The Stern-Gerlach Effect for
Electrons*
Herman Batelaan
Gordon Gallup
Julie Schwendiman
TJG
Behlen Laboratory of Physics
University of Nebraska
Lincoln, Nebraska 68588-0111
*Work funded by the NSF – Physics Division
Electron Polarization


P  tr ρ s pinσ 

N()  N()
P P 
N()  N()
example:
P = 0.3: 
65% spin-up
35% spin-down
Atomic Collisions
(from G.D.Fletcher et alii,
PRA 31, 2854 (1985))
A
f g cosθ
σ
Work done at NIST Gaithersburg by
M.R.Scheinfein et alii,
RSI 61, 2510 (1991)
From The Theory of Atomic
Collisions, N.F.Mott and
H.S,W. Massey
Anti-Bohr Devices
a)
N
+V
-V
S
(Knauer)
b)
(Darwin)
c)
N
(Brillouin)
1930 Solvay Conference – “Le Magnetism”
See e.g.,
•
•
•
•
•
•
Cohen-Tannoudji, Diu, et Laloë
Merzbacher
Mott & Massey
Baym
Keβler
Ohanian………..
Z
I
e-
Which ball arrives first ?
A) high road
B) low road
C) simultaneously
x
e-
Hy  0
Hz
vz
Hy  0
Hx
H
 z
x
z
Hx
Hx
Hz

x
z
vz
 1(!! )
Δv x
CALCULATIONS
 Hz
d  1 
i     B 
dt   2 
 H x  iH y
eigenenergies

spin
E
   B H ( x, y, z)
integrate
dp e

F
 (v  H )  Espin
dt c
(spin-flip probability < 10-3)
H x  iH y  1 
 
 H z   2 
CHOOSE INITIAL CONDITIONS
me (x)i (v)i   / 2
x(T )  (x)02  (v) 2 T 2
T
(x)i  (x)0 
2me
2a B B0
z spin 
me v z2
 1  z f
 tan 
 a

( zi  z f )a  2a B B0

1  zi  
 
  tan     2
2 
2
a
a

z
m
v



i
e z


require Δzspin ~ 1mm
use Bo = 10T, a = 1 cm (¡105A!)
→ vz ~ 105 m/s (30 meV)
→ t ~ 10μs
→ Δxi ~ 100 μm
H. Batelaan et al., PRL 79, 4518 (1997)
Landau States
En = (pz2/2m) + (2n + 1)μBB ± μBB
(n, ms)
E-(pz2/2m)
1, +1/2
2, -1/2
0, +1/2
1, -1/2
0
0, -1/2
n = (0,1,2,3….)
NB - The net
acceleration of the
(leading) spinbackward electrons
is zero.
Pauli Case
Landau Case
ΔrΔp ~ ħ/2
ΔrΔp ~ ħ/2
B
B
MAGNETIC BOTTLE FORCES
Bz  0

zˆ , ν e

(always || B )

L

S

L

B z
0
z

(always || B )
B


  
F      B

B z
Fz   μL  μB 
z
Fully quantum-mechanical calculation
(field due to a current loop)
Landau Hamiltonian
• KE
• ~ -μL·B
• ~ -μB·B
G.A.Gallup et alii, PRL 86. 4508 (2001)
S
W
F = S/W
Gedanken apparatus
106 Hz
~
φ
~
TDC
1m, 104 turns, 5A
2 cm bore, 10T
APERTURES
10μm
1μm
Landau States
En = (pz2/2m) + (2n + 1)μBB ± μBB
(n, ms)
E-(pz2/2m)
1, +1/2
2, -1/2
0, +1/2
1, -1/2
0
0, -1/2
n = (0,1,2,3….)
Δv
v
Δz; Δt = Δz/v
B
δ
δ
δ
• Gradient = B/ δ; Gradient force = ±(μBB/ δ); accel/decel = ±(μBB/ meδ) = ± a
• If 2aδ << v2, time lag = Δt = 2aδ/v3
• Let B = 1T, δ = 0.1m, Ebeam = 100 keV (β = 0.55) → Δt = 4 x 10-19 s (!)
• Since the transit time threough the magnet = 2 ns, R ~ 10-8
Conclusions
•
The Bohr-Pauli analysis of Brillouin’s proposal is wrong.
•
More generally, their prohibition against the spatial separation of electron
spin based on classical trajectories through macroscopic classical fields
fails.
•
A proper semi-classical analysis of Brillouin’s gedanken experiment yields
Rayleigh-resolved spin states.
•
A rigorous quantum-mechanical analysis (corresponding to reality) yields
complete and, in principle, arbitrarily large separation of spin states.
•
Experiments to observe such spin-spitting are feasible (i.e., not totally
insane), but would be very difficult.
y
x
e
-
Hz
vz
z
Hx
b
a
0
.
1
number of e- x(m)
0
.
0
0
2
0
.
1
5
0
0
.
0
0
2
5
0
0
0
0
.
9
9
9
7
.
0
0
0
3
0
.
9
9
9
1
.
0
0
0
z
(
m
)
z
(
m
)1
b
a
0
.
1
number of e- x(m)
0
.
0
0
2
0
.
1
5
0
0
.
0
0
2
5
0
0
0
0
.
9
9
9
7
.
0
0
0
3
0
.
9
9
9
1
.
0
0
0
z
(
m
)
z
(
m
)1
Feasibility ?