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.

Formulation

Scattering in quantum mechanics begins with a physical model based on the Schrödinger 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:

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}}

.

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

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 .}

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

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

′

.