Reminder: Spin precession
► Spin
part of Hamiltonian:
 Hspin = −µ·B = −γ S·B
►γ
= gq /2m (gyromagnetic ratio)
 Acts on “spin space” only, so really:
 Hspin = − γ Iwave fn  S·B
► So
we ignore spatial wave function
 For B in z direction, say Hspin = ω0Sz
►
ω0 = γ B
► Time-independent
t   e
iHt / 
Hamiltonian, so solution is
0  e
i0 S z t / 
0
Matrix Representation
t   e
► If
i0 S z t / 
 at   e i0t / 2

 
0 
Sz
 bt   0
0   a0 
  
i0t / 2  
e
  b0 
starting state is spin-left, i.e.
 a0 = b0 = 1/√2,
► then
we get precession with period 2π/ω0:
 at   e i0t / 2

 
 bt   0
► NB
0  1 1 1  e i0t / 2  e i0t / 2  1 

 i t / 2  
  
 i0t 
i0t / 2 


0
e
2e
2 e 
 2 1

after one period, overall sign reversed.
ATOMINSTITUT Vienna
Neutron interferometry
►
►
Beam split by Bragg diffraction
in vertical crystal planes
Whole interferometer carved
from a perfect crystal of silicon:
 Separation between elements
exact multiple of inter-atomic
spacing.
Credit: NIST, Boulder, Colorado
►
Only ~ a dozen successfully
made in 30 years!
2π rotation (Warner et al 1975)
►
g factor of neutron = −3.83
 i.e.  = ge/2mn even though no net
charge!
Precession frequency ω0=−geB/2mn
de Broglie λ = 0.1445 nm
Effective ℓ ≈ 2.7 cm allowing for
leakage of B outside magnet.
► Time in field t = ℓ/v = ℓmnλ/h
► Angle precessed: ω0t = −geBλℓ/2h
►
►
►
►
►
►
For 2π rotation we need B = 3.4 mT
= 34 gauss
Observed period ≈ 62 G
4π rotation needed to restore
original state: 2π rotation changes
sign, as predicted by QM.