Consider the center of mass system of two interacting identical fermions with
spin ½.

(a) What is the consequence of the Pauli exclusion principle on the
two-particle wave function?

(b) Let **S**_{1} and **S**_{2} be the spin operators of
the two individual neutrons.

Show that the operators P_{±} = ½ ± ¼ ± **S**_{1}**∙S**_{2}/ħ^{2}
are the projection operators of the triplet states and the singlet states of the
spin wave functions.

(c) Using the Pauli exclusion principle and the symmetry properties of the spin
and relative orbital angular momentum L, find the allowed values of L for any
bound triplet state of the two-particle system.

(d) Again using the Pauli exclusion principle and the symmetry properties of
the space coordinate, show that the particles in a triplet state can never
scatter through an angle of 90 degrees in their center of mass system.

Solution:

- Concepts:

Indistinguishable particles, identical fermions - Reasoning:

A state of many identical particles is totally**antisymmetric**if they are fermions. - Details of the calculation:

(a) The total state vector must be anti-symmetric under the exchange of the two particles. If the two particles are in the singlet state, then the space wave function is symmetric, if they are in the triplet state, then the space wave function is anti-symmetric.

(b) A set of basis vectors for the spin space are {|S,M_{S}> = |1,1>, |1,0>, |1,-1>, |0,0>}.

**S**_{1}∙**S**_{2}= ½(S^{2 }- S_{1}^{2 }- S_{2}^{2}).

S^{2}|S,M_{S}> = S(S + 1)ħ^{2}|S,M_{S}>.

S_{1}^{2}|S,M_{S}> = ¾ħ^{2}|S,M_{S}>.

S_{2}^{2}|S,M_{S}> = ¾ħ^{2}|S,M_{S}>.

P_{±}|S,M_{S}> = (½ ± ¼ ± (½(S(S + 1)/ħ^{2}- ¾ - ¾))|S,M_{S}>.

For S = 0, the singlet state:

P_{+}|0,0> = (½ + ¼ - ¾)|0,0> = 0.

P_{-}|0,0> = (½ - ¼ + ¾)|0,0> = |0,0>.

For S = 1, the triplet state:

P_{+}|1,M_{S}> = (½ + ¼ + 1 - ¾)|1,M_{S}> = |1,M_{S}>.

P_{-}|1,M_{S}> = (½ - ¼ - 1 + ¾)|1,M_{S}> = 0.

Any state vector for the system can be written as a linear combination of the basis vectors. Therefore P_{+}is the projector onto the triplet state and P_{-}is the projector onto the singlet state.

(c) The relative motion of two interacting particles in their CM frame is treated by treating the motion of a fictitious particle of reduced mass m in an external potential U(**r**). Here**r**points from particle 2 to particle 1,**r**=**r**_{1}-**r**_{2}.

Triplet state: The space wave function is anti-symmetic,

ψ(**r**_{12}) = -ψ(**r**_{21}) or ψ(**r**) = -ψ(-**r**).

We therefore need l = odd. (Property of the spherical harmonics.)

(d) For a scattering problem we have Φ_{k}(r,θ) = e^{ikz }+ f_{k}(θ)(e^{ikr})/r, σ_{k}(θ) = |f_{k}(θ)|^{2}.

For identical fermions the state vector must be anti-symmetric. For the triplet state the spin function is symmetric, so the space function must be anti-symmetric under exchange. We have

Φ_{k}(r,θ) = e^{ikz }- e^{-ikz }+ [f_{k}(θ) - f_{k}(π-θ)](e^{ikr})/r.

We therefore have σ_{k}(θ) = |f_{k}(θ) - f_{k}(π-θ)|^{2}. σ_{k}(θ) = 0 if θ = π/2.

The Coulomb scattering amplitude for two non-relativistic particles in the center of mass system is

f_{c}(θ) = -exp(iδ_{0})
γ exp[-iγ ln(sin^{2}(θ/2))]/(2ksin^{2}(θ/2)),

where
γ
= -μe^{2}Z_{1}Z_{2}/(ħ^{2}k), μ is the
reduced mass, and k is the wave number.

Find the differential cross
section in the center of mass if the two particles

(a) are distinguishable,

(b) are distinguishable, but the identity of the particles is not observed in
either bin,

(c) are identical spin-zero particles,

(d) are identical spin ½ particles in a spin-triplet state,

(e) are identical spin ½ particles in a spin-singlet state,

(f) are identical unpolarized spin ½ particles.

Evaluate each at θ = π/2.

You may ignore any relativistic spin effects for f_{c}(θ).

Solution:

- Concepts:

Scattering, indistinguishable particles - Reasoning:

The relative motion of two interacting particles in their CM frame is treated by treating the motion of a fictitious particle of reduced mass m in an external potential U(r).

