Scattering amplitude

Short description: Probability amplitude in quantum scattering theory

In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.[1]

## Contents

* 1 Formulation
* 2 Unitary condition
* 3 Partial wave expansion
* 4 X-rays
* 5 Neutrons
* 6 Quantum mechanical formalism
* 7 Measurement
* 8 See also
* 9 References

## Formulation

Scattering in quantum mechanics begins with a physical model based on the Schrodinger wave equation for probability amplitude ψ: −ℏ22μ∇2ψ+Vψ=Eψ where μ is the reduced mass of two scattering particles and E is the energy of relative motion. For scattering problems, a stationary (time-independent) wavefunction is sought with behavior at large distances (asymptotic form) in two parts. First a plane wave represents the incoming source and, second, a spherical wave emanating from the scattering center placed at the coordinate origin represents the scattered wave:[2]: 114 ψ(r->∞)∼ei𝐤i⋅𝐫+f(𝐤f,𝐤i)ei𝐤f⋅𝐫r The scattering amplitude, f(𝐤f,𝐤i), represents the amplitude that the target will scatter into the direction 𝐤f.[3]: 194 In general the scattering amplitude requires knowing the full scattering wavefunction: f(𝐤f,𝐤i)=−μ2πℏ2∫ψf*V(𝐫)ψid3r For weak interactions a perturbation series can be applied; the lowest order is called the Born approximation.

For a spherically symmetric scattering center, the plane wave is described by the wavefunction[4]

ψ(𝐫)=eikz+f(θ)eikrr,

where 𝐫≡(x,y,z) is the position vector; r≡|𝐫|; eikz is the incoming plane wave with the wavenumber k along the z axis; eikr/r is the outgoing spherical wave;  θ is the scattering angle (angle between the incident and scattered direction); and f(θ) is the scattering amplitude.

The dimension of the scattering amplitude is length. The scattering amplitude is a probability amplitude; the differential cross-section as a function of scattering angle is given as its modulus squared,

dσ=|f(θ)|2dΩ.

## Unitary condition

When conservation of number of particles holds true during scattering, it leads to a unitary condition for the scattering amplitude. In the general case, we have[4]

f(𝐧,𝐧′)−f*(𝐧′,𝐧)=ik2π∫f(𝐧,𝐧″)f*(𝐧,𝐧″)dΩ″

Optical theorem follows from here by setting 𝐧=𝐧′.

In the centrally symmetric field, the unitary condition becomes

Imf(θ)=k4π∫f(γ)f(γ′)dΩ″

where γ and γ′ are the angles between 𝐧 and 𝐧′ and some direction 𝐧″. This condition puts a constraint on the allowed form for f(θ), i.e., the real and imaginary part of the scattering amplitude are not independent in this case. For example, if |f(θ)| in f=|f|e2iα is known (say, from the measurement of the cross section), then α(θ) can be determined such that f(θ) is uniquely determined within the alternative f(θ)->−f*(θ).[4]

## Partial wave expansion

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,[5]

f=∑ℓ=0∞(2ℓ+1)fℓPℓ(cos⁡θ),

where fℓ is the partial scattering amplitude and Pℓ are the Legendre polynomials. The partial amplitude can be expressed via the partial wave S-matrix element Sℓ (=e2iδℓ) and the scattering phase shift δℓ as

fℓ=Sℓ−12ik=e2iδℓ−12ik=eiδℓsin⁡δℓk=1kcot⁡δℓ−ik.

Then the total cross section[6]

σ=∫|f(θ)|2dΩ,

can be expanded as[4]

σ=∑l=0∞σl,whereσl=4π(2l+1)|fl|2=4πk2(2l+1)sin2δl

is the partial cross section. The total cross section is also equal to σ=(4π/k)Imf(0) due to optical theorem.

For θ≠0, we can write[4]

f=12ik∑ℓ=0∞(2ℓ+1)e2iδlPℓ(cos⁡θ).

## X-rays

The scattering length for X-rays is the Thomson scattering length or classical electron radius, r0.

## Neutrons

The nuclear neutron scattering process involves the coherent neutron scattering length, often described by b.

## Quantum mechanical formalism

A quantum mechanical approach is given by the S matrix formalism.

## Measurement

The scattering amplitude can be determined by the scattering length in the low-energy regime.

## See also

* Levinson's theorem
* Plane wave expansion
* Veneziano amplitude

## References

1. ↑ Quantum Mechanics: Concepts and Applications By Nouredine Zettili, 2nd edition, page 623. ISBN 978-0-470-02679-3 Paperback 688 pages January 2009
2. ↑ Schiff, Leonard I. (1987). Quantum mechanics. International series in pure and applied physics (3. ed., 24. print ed.). New York: McGraw-Hill. ISBN 978-0-07-085643-1.
3. ↑ Baym, Gordon (1990). Lectures on quantum mechanics. Lecture notes and supplements in physics (3 ed.). Redwood City (Calif.) Menlo Park (Calif.) Reading (Mass.) [etc.]: Addison-Wesley. ISBN 978-0-8053-0667-5.
4. ↑ 4.0 4.1 4.2 4.3 4.4 Landau, L. D., & Lifshitz, E. M. (2013). Quantum mechanics: non-relativistic theory (Vol. 3). Elsevier.
5. ↑ Michael Fowler/ 1/17/08 Plane Waves and Partial Waves
6. ↑ Schiff, Leonard I. (1968). Quantum Mechanics. New York: McGraw Hill. pp. 119–120. https://archive.org/details/quantummechanics00schi_086.

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