Indistinguishable Particles

A(r) , yB(r): Two identical particles A and B in a certain states r

Since the particles are indistinguishable, this requires: |F(r1,r2)|2 = |F(r2,r1)|

2

Searching for a wave function F that describes both particles being in states r1 and r2

F(r1,r2) = F(r2,r1) or F(r1,r2) = – F(r2,r1)

FS (r1,r2) = 1/2 [yA(r1)yB(r2) + yB(r1)yA(r2)] - Bosons

FA (r1,r2) = 1/2 [yA(r1)yB(r2) – YB(r1)yA(r2)] - Fermions

Two particles in the same state (r1 = r2):

FS (r1,r1) = 1/2 [yA(r1)yB(r1)+yB(r1)yA(r1)]

= 1/2 ∙ 2yA(r1)yB(r1)

= 2 yA(r1)yB(r1)

|FS (r1,r1)|2 = 2 |yA(r1)|

2|yB(r1)|2

FA (r1,r1) = 1/2 [yA(r1)yB(r1) –yB(r1)yA(r1)]

= 0

|FS (r1,r1)|2 = 0

Bose-Einstein condensation Pauli-exclusion principle

Spinning Electron – Magnetic Dipole

B

LB

L

-

N

S

Stern-Gerlach Experiment

S

N

S

N

N

SS

N

S

N

Sz = ↑

Sz = ↓

Sy = ←/→

Sz = ↑

Sz = ↑

Sy = →

Understanding the Spin

N

S

B

F

Fr

r

torque:M = 2 r × F

-

L

M = dL/dt(analogue to F = dp/dt)

M

M

Sz

welldefined

Sx, Sy

changing

S

N

S

N

N

SS

N

S

N

Sz = + ½ ħ

Sz = – ½ ħ

Sz = + ½ ħ

Sz = + ½ ħ

Sy = + ½ ħ

-

SS

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Stern-Gerlach Experiment