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1 Introduction and Foundational Concepts
1.1 Why Scattering?
Scattering experiments are fundamental to understanding particle interactions:
Experimental Motivation
Key Idea: We cannot directly observe particles colliding at microscopic scales.
Instead, we:
- Send a beam of particles toward a target
- Measure the angular distribution of outgoing particles
- Measure the energy of scattered particles
- Extract information about the underlying interaction potential
Focus: We concentrate on:
- Elastic scattering: Particles same before and after (internal states unchanged)
- Central potentials: V = V (r), angular momentum is conserved
- Single scattering: Thin target, particles scatter at most once
1.2 Simplifying Assumptions
For tractable treatment:
- Spinless particles: Ignore spin; include when needed via spinor wavefunctions
- Elastic only: No inelastic processes or excitation
- Single scattering: Target thin enough to neglect multiple scattering
- Incoherent target: Ignore interference between di!erent scatterers
- Potential scattering: Time-independent, short-range potential V (r)
Remark: Long-range forces (Coulomb) require modified asymptotic forms and Ruther-
ford scattering formulas.
1.3 Classical Scattering Picture
1.3.1 Impact Parameter and Scattering Angle
For an azimuthally symmetric potential:
Classical Di!erential Cross Section
b = impact parameter (perpendicular o!set from scattering center) (1)
ω = scattering angle (deflection angle) (2)
dε
d”
b(ω)
sin ω
db
dω
The classical di!erential cross section relates the entrance area element to the solid-
angle element:
dϑ
d”
b(ω)
sin ω
db
dω
1.3.2 Example: Hard Sphere
For a hard sphere of radius R:
- Impact parameter: b = R cos(ω/2)
- Derivative:
db dω
R 2
sin(ω/2)
- Di!erential cross section:
dε d!
R 2 4
(isotropic!)
- Total cross section: ε (^) tot = ϖR
2 (geometric cross section)
1.4 Quantum Di!erential Cross Section
In quantum mechanics, there are no definite trajectories. Instead:
Quantum Definition
If incident flux density is Fi (particles per unit area per unit time), then particles
scattered into solid angle d” at angle (ω, ϱ) per unit time is:
dn
dt
dε
d”
(ω, ϱ) · Fi · d” (5)
Total cross section:
εtot =
dε
d”
d” (6)
Units: Cross sections have dimensions of area. Common unit: barn = 10
→ 24 cm
2
10
→ 28 m
2 .
2.4 Probability Current and Di!erential Cross Section
The probability current density is:
j =
m
Im[ς
↓ ↑ς] (12)
The radial component of the scattered current at large r:
j
sc r =^
⊋k
m
|f (ω, ϱ)|
2
r
2
The incident plane wave has flux density F (^) i = ⊋k/m.
Di!erential Cross Section from Amplitude
dε
d”
(ω, ϱ) = |f (ω, ϱ)|
2 (14)
This is a fundamental result: the di!erential cross section equals the modulus
squared of the scattering amplitude.
3 Born Approximation
3.1 The Born Series
Iterating the Lippmann-Schwinger equation generates the Born series:
ς(r) = ς 0 (r) +
G
U ς 0 d
3 r
↑ (1st order) (15)
G
U G
U ς 0 d
3 r
↑ d
3 r
↑↑
The n-th term is proportional to U
n .
3.2 First-Order Born Approximation
Approximation: Replace ς(r
↑ ) in the Lippmann-Schwinger equation with ς 0 (r
↑ ) =
e
ik·r ↑ .
For an incident plane wave along ˆz with k = kˆz:
Born Scattering Amplitude
f (^) B (ω, ϱ) = →
4 ϖ
e
→i(k ↑^ →k)·r U (r)d
3 r (17)
or in terms of the original potential:
f (^) B (ω, ϱ) = →
m
2 ϖ⊋
2
e
→iq·r V (r)d
3 r (18)
where the momentum transfer is:
q = k → k
↑ = 2k sin(ω/2)ˆq, |q| = 2k sin(ω/2) (19)
Physical interpretation: The Born amplitude is the Fourier transform of the po-
tential at momentum transfer q.
