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Transcript of WKB Report
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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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Quantum Mechanics II
(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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Cody Arceneaux
Quantum Mechanics II
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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Cody Arceneaux
Quantum Mechanics II
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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Cody Arceneaux
Quantum Mechanics II
()
(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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Quantum Mechanics II
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.
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