Advanced Quantum Mechanics

Scattering Theory

Rajdeep Sensarma

sensarma@theory.tifr.res.in

Ref : Sakurai, Modern Quantum Mechanics

Taylor, Quantum Theory of Non-Relativistic Collisions

Landau and Lifshitz, Quantum Mechanics

mailto:sensarma@theory.tifr.res.in

Recap of Previous Classes

Free particle with

momentum k

and energy E

Detector

Free par

ticle wit

h

momentu

m k’

and ene

rgy E

ẑ

(✓,�)

d⌦

S Matrix and decomposition of

scattered state in free particle

basis

S↵� = h�↵| (+)� (t ! 1)i Propagator in presence of scatterer

Probability conservation ----> Unitarity of S

GR = GR0 +GR0 V G

R0 +G

R0 V G

R0 V G

R0 + . . . = G

R0 +G

R0 V G

R

=G G0

+G0VG0

+G0VG0VG0

+ ....= +

Recap of Previous ClassesFree Prop.

with G0

Free Pro

p

with G0

T captures all the

effects of V incl. multiple scatterings

TT = V [1�G0V ]�1 = V + V G0V + V G0V G0V + . . . = V + V G0T

=+

V

+

VG0VG0V

+ ....

VG0VT

+=

(+)(~r) =1

(2⇡)3/2

ei~p·~r +

eipr

rf(~p0, ~p)

�

3D Scattered State

1D Scattered StateA eikx C eikx

B e-ikx D e-ikx

Recap of Previous ClassesBorn Approximation T = V + V G0V + V G0V G0V + ....

Born. Approx/

1st Born Approx.

2nd Born Approx.

3rd Born Approx.Validity: • Weak potentials

• High Energy of Incident particles

V (r) = V0 0 < r < a

= 0 r > aA) Square Well f(✓) = 2mV0

q2

sin(qa)

q� a cos(qa)

�

|f(θ)|2 /a2qa=0.1, 0.5,1.0, 1.5

2mV0a2=1

Differential Scattering Cross Section

goes down monotonically

0 p2

p0

0.05

0.1

0.15

B) Yukawa PotentialV (r) =

V0e��r

�r

f(✓) = �2mV0�

1

q2 + �2

θ0 p

2p

0

0.5

1.

|f(θ)|2 λ2

p/λ=0.1, 0.5,1.0, 1.5

2mV0/λ2=1

As λ ---> 0, with V0/λ fixed, the Yukawa potential goes over to the Coulomb potential.

In this limit, recover the Rutherford cross-section. d�

d⌦=

(2m)2(ZZ 0e2)2

16p4 sin4(✓/2)

q = |~p0 � ~p| = 2p sin ✓2

Recap of Previous ClassesExpansion in Partial waves

f(~p, ~p0) = �4⇡2

p

X

lm

Tl(E) Yml (p̂

0)Y m⇤l (p̂) f(~p,~p0) =

1X

l=0

(2l + 1)fl(p)Pl(cos ✓) θ

Tl(p) = �ei�l(p) sin �l(p)

⇡=

1

⇡

1

cot �l(p)� iT Matrix:

=

1

(2⇡)3/2

X

l

(2l + 1)Pl(cos ✓)

2ip

✓[1 + 2ipfl(p)]

eipr

r� e

�i(pr�l⇡)

r

◆ (+)(~r)

Scattered Wavefunction:

incoming spherical waveoutgoing spherical wave

Sl(p) = 1 + 2ipfl(p)S Matrix : |Sl(p)| = 1 ) Sl(p) = e2i�l(p)

phase shift in l channel

Conservation of Angular Momentum ——> Unitarity of Sl

Recap of Previous ClassesExpansion in Partial waves

f(~p, ~p0) = �4⇡2

p

X

lm

Tl(E) Yml (p̂

0)Y m⇤l (p̂) f(~p,~p0) =

1X

l=0

(2l + 1)fl(p)Pl(cos ✓) θ

Scattering Amplitude:

f(✓) =X

l

(2l + 1)Pl(cos ✓)ei�l(p) sin �l(p)

p

d�

d⌦= |f(✓)|2 =

X

ll0

(2l + 1)(2l0 + 1)

p2Pl(cos ✓)Pl0(cos ✓) sin �l(p) sin �l0(p)e

i[�l(p)��l0 (p)]

Differential Cross Section:

interference of different l channels

=4⇡

p2

X

l

(2l + 1) sin2 �l(p)�Total Cross Section:

interference washed out

Low Energy Scattering and few Partial WavesUse the expansion of a plane wave into spherical waves in 3D

Use orthonormality of Ylm

Y ml (✓, 0) =

r2l + 1

4⇡Pl(cos ✓)�m0Use the fact that k is along z and

Use for x

Low Energy Scattering and few Partial WavesLet us use the self-consistent eqn. for T matrix

