FOUNDATIONS OF QUANTUM MECHANICS

IV. 4. NEUTRON INTERFEROMETRY 95
all neutrons have, at entry in the interferometer, spin up in the z - direction. We place the complete
setup in a homogeneous magnetic field which ensures that spin up and spin down have a different
energy ω 0 . In one of the paths we place a ‘spin flipper’, a small coil through which an alternating
current runs having exactly the resonance frequency ω 0 . At a suitable choice of the length of the
coil the spin of every neutron which travels through it will be flipped over. Subsequently, we place
spin analyzers in front of the detectors, so that we can not only observe in which emanating beam the
neutron is located but also its spin in the z - direction.
In this setup we can therefore uncover exactly along which path the particle has traveled; spin up
means the path without the spin flipper has been chosen, spin down means the neutron traveled along
the path with the spin flipper. But in this setup no more interference is seen! The intensity is equal in
both detectors and independent of the phase difference.
We can describe this as follows. The wavepath function |ϕ 0 ⟩ ∈ L 2 (R 2 ) of an emanating neutron
exists of four terms,
|ϕ 0 ⟩ = 1 2
(
|ϕ1A ⟩ + |ϕ 1B ⟩ + e i χ |ϕ 2A ⟩ + e i χ |ϕ 2B ⟩ ) . (IV. 14)
Here ϕ iA and ϕ iB represent the wave functions ending up in the detectors A and B, respectively, 1
and 2 refer to the two possible paths through the interferometer, as can be seen in figure IV. 5 a. The
factor e iχ corresponds to the phase shift by the aluminium. If χ = 0, there is maximum constructive
interference in A and total destructive interference in B, from which it follows that
|ϕ 1A ⟩ = |ϕ 2A ⟩ and |ϕ 1B ⟩ = − |ϕ 2B ⟩. (IV. 15)
The intensity in detector A is given by the expectation value of a projection P A , where
P A |ϕ iA ⟩ = |ϕ iA ⟩ and P A |ϕ iB ⟩ = 0, analogously for P B . Therefore, we find for the intensity I A
of the neutron beam that encounters detector A, quantum mechanically expressed as the probability
to find a neutron in detector A,
I A = ⟨ϕ 0 | P A |ϕ 0 ⟩ = 1 (
4 ⟨ϕ1A | + ⟨ϕ 2A | e − i χ) ( |ϕ 1A ⟩ + e i χ |ϕ 2A ⟩ )
and likewise for I B ,
= 1 2
I B = ⟨ϕ 0 | P B |ϕ 0 ⟩ = 1 4
= 1 2
(1 + cos χ), (IV. 16)
(
⟨ϕ1B | + ⟨ϕ 2B | e − i χ) ( |ϕ 1B ⟩ + e i χ |ϕ 2B ⟩ )
(1 − cos χ). (IV. 17)
In this experiment the neutrons are polarized, therefore we can add the spin state to the wavepath
function and thus get a Pauli spinor,
( 1
|ϕ i, tot ⟩ = |ϕ 0 ⟩ ⊗ |z ↑⟩ = ϕ(⃗q) =
0)
( ) ϕ(⃗q)
0
∈ L 2 (R 3 ) ⊗ C 2 . (IV. 18)
The functioning of the spin flipper, which we assume to be completely ideal, can now be described as
follows. The component of the state traveling along path 1 does not meet a spin flipper, which means