PHY456H1F: Quantum Mechanics II. Lecture L23 (Taught by Prof J.E. Sipe). 3D Scattering. Originally appeared at: http://sites.google.com/site/peeterjoot2/math2011/qmTwoL23.pdf Peeter Joot — [email protected] Nov 30, 2011

qmTwoL23.tex

Contents 1

Disclaimer.

1

2

3D Scattering. 2.1 Seeking a post scattering solution away from the potential 2.2 The radial equation and its solution. . . . . . . . . . . . . . 2.3 Limits of spherical Bessel and Neumann functions . . . . . 2.4 Back to our problem. . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

3

Scattering geometry and nomenclature.

4

Appendix 4.1 Q: Are Bessel and Neumann functions orthogonal? . . . . . . . . . . . . 4.2 Deriving the large limit Bessel and Neumann function approximations. . 4.3 Deriving the small limit Bessel and Neumann function approximations. 4.4 Verifying the solution to the spherical Bessel equation. . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

1 2 3 5 5 6

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

7 . 7 . 8 . 8 . 10

1. Disclaimer. Peeter’s lecture notes from class. May not be entirely coherent. 2. 3D Scattering. READING: §20, and §4.8 of our text [1]. We continue to consider scattering off of a positive potential as depicted in figure (1) Here we have V (r ) = 0 for r > r0 . The wave function eiknˆ ·r

(1)

¨ is found to be a solution of the free particle Schrodinger equation.

−

h¯ 2 2 iknˆ ·r h¯ 2 k2 iknˆ ·r ∇ e = e 2µ 2µ

1

(2)

Figure 1: Radially bounded potential. 2.1. Seeking a post scattering solution away from the potential What other solutions can be found for r > r0 , where our potential V (r ) = 0? We are looking for Φ(r) such that h¯ 2 2 h¯ 2 k2 ∇ Φ (r ) = Φ (r ) 2µ 2µ

−

(3)

What can we find? We split our Laplacian into radial and angular components as we did for the hydrogen atom

−

h¯ 2 ∂2 L2 ( rΦ ( r )) + Φ(r) = EΦ(r), 2µ ∂r2 2µr2

(4)

where 2

L = −h¯

2



∂2 1 ∂ + 1 ∂2 + ∂θ 2 tan θ ∂θ sin2 θ ∂φ2

 (5)

Assuming a solution of Φ(r) = R(r )Ylm (θ, φ),

(6)

L2 Ylm (θ, φ) = h¯ 2 l (l + 1)Ylm (θ, φ),

(7)

and noting that

we find that our radial equation becomes 2

−

h¯ 2 ∂2 h¯ 2 l (l + 1) h¯ 2 k2 ( rR ( r )) + R ( r ) = ER ( r ) = R (r ). 2µr ∂r2 2µr2 2µ

(8)

Writing R (r ) =

u (r ) , r

(9)

we have

−

h¯ 2 k2 u(r ) h¯ 2 ∂2 u(r ) h¯ 2 l (l + 1) + u ( r ) = , 2µr ∂r2 2µr 2µ r

(10)

or 

d2 l ( l + 1) + k2 − dr2 r2

 u (r ) = 0

(11)

u (r ) = 0

(12)

Writing ρ = kr, we have 

l ( l + 1) d2 +1− 2 dρ ρ2



2.2. The radial equation and its solution. With a last substitution of u(r ) = U (kr ) = U (ρ), and introducing an explicit l suffix on our eigenfunction U (ρ) we have   d2 l ( l + 1) − 2+ Ul (ρ) = Ul (ρ). (13) dρ ρ2 We’d not have done this before with the hydrogen atom since we had only finite E = h¯ 2 k2 /2µ. Now this can be anything. Making one final substitution, Ul (ρ) = ρ f l (ρ) we can rewrite 13 as   2 d 2 d 2 ρ + 2ρ + (ρ − l (l + 1)) f l = 0. (14) dρ2 dρ This is the spherical Bessel equation of order l and has solutions called the Bessel and Neumann functions of order l, which are    1 d l sin ρ jl (ρ) = (−ρ) ρ dρ ρ  l   1 d cos ρ l nl (ρ) = (−ρ) − . ρ dρ ρ l



(15a) (15b)

We can easily calculate U0 (ρ) = ρj0 (ρ) = sin ρ

(16)

sin ρ U1 (ρ) = ρj1 (ρ) = − cos ρ + ρ

(17)

3

and can plug these into 13 to verify that they are a solution. A more general proof looks a bit trickier. Observe that the Neumann functions are less well behaved at the origin. To calculate the first few Bessel and Neumann functions we first compute   1 d sin ρ 1 cos ρ sin ρ = − 2 ρ dρ ρ ρ ρ ρ cos ρ sin ρ = 2 − 3 ρ ρ 