3.3 Born Di!erential Cross Section
dε
d”
B
= |f (^) B (ω)|
2
m
2 ϖ⊋ 2
2
e
→iq·r V (r)d
3 r
2
(20)
3.4 Validity of Born Approximation
The Born approximation is valid when:
- Weak potential: |V (r)| ⇐
⊋ 2 k 2 2 m
(weak compared to kinetic energy)
- High energy: Large momentum k ensures weak distortion of incident plane wave
- Weak scattering parameter:
m 2 ϑ⊋ 2 k
| V˜ (q)| ⇐ 1
Breaks down for:
- Strong potentials
- Low energies
- Long-range interactions (Coulomb)
- Resonances or bound states
3.5 Example: Yukawa Potential
The Yukawa potential:
V (r) = →
g
2
4 ϖ
e
→μr
r
The Fourier transform of e
→μr /r is:
e
→iq·r e^
→μr
r
d
3 r =
4 ϖ
q 2 + μ 2
Born amplitude:
f (^) B (ω) =
mg
2
2 ϖ⊋
2
q
2
2
, q = 2k sin(ω/2) (23)
Di!erential cross section:
dε
d”
mg
2
2 ϖ⊋ 2
(q 2 + μ 2 ) 2
Limits:
dε d!
⇒ 1 / sin
4 (ω/2) (Rutherford formula)
- Large q (backscattering): Suppressed by factor (q
2
2 )
→ 2
where j (^) ϖ is the spherical Bessel function, with far-field behavior:
j (^) ϖ (↽)
ϱ↔↗ →→→↓
2 i↽
[
e
i(ϱ→ϖϑ/2) → e
→i(ϱ→ϖϑ/2)
]
This is a superposition of outgoing and incoming spherical waves.
4.4 Phase Shift Definition
For a short-range potential, as r ↓ ↔:
Asymptotic Form and Phase Shift
uk,ϖ(r)
r↔↗ →→→↓ Aϖ sin
kr →
↼ϖ
where φϖ(k) is the phase shift:
- The phase advance induced by the potential
- Real number for elastic scattering (unitarity)
- Depends on energy k and angular momentum ↼
- Contains all scattering information for each ↼
4.5 Plane Wave Expansion in Partial Waves
The incident plane wave decomposes as:
e
ikz
∑^ ↗
ϖ=
i
ϖ (2↼ + 1)j (^) ϖ (kr)Pϖ (cos ω) (31)
where Pϖ (cos ω) is the Legendre polynomial. Each term contains both incoming and
outgoing spherical components:
j (^) ϖ (kr) =
2 ikr
[
e
i(kr→ϖϑ/2) → e
→i(kr→ϖϑ/2)
]
The potential shifts the phase by φ (^) ϖ while preserving the incoming component.
4.6 Scattering Amplitude from Phase Shifts
Assembling the full stationary scattering state from partial waves:
Scattering Amplitude (Partial Waves)
f (ω) =
∑^ ↗
ϖ=
2 ik
e
2 iςω → 1
Pϖ(cos ω) (33)
Alternative form:
f (ω) =
k
∑^ ↗
ϖ=
(2↼ + 1)e
iςω sin φϖ Pϖ(cos ω) (34)
Each partial wave contributes:
f (^) ϖ =
2 ik
(e
2 iς (^) ω → 1) =
k
e
iς (^) ω sin φ (^) ϖ (35)
4.7 Cross Sections from Phase Shifts
4.7.1 Di!erential Cross Section
dε
d”
= |f (ω)|
2
↗ ∑
ϖ=
2 ik
(e
2 iς (^) ω → 1)P (^) ϖ (cos ω)
2
4.7.2 Total Cross Section
Using orthonormality of Legendre polynomials:
Total Cross Section
ε (^) tot =
4 ϖ
k
2
∑^ ↗
ϖ=
(2↼ + 1) sin
2 φ (^) ϖ (k) (37)
Each partial wave contributes independently:
ε (^) ϖ =
4 ϖ(2↼ + 1)
k 2
sin
2 φ (^) ϖ (38)
Maximum contribution: When sin
2 φ (^) ϖ = 1 (resonance at φ (^) ϖ = ϖ/2):
ε
max ϖ =
4 ϖ(2↼ + 1)
k 2
For ↼ = 0 (s-wave): ε
max 0 =^
4 ϑ k 2
4.8 The Optical Theorem
A fundamental relation between forward scattering and total cross section:
Optical Theorem
ε (^) tot =
4 ϖ
k
Im[f (0)] (40)
where f (0) is the forward scattering amplitude (ω = 0).
This follows from unitarity and is a powerful consistency check.
For partial waves:
Im[f (0)] =
k
↗ ∑
ϖ=
(2↼ + 1) sin
2 φ (^) ϖ (41)
5.2 Hard Sphere Example (Low Energy)
5.2.1 Phase Shift Calculation
For a hard sphere: V (r) = ↔ for r < a, so u 0 (a) = 0.
