg=PA567&dq=Liouville-Green+WKBJ+WKB+LGhttp://books.google.com/?id=a36iXw5_VzcC&pg=PA567&dq=Liouville-Green+WKBJ+WKB+LG

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