Scattering amplitude

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]

Scattering in quantum mechanics begins with a physical model based on the Schrodinger wave equation for probability amplitude ψ {\displaystyle \psi } : − ℏ 2 2 μ ∇ 2 ψ + V ψ = E ψ {\displaystyle -{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}\psi +V\psi =E\psi } where μ {\displaystyle \mu } 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 → ∞ ) ∼ e i k i ⋅ r + f ( k f , k i ) e i k f ⋅ r r {\displaystyle \psi (r\rightarrow \infty )\sim e^{i\mathbf {k} _{i}\cdot \mathbf {r} }+f(\mathbf {k} _{f},\mathbf {k} _{i}){\frac {e^{i\mathbf {k} _{f}\cdot \mathbf {r} }}{r}}} The scattering amplitude, f ( k f , k i ) {\displaystyle f(\mathbf {k} _{f},\mathbf {k} _{i})} , represents the amplitude that the target will scatter into the direction k f {\displaystyle \mathbf {k} _{f}} .^[3]: 194 In general the scattering amplitude requires knowing the full scattering wavefunction: f ( k f , k i ) = − μ 2 π ℏ 2 ∫ ψ f ∗ V ( r ) ψ i d 3 r {\displaystyle f(\mathbf {k} _{f},\mathbf {k} _{i})=-{\frac {\mu }{2\pi \hbar ^{2}}}\int \psi _{f}^{*}V(\mathbf {r} )\psi _{i}d^{3}r} 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 ) = e i k z + f ( θ ) e i k r r , {\displaystyle \psi (\mathbf {r} )=e^{ikz}+f(\theta ){\frac {e^{ikr}}{r}}\;,}

where r ≡ ( x , y , z ) {\displaystyle \mathbf {r} \equiv (x,y,z)} is the position vector; r ≡ | r | {\displaystyle r\equiv |\mathbf {r} |} ; e i k z {\displaystyle e^{ikz}} is the incoming plane wave with the wavenumber k along the z axis; e i k r / r {\displaystyle e^{ikr}/r} is the outgoing spherical wave; θ is the scattering angle (angle between the incident and scattered direction); and f ( θ ) {\displaystyle f(\theta )} 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,

d σ = | f ( θ ) | 2 d Ω . {\displaystyle d\sigma =|f(\theta )|^{2}\;d\Omega .}

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 ) = i k 2 π ∫ f ( n , n ″ ) f ∗ ( n , n ″ ) d Ω ″ {\displaystyle f(\mathbf {n} ,\mathbf {n} ')-f^{*}(\mathbf {n} ',\mathbf {n} )={\frac {ik}{2\pi }}\int f(\mathbf {n} ,\mathbf {n} '')f^{*}(\mathbf {n} ,\mathbf {n} '')\,d\Omega ''}

Optical theorem follows from here by setting n = n ′ . {\displaystyle \mathbf {n} =\mathbf {n} '.}

In the centrally symmetric field, the unitary condition becomes

I m f ( θ ) = k 4 π ∫ f ( γ ) f ( γ ′ ) d Ω ″ {\displaystyle \mathrm {Im} f(\theta )={\frac {k}{4\pi }}\int f(\gamma )f(\gamma ')\,d\Omega ''}

where γ {\displaystyle \gamma } and γ ′ {\displaystyle \gamma '} are the angles between n {\displaystyle \mathbf {n} } and n ′ {\displaystyle \mathbf {n} '} and some direction n ″ {\displaystyle \mathbf {n} ''} . This condition puts a constraint on the allowed form for f ( θ ) {\displaystyle f(\theta )} , i.e., the real and imaginary part of the scattering amplitude are not independent in this case. For example, if | f ( θ ) | {\displaystyle |f(\theta )|} in f = | f | e 2 i α {\displaystyle f=|f|e^{2i\alpha }} is known (say, from the measurement of the cross section), then α ( θ ) {\displaystyle \alpha (\theta )} can be determined such that f ( θ ) {\displaystyle f(\theta )} is uniquely determined within the alternative f ( θ ) → − f ∗ ( θ ) {\displaystyle f(\theta )\rightarrow -f^{*}(\theta )} .^[4]

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

f = ∑ ℓ = 0 ∞ ( 2 ℓ + 1 ) f ℓ P ℓ ( cos ⁡ θ ) {\displaystyle f=\sum _{\ell =0}^{\infty }(2\ell +1)f_{\ell }P_{\ell }(\cos \theta )} ,

where f_ℓ is the partial scattering amplitude and P_ℓ are the Legendre polynomials. The partial amplitude can be expressed via the partial wave S-matrix element S_ℓ ( = e 2 i δ ℓ {\displaystyle =e^{2i\delta _{\ell }}} ) and the scattering phase shift δ_ℓ as

f ℓ = S ℓ − 1 2 i k = e 2 i δ ℓ − 1 2 i k = e i δ ℓ sin ⁡ δ ℓ k = 1 k cot ⁡ δ ℓ − i k . {\displaystyle f_{\ell }={\frac {S_{\ell }-1}{2ik}}={\frac {e^{2i\delta _{\ell }}-1}{2ik}}={\frac {e^{i\delta _{\ell }}\sin \delta _{\ell }}{k}}={\frac {1}{k\cot \delta _{\ell }-ik}}\;.}

Then the total cross section^[6]

σ = ∫ | f ( θ ) | 2 d Ω {\displaystyle \sigma =\int |f(\theta )|^{2}d\Omega } ,

can be expanded as^[4]

σ = ∑ l = 0 ∞ σ l , where σ l = 4 π ( 2 l + 1 ) | f l | 2 = 4 π k 2 ( 2 l + 1 ) sin 2 ⁡ δ l {\displaystyle \sigma =\sum _{l=0}^{\infty }\sigma _{l},\quad {\text{where}}\quad \sigma _{l}=4\pi (2l+1)|f_{l}|^{2}={\frac {4\pi }{k^{2}}}(2l+1)\sin ^{2}\delta _{l}}

is the partial cross section. The total cross section is also equal to σ = ( 4 π / k ) I m f ( 0 ) {\displaystyle \sigma =(4\pi /k)\,\mathrm {Im} f(0)} due to optical theorem.

For θ ≠ 0 {\displaystyle \theta \neq 0} , we can write^[4]

f = 1 2 i k ∑ ℓ = 0 ∞ ( 2 ℓ + 1 ) e 2 i δ l P ℓ ( cos ⁡ θ ) . {\displaystyle f={\frac {1}{2ik}}\sum _{\ell =0}^{\infty }(2\ell +1)e^{2i\delta _{l}}P_{\ell }(\cos \theta ).}

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

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

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

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

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