Note that we want k, k’ to be small, but q sum is unrestricted

However, we have

So, Expand the series to show that each term has the form Vkq …… Vpk’

and

Now use the fact that for elastic scattering k=k’ and Tl(E)/k ~ Tkk’(E)

For low E 0, s-wave scattering

Calculating Phase Shifts in simple potentialsThe straight forward approach (works only in few selective cases) is to solve the Schrodinger equation with the potential, and obtain the phase shift from the asymptotic form of the wave-function far from the origin by comparing it with

(+)(r) =1

(2⇡)3/2

X

l

(2l + 1)Pl(cos ✓)

2ip

✓e2i�l(p)

eipr

r� e

�i(pr�l⇡)

r

◆

Spherically Symmetric Square Potential Well/Barrier:

V(r)

ra

V0

0

V(r)

r

a

V0

0

V(r)

r

a0

Potential Barrier Potential Well Hard Sphere

V (r) = V0 0 < r < a

= 0 r > a

•Scattering states behave like free particle far from the potential region.

•Since they have KE, we are looking for states with E>0

•The potential well can sustain bound states, but we are not interested in them (for now).

Use spherical co-ord

angular part given by Y0l

Spherically Symmetric Potential Well/Barrier: (+)(r, ✓) =

X

l

il(2l + 1)Rl(r)Pl(cos ✓) =X

l

il(2l + 1)1

rul(r)Pl(cos ✓)

r > a

Spherical Bessel Functions

Free particle solutions far from origin

Spherical Hankel Functions

h(1(2))l (pr) ⇠ ±e±i(pr�l⇡/2)

ipr

pr � 1

h(1(2))l (pr) = jl(pr)± inl(pr)Rl(r) = c1jl(pr) + c2nl(pr) = c

(1)h(1)l (pr) + c(2)h(2)l (pr)

Radial Equation: V (r) = V0 0 < r < a= 0 r > a

�d2ul(r)

dr2+

2mV (r)� p2 + l(l + 1)

r2

�ul(r) = 0

(+)(r) =1

(2⇡)3/2

X

l

(2l + 1)Pl(cos ✓)

2ip

✓e2i�l(p)

eipr

r� e

�i(pr�l⇡)

r

◆Comparing with

c(1)l =1

2e2i�l(p) c(2)l =

1

2Rl(r) = e

i�l[cos �ljl(pr)� sin �lnl(pr)]

Spherically Symmetric Potential Well/Barrier:

Spherical Bessel Functions Spherical Hankel Functions

Rl(r) = c1jl(pr) + c2nl(pr) = c(1)h(1)l (pr) + c

(2)h(2)l (pr)

(+)(r) =1

(2⇡)3/2

X

l

(2l + 1)Pl(cos ✓)

2ip

✓e2i�l(p)

eipr

r� e

�i(pr�l⇡)

r

◆Comparing with

c(1)l =1

2e2i�l(p) c(2)l =

1

2Rl(r) = e

i�l[cos �ljl(pr)� sin �lnl(pr)]

c(1) and c(2) are obtained from the continuity of the logarithmic derivative at r=a.

�l =

r

Rl

dRldr

�����r=a tan �l(p) =

paj0

l (pa)� �ljl(pa)pan

0l(pa)� �lnl(pa)

Note that till now we have not used the specific square-wave form

of the potential. This result is valid for any potential that vanishes at

a finite range

Spherically Symmetric Potential Well/Barrier:To find the parameter βl, we need the solution inside the potential region

A) Hard Sphere Potential

V(r)

ra0

Negative phase shift for

repulsive potential

Generically true for finite

potential barrier as well.

Indicates that the wave-fn

is pushed out

Boundary Condition : wfn. vanishes at r=a

�l ! 1�l =r

Rl

dRldr

�����r=a

tan �l(p) =jl(pa)

nl(pa)

tan �l(p) =paj

0

l (pa)� �ljl(pa)pan

0l(pa)� �lnl(pa)

�0(p) = �pas-wave phase shift

d�

d⌦ 0=

sin2 �0k

2' a2 for ka ⌧ 1

Spherically Symmetric Potential Well/Barrier:To find the parameter βl, we need the solution inside the potential region

B) Potential Well/ Barrier

V(r)

ra

V0

0

Potential Well

�l =qaj

0

l (qa)

jl(qa)tan �l(p) =

pj0

l (pa)� qj0

l (qa)jl(pa)/jl(qa)

pn0l(pa)� qj

0l (qa)nl(pa)/jl(qa)