1 d ρ dρ

2

sin ρ 1 = ρ ρ



cos ρ cos ρ sin ρ sin ρ − 2 −2 3 − 3 +3 4 ρ ρ ρ ρ   3 1 cos ρ = sin ρ − 3 + 5 − 3 4 ρ ρ ρ



and   cos ρ 1 sin ρ cos ρ 1 d − = + 2 ρ dρ ρ ρ ρ ρ sin ρ cos ρ = 2 + 3 ρ ρ 

1 d ρ dρ

2



sin ρ sin ρ cos ρ cos ρ −2 3 − 3 −3 4 2 ρ ρ ρ ρ   1 3 sin ρ = cos ρ − 5 −3 4 3 ρ ρ ρ

cos ρ 1 − = ρ ρ



so we find j0 (ρ) =

sin ρ ρ sin ρ ρ2

− cosρ ρ  j2 (ρ) = sin ρ − ρ1 + j1 (ρ) =

cos ρ ρ ρ − cos − sinρ ρ ρ2   cos ρ 1ρ − ρ33

n0 ( ρ ) = − n1 ( ρ ) = 3 ρ3



+ cos ρ



− ρ32



n2 ( ρ ) =

(18)

+ sin ρ



− ρ32



Observe that our radial functions R(r ) are proportional to these Bessel and Neumann functions u (r ) r U (kr ) = (r

R (r ) =

= (

=

4

jl (ρ)ρ r nl (ρ)ρ r jl (ρ)kr r nl (ρ)kr r

Or R(r ) ∼ jl (ρ), nl (ρ).

(19)

2.3. Limits of spherical Bessel and Neumann functions With n!! denoting the double factorial, like factorial but skipping every other term n!! = n(n − 2)(n − 4) · · · ,

(20)

we can show that in the limit as ρ → 0 we have jl (ρ) →

ρl (2l + 1)!!

nl (ρ) → −

(2l − 1)!! , ρ ( l +1)

(21a) (21b)

(for the l = 0 case, note that (−1)!! = 1 by definition). Comparing this to our explicit expansion for j1 (ρ) in 18 where we appear to have a 1/ρ dependence for small ρ it is not obvious that this would be the case. To compute this we need to start with a power series expansion for sin ρ/ρ, which is well behaved at ρ = 0 and then the result follows (done later). It is apparently also possible to show that as ρ → ∞ we have   1 lπ jl (ρ) → sin ρ − (22a) ρ 2   1 lπ nl (ρ) → − cos ρ − . (22b) ρ 2 2.4. Back to our problem. For r > r0 we can construct (for fixed k) a superposition of the spherical functions

∑ ∑ ( Al jl (kr) + Bl nl (kr)) Ylm (θ, φ) l

(23)

m

we want outgoing waves, and as r → ∞, we have   sin kr − lπ 2 jl (kr ) → kr   cos kr − lπ 2 nl (kr ) → − kr Put Al /Bl = −i for a given l we have      lπ lπ sin kr − cos kr − 2 2 1   ∼ 1 ei(kr−πl/2) −i − kr kr kr kr

5

(24a)

(24b)

(25)

For 1

∑ ∑ Bl kr ei(kr−πl/2) Ylm (θ, φ). l

(26)

m

Making this choice to achieve outgoing waves (and factoring a (−i )l out of Bl for some reason, we have another wave function that satisfies our Hamiltonian equation eikr kr

∑ ∑(−1)l Bl Ylm (θ, φ). l

(27)

m

The Bl coefficients will depend on V (r ) for the incident wave eik·r . Suppose we encapsulate that dependence in a helper function f k (θ, φ) and write eikr f k (θ, φ) r

(28)

We seek a solution ψk (r) h¯ 2 − ∇2 + V ( r ) 2µ

! ψk (r) =

h¯ 2 k2 ψk (r), 2µ

(29)

where as r → ∞ eikr f k (θ, φ). (30) r Note that for r < r0 in general for finite r, ψk (r), is much more complicated. This is the analogue of the plane wave result ψk (r) → eik·r +

ψ( x ) = eikx + β k e−ikx

(31)

3. Scattering geometry and nomenclature. We can think classically first, and imagine a scattering of a stream of particles barraging a target as in figure (2) Here we assume that dΩ is far enough away that it includes no non-scattering particles. Write P for the number density P=

number of particles , unit volume

(32)

and Number of particles flowing through (33) a unit area in unit time We want to count the rate of particles per unit time dN through this solid angle dΩ and write   dσ(Ω) dN = J dΩ. (34) dΩ J = Pv0 =

The factor 6

Figure 2: Scattering cross section.