The s-wave phase shift:
tan φ 0 =
j 0 (ka)
n 0 (ka)
sin(ka)/ka
→ cos(ka)/ka
= → tan(ka) (50)
For ka ⇐ 1:
φ 0 ≃ →ka (51)
5.2.2 Cross Sections
Scattering amplitude:
f (ω) ≃ →a (52)
Di!erential cross section:
dε
d”
= a
2 (isotropic) (53)
Total cross section:
ε (^) tot = 4ϖa
2 (54)
Key observation: Total cross section is 4 times the geometric cross section
ϖa
2 !
Why 4 times? Quantum di!raction contributes equally to forward and backward
scattering.
5.2.3 Verification via Optical Theorem
f (0) = →a + ia
2 k (keeping O(k) term) (55)
Im[f (0)] = a
2 k (56)
ε (^) tot =
4 ϖ
k
Im[f (0)] =
4 ϖ
k
· a
2 k = 4ϖa
2 ↭ (57)
6 Resonances and Phase Shifts
6.1 Resonance Condition
A resonance occurs when the phase shift passes through φ (^) ϖ = ϖ/2:
Resonance in Scattering
At resonance (φ (^) ϖ = ϖ/2):
sin φ (^) ϖ = 1 ↗ ε
max ϖ =
4 ϖ(2↼ + 1)
k
2
The cross section peaks dramatically. For s-wave (↼ = 0):
ε
max 0 =
4 ϖ
k
2
This is the unitarity limit: the maximum possible cross section.
6.2 Physical Interpretation
A resonance corresponds to a quasi-bound state:
- An attractive potential can temporarily trap the particle
- The particle ”dwells” in the potential region
- It eventually escapes with a phase delay
- This phase delay causes maximum scattering
Example: In the s-channel, a resonance indicates a nearly bound state with energy
just below the scattering threshold.
6.3 Breit-Wigner Formula
Near a resonance at energy E 0 , the cross section follows:
ε(E) =
4 ϖ
k 2
2 / 4
(E → E 0 ) 2 + # 2 / 4
where:
- E 0 = resonance energy
= resonance width (related to decay rate)
- Lifetime: τ = ⊋/#
FWHM: The width # is the full width at half maximum.
7 Worked Examples and Applications
7.1 Example 1: Hard Sphere (Complete)
7.1.1 Setup
Potential: V (r) = ↔ for r < a, V (r) = 0 for r > a
Energy: Low energy limit, ka ⇐ 1
7.1.6 Optical Theorem Check
To verify, keep next order in ka:
f (ω) ≃ →a + ia
2 k (73)
Im[f (0)] = a
2 k (74)
ε =
4 ϖ
k
Im[f (0)] = 4ϖa
2 ↭ (75)
7.2 Example 2: Square Well Potential
7.2.1 Setup
V (r) =
→V 0 r < a
0 r > a
, V 0 > 0 (76)
7.2.2 Wave Numbers
Inside (r < a):
k
2 0 =^ k^
2
2 mV (^0)
2
Outside (r > a):
k
2
2 mE
2
7.2.3 Solutions
Inside:
u 0 (r) = A sin(k 0 r) (79)
Outside:
u 0 (r) = B
sin(kr + φ 0 )
kr
7.2.4 Boundary Conditions at r = a
Continuity of u 0 :
A sin(k 0 a) = B
sin(ka + φ 0 )
ka
Continuity of du 0 /dr:
Ak 0 cos(k 0 a) = B
[
cos(ka + φ 0 )
a
sin(ka + φ 0 )
ka 2
]
7.2.5 Phase Shift
Eliminating A and B:
k (^0)
k
cot(k 0 a) = cot(ka + φ 0 ) (83)
φ 0 (k) = arccot
[
k (^0)
k
cot(k 0 a)
]
→ ka (84)
7.2.6 Low-Energy Behavior
As k ↓ 0, k 0 remains fixed. The phase shift approaches:
φ 0 ≃ →ka (^) s (85)
where the scattering length is:
a (^) s =
k (^0)
tan(k 0 a) → a (86)
This depends on the well depth and radius, showing how the potential determines the
low-energy cross section.
7.3 Example 3: Neutron-Proton Scattering
7.3.1 Physical Situation
At low energy (E ⇔ 1 eV), neutron-proton scattering is dominated by s-wave.