Inside Soln.: regular at r=0 jl(qr)q2

2m= E � V0

V(r)

ra

V0

0

Potential

Barrier

�l(p) ⇠ p2l+1

�l =

r

Rl

dRldr

�����r=a

Square Potential Well: s-wave ScatteringConsider the l=0 channel in the low energy limit tan �l(p) ⇠ �pas

s-wave scattering length

For square well j0(x) =sinx

x

�l =qaj

0

l (qa)

jl(qa)Using �0 = qa cot(qa)� 1

2ma2V0

β0

-40 -20 0 20 40-15-10-505101520

-40 -20 0 20 400.0

0.5

1.0

1.5

2ma2V0

as

Scattering

Length

as =a�0

1 + �0�0 = 4⇡a

2s

Scattering

Cross-Section

as diverges when

qa = (2n+1) π/2

f0(p) =1

p cot �0(p)� ip=

�11as

+ ip= � as

1 + ipas

Scattering

Amplitude

f0(0) = �as

f0 does not diverge when qa = (2n+1) π/2, f0 is -1/ ip at this point --- Unitary limit

s-wave Scattering LengthFor large r u0(r) ⇠ ei�0 sin(pr + �0) ⇠ ei�0 sin p(r � as) ⇠ ei�0p(r � as)

So as has the interpretation of the first point in space where the extrapolation of the far solution hits zero. Note that it is not a zero of the actual solution.

ra

U(r)

For purely repulsive potential

curvature is away from axis.

Scattering Length is always positive

�d2ul(r)

dr2+

2mV (r)� p2 + l(l + 1)

r2

�ul(r) = 0

Consider p=0, l=0u(r)

s-wave Scattering LengthFor large r u0(r) ⇠ ei�0 sin(pr + �0) ⇠ ei�0 sin p(r � as) ⇠ ei�0p(r � as)

So as has the interpretation of the first point in space where the extrapolation of the far solution hits zero. Note that it is not a zero of the actual solution.

a a

u (r) u (r) u (r)

U(r) U(r) U(r)

00

0

(b) (c)(a)

For attractive potential wells, the scattering length is initially negative

As we increase the well depth,the scattering length becomes more

and more negative till it reaches -∞

Beyond this point, the scattering length starts at + ∞ and keeps decreasing

This is the point where we have the first bound state in the system

V(r) V(r)V(r)

Effective Range Expansion H. A. Bethe, Phys. Rev. 76, 38 (1949)

u2d2u1dr2

� u1d2u2dr2

= (p21 � p22)u1u2 u2du1dr

� u1du2dr

����R

0

= (p21 � p22)Z R

0dru1u2

What happens when we go to larger energies, aka what is the next term in f

f0(p) =1

p cot �0(p)� ip=

�11as

+ ip= � as

1 + ipas

�d2ul(r)

dr2+

2mV (r)� p2 + l(l + 1)

r2

�ul(r) = 0

d2u1dr2

+ [p21 � V (r)]u1(r) = 0d2u2dr2

+ [p22 � V (r)]u2(r) = 0

Schrodinger Eqn for 2 different momenta

Consider the asymptotic form of the solutions at large r

p(r) =sin[pr + �0(p)]

sin �0(p)

2d 1dr

� 1d 2dr

����R

0

= (p21 � p22)Z R

0dr 1 2

The asymptotic soln. also follows similar eqns as u

Subtract the equations for u and ψ,

p1 ! 0, p2 ! p p cot �0(p) = �1

as� p2

Z 1

0dr 0 p � u0up

Effective range of potential r0

p cot �0(p) =�1as

� r0p2 f0(p) =�1

1as

+ ip+ r0p2 Effective range expansion

Effective Range Expansion H. A. Bethe, Phys. Rev. 76, 38 (1949)

2d 1dr

� u2du1dr

� 1d 2dr

+ u1du2dr

����R

0

= (p21 � p22)Z R

0dr 1 2 � u1u2

At r=R, LHS vanishes by continuity eqn.s.

At r=0, terms in LHS involving u vanish as u(0)=0.

Int extended to ∞ since the integrand vanishes outside

the range of potential 1

d 2dr

� 2d 1dr

����0

= (p21 � p22)Z 1

0dr 1 2 � u1u2

p2 cot �0(p2)� p1 cot �0(p1) = (p21 � p22)Z 1

0dr 1 2 � u1u2Using explicit form ψ at r=0,

' � 1as

� p2Z 1

0dr 20 � u20

Universality of low energy scatteringWe have seen that the low energy scattering from a potential can be characterized

by a few parameters

E.g. s-wave scattering can be parametrized by as, r0, etc.

Clearly this cannot depend on all the details of the shape of the potential

So we can have many different potentials at the microscopic level, whose low energy

scattering (say as, r0) are same.

E.g. can choose different V0 and a for a square well so that qa is fixed. Low energy

scattering is same for both. We can even get away with a simpler potential (say delta fn) provided we manage to get the correct scattering length

This is your first glimpse into the general phenomenon of universality:

Many systems which look different on a microscopic scale (i.e. different V) can show

same phenomena at low energy. This is at the heart of theoretical endeavours to

calculate properties of complicated systems.

For square well �0 = qa cot(qa)� 1 Scattering Length as =

a�01 + �0

q2

2m= E � V0