PHY456H1F: Quantum Mechanics II. Lecture 12 (Taught by Mr. Federico Duque Gomez). WKB Method Originally appeared at: http://sites.google.com/site/peeterjoot/math2011/qmTwoL12.pdf Peeter Joot —
[email protected] Oct 19, 2011
qmTwoL12.tex
Contents 1
Disclaimer.
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2
WKB (Wentzel-Kramers-Brillouin) Method.
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3
Examples
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1. Disclaimer. Peeter’s lecture notes from class. May not be entirely coherent. 2. WKB (Wentzel-Kramers-Brillouin) Method. This is covered in §24 in the text [1]. Also §8 of [2]. ¨ We start with the 1D time independent Schrodinger equation h¯ 2 d2 U − + V ( x )U ( x ) = EU ( x ) 2m dx2
(1)
which we can write as d2 U 2m + 2 ( E − V ( x ))U ( x ) = 0 dx2 h¯ Consider a finite well potential as in figure (1) With 2m( E − V ) , h¯ 2m(V − E) κ= , h¯ k=
(2)
E>V
(3)
V > E,
(4)
we have for a bound state within the well U ∝ e±ikx and for that state outside the well
1
(5)
Figure 1: Finite well potential
U ∝ e±κx
(6)
In general we can hope for something similar. Let’s look for that something, but allow the constants k and κ to be functions of position 2m( E − V ( x )) , h¯ 2m(V ( x ) − E) κ2 (x) = , h¯ k2 ( x ) =
E>V
(7)
V > E.
(8)
¨ In terms of k Schrodinger’s equation is just d2 U ( x ) + k2 ( x )U ( x ) = 0. dx2
(9)
U ( x ) = Aeiφ(x) ,
(10)
φ( x ) = φR ( x ) + iφ I ( x ).
(11)
We use the trial solution
allowing φ( x ) to be complex
We need second derivatives
(eiφ )00 = (iφ0 eiφ )0 = (iφ0 )2 eiφ + iφ00 eiφ , ¨ and plug back into our Schrodinger equation to obtain
− (φ0 ( x ))2 + iφ00 ( x ) + k2 ( x ) = 0.
(12)
For the first round of approximation we assume φ00 ( x ) ≈ 0, 2
(13)
and obtain
or
(φ0 ( x ))2 = k2 ( x ),
(14)
φ 0 ( x ) = ± k ( x ).
(15)
A second round of approximation we use 15 and obtain φ00 ( x ) = ±k0 ( x )
(16)
− (φ0 ( x ))2 ± ik0 ( x ) + k2 ( x ) = 0,
(17)
Plugging back into 12 we have
or φ0 ( x ) = ±
q
±ik0 ( x ) + k2 ( x ) s k0 ( x ) = ±k( x ) 1 ± i 2 . k (x)
(18)
If k0 is small compared to k2 k0 ( x ) 1, k2 ( x )
(19)
we have φ0 ( x ) = ±k ( x ) ± i
k0 ( x ) 2k ( x )
Integrating
φ( x ) = ±
=±
Z Z
dxk( x ) ± i
Z
dx
k0 ( x ) + const 2k ( x )
1 dxk( x ) ± i ln k ( x ) + const 2
Going back to our wavefunction, if E > V ( x ) we have U ( x ) ∼ Aeiφ(x) Z 1 = exp i ± dxk( x ) ± i ln k( x ) + const 2 Z 1 ∼ exp i ± dxk( x ) ± i ln k( x ) 2
= e ±i
R
dxk ( x ) ∓ 12 ln k ( x )
e
3
(20)
or U (x) ∝ p
1 k( x)
e ±i
R
dxk( x )
(21)
R Question: the ± on the real exponential got absorbed here, but would not U ( x ) ∝ p FIXME: ± i dxk ( x ) also be a solution? If so, why is that one excluded? k( x )e Similarly for the E < V ( x ) case we can find
U (x) ∝ p
1 κ (x)
e ±i
R
dxκ ( x )
.
(22)
Validity 1. V(x) changes very slowly =⇒ k0 ( x ) small, and k ( x ) =
p
2m( E − V ( x ))/¯h.
2. E very far away from the potential |( E − V ( x ))/V ( x )| 1. 3. Examples
Figure 2: Example of a general potential
Figure 3: Turning points where WKB won’t work WKB won’t work at the turning points in this figure since our main assumption was that 0 k (x) (23) k2 ( x ) 1 4
Figure 4: Diagram for patching method discussion so we get into trouble where k ( x ) ∼ 0. There are some methods for dealing with this. Our text as well as Griffiths give some examples, but they require Bessel functions and more complex mathematics. The idea is that one finds the WKB solution in the regions of validity, and then looks for a polynomial solution in the patching region where we are closer to the turning point, probably requiring lookup of various special functions. This power series method is also outlined in [3], where solutions to connect the regions are expressed in terms of Airy functions. References [1] BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009. 2 [2] D.J. Griffiths. Introduction to quantum mechanics, volume 1. Pearson Prentice Hall, 2005. 2 [3] Wikipedia. Wkb approximation — wikipedia, the free encyclopedia, 2011. [Online; accessed 19-October-2011]. Available from: http://en.wikipedia.org/w/index.php?title= WKB_approximation&oldid=453833635. 3
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