dσ(Ω) , dΩ is called the differential cross section, and has “units” of area steradians (recalling that steradians are radian like measures of solid angle [2]). The total number of particles through the volume per unit time is then

(35)

(36)

dσ(Ω) dσ(Ω) dΩ = J dΩ = Jσ (37) dΩ dΩ where σ is the total cross section and has units of area. The cross section σ his the effective size of the area required to collect all particles, and characterizes the scattering, but isn’t necessarily entirely geometrical. For example, in photon scattering we may have frequency matching with atomic resonance, finding σ ∼ λ2 , something that can be much bigger than the actual total area involved. Z

J

Z

4. Appendix 4.1. Q: Are Bessel and Neumann functions orthogonal? Answer: There is an orthogonality relation, but it is not one of plain old multiplication. Curious about this, I find an orthogonality condition in [3]
Z ∞ dz 2 sin π2 (α − β) Jα (z) Jβ (z) = , z π α2 − β2 0 from which we find for the spherical Bessel functions

7

(38)

Z ∞ 0


sin π2 (l − m) . jl (ρ) jm (ρ)dρ = (l + 1/2)2 − (m + 1/2)2

(39)

Is this a satisfactory orthogonality integral? At a glance it doesn’t appear to be well behaved for l = m, but perhaps the limit can be taken? 4.2. Deriving the large limit Bessel and Neumann function approximations. For 22 we are referred to any “good book on electromagnetism” for details. I thought that perhaps the weighty [4] would be to be such a book, but it also leaves out the details. In §16.1 the spherical Bessel and Neumann functions are related to the plain old Bessel functions with r π jl ( x ) = J (x) (40a) 2x l +1/2 r π N (x) (40b) nl ( x ) = 2x l +1/2 Referring back to §3.7 of that text where the limiting forms of the Bessel functions are given r  2 νπ π  Jν ( x ) → (41a) cos x − − πx 2 4 r  2 νπ π  Nν ( x ) → sin x − − (41b) πx 2 4 This does give us our desired identities, but there’s no hint in the text how one would derive 41 from the power series that was computed by solving the Bessel equation. 4.3. Deriving the small limit Bessel and Neumann function approximations. Writing the sinc function in series form ∞ sin x x2k = ∑ (−1)k , x (2k + 1)! k =0

(42)

we can differentiate easily ∞ 1 d sin x x2k−2 = ∑ (−1)k (2k) x dx x (2k + 1)! k =1 ∞

= (−1) ∑ (−1)k (2k + 2) k =0 ∞

= (−1) ∑ (−1)k k =0

x2k (2k + 3)!

1 x2k 2k + 3 (2k + 1)!

Performing the derivative operation a second time we find

8

(43)



1 d x dx

2

∞ sin x 1 x2k−2 = (−1) ∑ (−1)k (2k) x 2k + 3 (2k + 1)! k =1 ∞

1 x2k 1 = ∑ (−1) 2k + 5 2k + 3 (2k + 1)! k =0

(44)

k

It appears reasonable to form the inductive hypotheses 

1 d x dx

l

∞ sin x (2k + 1)!! x2k = (−1)l ∑ (−1)k , x (2(k + l ) + 1)!! (2k + 1)! k =0

(45)

and this proves to be correct. We find then that the spherical Bessel function has the power series expansion of ∞

jl ( x ) =

∑ (−1)k

k =0

x2k+l (2k + 1)!! (2(k + l ) + 1)!! (2k + 1)!

(46)

and from this the Bessel function limit of 21a follows immediately. Finding the matching induction series for the Neumann functions is a bit harder. It’s not really any more difficult to write it, but it is harder to put it in a tidy form that is. We find

−

∞ cos x x2k−1 = − ∑ (−1)k x (2k)! k =0

(47)

∞ 1 d cos x 2k − 1 x2k−3 − = − ∑ (−1)k x dx x 2k (2k − 2)! k =0  2 ∞ 1 d cos x (2k − 1)(2k − 3) x2k−3 − = − ∑ (−1)k x dx x 2k (2k − 2) (2k − 4)! k =0

(48) (49)

The general expression, after a bit of messing around (and I got it wrong the first time), can be found to be 

1 d x dx

l

−

l −1 l −1 cos x x2(k−l )−1 = (−1)l +1 ∑ ∏ |2(k − j) − 1| x (2k)! k =0 j =0

+ (−1)

l +1

∞

∑ (−1)

+ l ) − 1)!! x2k−1 . (2k − 1)!! (2(k + l )!

k (2( k

k =0

(50)

We really only need the lowest order term (which dominates for small x) to confirm the small limit 21b of the Neumann function, and this follows immediately. For completeness, we note that the series expansion of the Neumann function is l −1 l −1

nl ( x ) = −

∑ ∏ |2( k − j ) − 1|

k =0 j =0 ∞

x2k−l −1 (2k)!

