The Stern-Gerlach Effect for Electrons*

The Stern-Gerlach Effect for Electrons

Herman BatelaanGordon Gallup

Julie SchwendimanTJG

Behlen Laboratory of PhysicsUniversity of Nebraska

Lincoln Nebraska 68588-0111

Work funded by the NSF ndash Physics Division

Electron Polarization

σρtrP spin

)N()N(

)N()N(PP

example

P = 03

65 spin-up

35 spin-down

Atomic Collisionsσ

cosθgfA

(from GDFletcher et alii PRA 31 2854 (1985))

Work done at NIST Gaithersburg by MRScheinfein et alii

RSI 61 2510 (1991)

From The Theory of Atomic Collisions NFMott and HSW Massey

-V

N

S

+V

Anti-Bohr Devices

a)

(Knauer)

b)

(Darwin)

N

c)

(Brillouin)

1930 Solvay Conference ndash ldquoLe Magnetismrdquo

See eg

bull Cohen-Tannoudji Diu et Laloeuml

bull Merzbacher

bull Mott amp Massey

bull Baym

bull Keβler

bull Ohanianhelliphelliphellip

I

Z

e-

Which ball arrives first A) high roadB) low roadC) simultaneously

z

H

x

H

0H

zx

y

Hz

Hx

vz

xe-

)1(Δv

v

x

z

z

H

x

H

0H

zx

y

CALCULATIONS

2

1

2

1

zyx

yxzB HiHH

iHHH

dt

di

spinEHvc

e

dt

pdF )(

eigenenergies

integrate

(spin-flip probability lt 10-3)

)( zyxHE Bspin

CHOOSE INITIAL CONDITIONS

2220 )()()( TvxTx

2)()( iie vxm

ei m

Txx

2)()( 0

require Δzspin ~ 1mm

use Bo = 10T a = 1 cm (iexcl105A)

rarr vz ~ 105 ms (30 meV)

rarr t ~ 10μs

rarr Δxi ~ 100 μm

20

2211

20 2)(

tantan2

ze

B

i

fiif

ze

Bspin vm

Ba

za

azz

a

z

a

z

vm

Baz

H Batelaan et al PRL 79 4518 (1997)

Landau States

0 +12

1 -12

0 -12

1 +12

2 -12

(n ms)E-(pz

22m)

0

En = (pz22m) + (2n + 1)μBB plusmn μBB

n = (0123hellip)

NB - The net acceleration of the (leading) spin-backward electrons is zero

B

Pauli Case

ΔrΔp ~ ħ2

Landau Case

ΔrΔp ~ ħ2

B

MAGNETIC BOTTLE FORCES

z

BμμF z

BLz

B F

BL

L

S

(always || )B

(always || )B

0z

Bz

eνz

ˆ

0Bz

Fully quantum-mechanical calculation

(field due to a current loop)

Landau Hamiltonian

bull KE

bull ~ -μLB

bull ~ -μBB

GAGallup et alii PRL 86 4508 (2001)

S

W

F = SW

Gedanken apparatus

~~ φ TDC

1m 104 turns 5A

2 cm bore 10T

APERTURES

10μm 1μm

106 Hz

Landau States

0 +12

1 -12

0 -12

1 +12

2 -12

(n ms)E-(pz

22m)

0

En = (pz22m) + (2n + 1)μBB plusmn μBB

n = (0123hellip)

Δz Δt = Δzv

Δv

v

B

δ δ δ

bull Gradient = B δ Gradient force = plusmn(μBB δ) acceldecel = plusmn(μBB meδ) = plusmn a

bull If 2aδ ltlt v2 time lag = Δt = 2aδv3

bull Let B = 1T δ = 01m Ebeam = 100 keV (β = 055) rarr Δt = 4 x 10-19 s ()

bull Since the transit time threough the magnet = 2 ns R ~ 10-8

Conclusions

bull The Bohr-Pauli analysis of Brillouinrsquos proposal is wrong

bull More generally their prohibition against the spatial separation of electron spin based on classical trajectories through macroscopic classical fields fails

bull A proper semi-classical analysis of Brillouinrsquos gedanken experiment yields Rayleigh-resolved spin states

bull A rigorous quantum-mechanical analysis (corresponding to reality) yields complete and in principle arbitrarily large separation of spin states

bull Experiments to observe such spin-spitting are feasible (ie not totally insane) but would be very difficult

y

z

Hz

Hx

vz

xe-

-0002

0002

b

-01

01x

(mm

)

a

0999 1000z (m)0

50

09997 10003z (m)0

50

num

ber

of e

-

-0002

0002

b

-01

01x

(mm

)

a

0999 1000z (m)0

50

09997 10003z (m)0

50

num

ber

of e

-

Feasibility