indistinguishable particles a (r), b (r): two identical particles a and b in a certain states r...
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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
--
Stern-Gerlach Experiment