Impact parameter for nuclear force: b ≃ 1 .4 fm
Wave number: k ≃ 0 .007 fm
→ 1
Therefore: ka ≃ 0. 01 ⇐ 1 ↗ s-wave only
7.3.2 Spin Considerations
Both neutron and proton have spin 1/2. The total spin can be:
- Triplet (S = 1): 3 magnetic sublevels, probability 3/
- Singlet (S = 0): 1 sublevel, probability 1/
The nuclear interaction is di!erent for each spin state:
sin
2 φ
(t) 0 ≃^0.^8 (triplet)^ (87)
sin
2 φ
(s) 0 ≃^0.^3 (singlet)^ (88)
7.3.3 Total Cross Section
ε =
4 ϖ
k
2
[
sin
2 φ
(t) 0 +
sin
2 φ
(s) 0
]
ε =
4 ϖ
k
2
[3 ↖ 0 .8 + 0.3] ↖
4 ϖ
k
2
With k ≃ 0 .007 fm
→ 1 :
ε ≃ 20 barns = 20 ↖ 10
→ 24 cm
2 (91)
Comparison with experiment: This agrees well with measured values, validating
the partial-wave analysis!
9 Key Formulas Summary
9.1 Fundamental Definitions
Essential Formulas
Scattering Amplitude:
f (ω) =
k
↗ ∑
ϖ=
(2↼ + 1)e
iς (^) ω sin φ (^) ϖ P (^) ϖ (cos ω) (94)
Di!erential Cross Section: dε
d”
= |f (ω)|
2 (95)
Total Cross Section:
ε (^) tot =
4 ϖ
k
2
∑^ ↗
ϖ=
(2↼ + 1) sin
2 φ (^) ϖ (96)
Optical Theorem:
ε (^) tot =
4 ϖ
k
Im[f (0)] (97)
S-Matrix:
S (^) ϖ = e
2 iς (^) ω (98)
Scattering Length:
a (^) s = → lim k↔ 0
φ (^0)
k
Hard Sphere (s-wave, ka ⇐ 1 ):
φ 0 ≃ →ka, ε = 4ϖa
2 (100)
9.2 Born Approximation
f (^) B (ω) = →
m
2 ϖ⊋
2
e
→iq·r V (r)d
3 r (101)
with q = 2k sin(ω/2).
10 Exam Preparation Checklist
10.1 Conceptual Understanding
↫ Why scattering theory is important (experimental access to forces)
↫ Lippmann-Schwinger equation and its physical meaning
↫ Scattering amplitude as Fourier transform of potential (Born)
↫ Angular momentum conservation in central potentials
↫ Centrifugal barrier suppresses high-↼ at low energy
↫ Phase shift as the fundamental scattering parameter
↫ Resonance condition (φ (^) ϖ = ϖ/2) and quasi-bound states
10.2 Mathematical Skills
↫ Set up radial Schr¨odinger equation for given potential
↫ Apply boundary conditions (continuity, regularity at origin)
↫ Solve for phase shifts from boundary conditions
↫ Calculate scattering amplitude from phase shifts
↫ Compute di!erential and total cross sections
↫ Verify results using optical theorem
↫ Perform Born approximation calculations
10.3 Problem-Solving Strategy
- Calculate ka to determine dominant energy regime
- Identify which partial waves contribute significantly
- For low energy (ka ⇐ 1): use s-wave dominance
- For high energy (ka ↘ 1): sum many partial waves or use Born
- Always check with optical theorem as consistency check
- Interpret physics: resonances, unitarity limits, phase behavior
11 Common Mistakes to Avoid
Mistake 1
WRONG: ”Phase shift is independent of energy”
RIGHT: φ (^) ϖ (k) strongly depends on k. Di!erent energies have di!erent phase shifts!
Mistake 2
WRONG: ”All partial waves contribute equally to scattering”
RIGHT: High-↼ suppressed at low energy by centrifugal barrier. Only few waves
matter at ka ⇐ 1.
Mistake 3
WRONG: ”Born approximation always works”
RIGHT: Born only valid for high energies, weak potentials. Fails at low energy,
resonances, and strong interactions.
12.3 Good Luck on Your Exam!
Remember that scattering theory is one of the most important applications of quantum
mechanics. The concepts and techniques you learn here appear everywhere in modern
physics:
- Nuclear physics (nucleon-nucleon scattering)
- Atomic physics (electron-atom collisions)
- Particle physics (fundamental particle interactions)
- Solid state physics (electron-defect scattering)
- Quantum computing (qubit scattering)
The framework is robust and beautiful. Mastering partial-wave analysis gives you
powerful tools for understanding quantum dynamics!