(2k + 3l − 1)!! x2k−1 − ∑ (−1)k . (2k − 1)!! (2(k + l )! k =0 9

(51)

4.4. Verifying the solution to the spherical Bessel equation. One way to verify that 15a is a solution to the Bessel equation 14 as claimed should be to substitute the series expression and verify that we get zero. Another way is to solve this equation directly. We have a regular singular point at the origin, so we look for solutions of the form ∞

∑ ak x k

(52)

d d2 + 2x + x2 − l (l + 1), 2 dx dx

(53)

f = xr

k =0

Writing our differential operator as L = x2 we get 0 = Lf ∞

∑ ak

=



 (k + r )(k + r − 1) + 2(k + r ) − l (l + 1) x k+r + ak x k+r+2

k =0

  = a0 r (r + 1) − l ( l + 1) x r   + a1 (r + 1)(r + 2) − l (l + 1) xr+1   ∞ + ∑ ak (k + r )(k + r − 1) + 2(k + r ) − l (l + 1) + ak−2 x k+r k =2

Since we require this to be zero for all x including non-zero values, we must have constraints on r. Assuming first that a0 is non-zero we must then have 0 = r (r + 1) − l ( l + 1).

(54)

One solution is obviously r = l. Assuming we have another solution r = l + k for some integer k we find that r = −l − 1 is also a solution. Restricting attention first to r = l, we must have a1 = 0 since for non-negative l we have (l + 1)(l + 2) − l (l + 1) = 2(l + 1) 6= 0. Thus for non-zero a0 we find that our function is of the form f =

∑ a2k x2k+l .

(55)

k

It doesn’t matter that we started with a0 6= 0. If we instead start with a1 6= 0 we find that we must have r = l − 1, −l − 2, so end up with exactly the same functional form as 55. It ends up slightly simpler if we start with 55 instead, since we now know that we don’t have any odd powered ak ’s to deal with. Doing so we find 0 = Lf ∞

= =

∑

  a2k (2k + l )(2k + l − 1) + 2(2k + l ) − l (l + 1) x2k+l + a2k x2k+l +2

k =0 ∞ 

∑

 a2k 2k(2(k + l ) + 1) + a2(k−1) x2k+l

k =1

10

We find a2k −1 = . a 2( k −1) 2k (2(k + l ) + 1)

(56)

Proceeding recursively, we find ∞

f = a0 (2l + 1)!!

(−1)k

∑ (2k)!!(2(k + l ) + 1)!! x2k+l .

(57)

k =0

With a0 = 1/(2l + 1)!! and the observation that

(2k + 1)!! 1 = , (2k)!! (2k + 1)!

(58)

we have f = jl ( x ) as given in 46. If we do the same for the r = −l − 1 case, we find a2k a 2( k −1)

=

−1 , 2k (2(k − l ) − 1)

(59)

and find

(−1)k a2k = . a0 (2k)!!(2(k − l ) − 1)(2(k − l ) − 3) · · · (−2l + 1)

(60)

Flipping signs around, we can rewrite this as a2k 1 = . a0 (2k)!!(2(l − k) + 1)(2(l − k) + 3) · · · (2l − 1)

(61)

For those values of l > k we can write this as

(2(l − k) − 1)!! a2k = . a0 (2k)!!(2l − 1)!!

(62)

Comparing to the small limit 21b, the k = 0 term, we find that we must have a0 = −1. (2l − 1)!!

(63)

After some play we find a2k =

  − (2(l −k)−1)!! (2k)!!

if l ≥ k



if l ≤ k

(−1)k−l +1

(2k)!!(2(k−l )−1)!!

(64)

Putting this all together we have nl ( x ) = −

∑

0≤ k ≤ l

(2(l − k) − 1)!!

x2k−l −1 (−1)k−l x2k−l −1 −∑ (2k)!! (2(k − l ) − 1)!! (2k)!! l
FIXME: check that this matches the series calculated earlier 51.

11

(65)

References [1] BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009. 2 [2] Wikipedia. Steradian — wikipedia, the free encyclopedia [online]. 2011. [Online; accessed 4-December-2011]. Available from: http://en.wikipedia.org/w/index.php?title= Steradian&oldid=462086182. 3 [3] Wikipedia. Bessel function — wikipedia, the free encyclopedia [online]. 2011. [Online; accessed 4-December-2011]. Available from: http://en.wikipedia.org/w/index.php?title= Bessel_function&oldid=461096228. 4.1 [4] JD Jackson. Classical Electrodynamics Wiley. John Wiley and Sons, 2nd edition, 1975. 4.2

12