Let**r**points from particle 2 to particle 1,**r**=**r**_{1}-**r**_{2}and let ψ(**r**) be the space part of the wave function.

In a central potential, stationary state solutions of the eigenvalue equation HΦ_{k}(**r**) = E_{k}Φ_{k}(**r**) with the asymptotic formΦ

_{k}(**r**) = e^{ikz}+ f_{k}(θ) e^{ikr}/r exist.

The differential scattering cross section is σ_{k}(θ) = |f_{k}(θ)|^{2} - Details of the calculation:

(a) Let Φ_{k}(r,θ) = e^{ikz}+ f_{k}(θ)(e^{ikr})/r as r --> ∞. Let f_{k}(θ) = f_{c}(θ).

σ_{c}(θ)_{a}= |f_{c}(θ)|^{2}is the differential scattering cross section for detecting particle 1 in bin 1 and detecting particle 2 in bin 2.

σ_{c}(θ)_{a}= γ^{2}/(4k^{2}sin^{4}(θ/2)).

At θ = π/2 we have σ_{c}(θ)_{b}= γ^{2}/k^{2}.

(b) σ_{c}(θ)_{b}= |f_{c}(θ)|^{2}+ |f_{c}(π - θ)|^{2}is the differential scattering cross section for detecting one particle in bin 1 and one particle in bin 2.

σ_{c}(θ)_{b}= γ^{2}/(4k^{2})[1/sin^{4}(θ/2) + 1/sin^{4}((π - θ)/2)].

At θ = π/2 we have σ_{c}(θ)_{b}= 2γ^{2}/k^{2}.

(c) For the spin zero particles the space wave functions must be symmetric under the exchange of the two particles.

Φ_{k}(**r**) = e^{ikz}+ e^{-ikz}+ [f_{k}(θ) + f_{k}(π - θ)](e^{ikr})/r as r --> ∞.

σ_{c}(θ) = |f_{c}(θ) + f_{c}(π - θ)|^{2}. σ_{c}(θ) = |f_{c}(θ)|^{2}+ |f_{c}(π - θ)|^{2}+ 2Re[f_{c}(θ) f_{c}*(π - θ)].

Therefore σ_{c}(θ)_{c}= σ_{c}(θ)_{b}+ 2Re[f_{c}(θ) f_{c}*(π - θ)].

f_{c}(θ) f_{c}*(π - θ) = [γ^{2}/(4k^{2}sin^{2}(θ/2) sin^{2}((π - θ)/2)] exp[-iγ (ln(sin^{2}(θ/2)) - ln(sin^{2}((π - θ)/2)].

Re[f_{c}(θ) f_{c}*(π - θ)] = [γ^{2}/(4k^{2}sin^{2}(θ/2) sin^{2}((π - θ)/2)] cos[γ ln{sin^{2}(θ/2)/sin^{2}((π - θ)/2)})].

At θ = π/2 we have Re[f_{c}(θ) f_{c}*(π - θ)] = γ^{2}/k^{2}and σ_{c}(θ)_{c}= 4γ^{2}/k^{2}.

The cross section is enhanced through constructive interference.

(d) For the spin 1/2 particles in the triplet state the spin function is symmetric under exchange, so the space wave functions must be anti-symmetric under the exchange of the two particles.

Φ_{k}(**r**) = e^{ikz}- e^{-ikz}+ [f_{k}(θ) - f_{k}(π - θ)](e^{ikr})/r as r --> ∞.

σ_{c}(θ)_{d}= |f_{c}(θ)|^{2}+ |f_{c}(π - θ)|^{2}- 2Re[f_{c}(θ) f_{c}*(π - θ)] = σ_{c}(θ)_{b}- 2Re[f_{c}(θ) f_{c}*(π - θ)].

At θ = π/2 we have destructive interference, σ_{d}(π/2) = 0.

(e) For the spin 1/2 particles in the singlet state the spin function is anti-symmetric under exchange, so the space wave functions must be symmetric under the exchange of the two particles.

We have σ_{c}(θ)_{e}= σ_{c}(θ)_{c}.

(f) We have a statistical distribution, 1/4 for spin singlet and 3/4 for spin triplet.

σ_{c}(θ)_{f}= (1/4)σ_{c}(θ)_{e}+ (3/4)σ_{c}(θ)_{f}= |f_{c}(θ)|^{2}+ |f_{c}(π - θ)|^{2}- Re[f_{c}(θ) f_{c}*(π - θ)].

At θ = π/2 we have Re[f_{c}(θ) f_{c}*(π - θ)] = γ^{2}/k^{2}and σ_{c}(θ)_{c}= γ^{2}/k^{2}