Magnetism 1

Also in general we have to watch out because A and ∇ do not commute.
∇A−A∇≠0
∇A Ψ −A∇ Ψ ≠0
The good thing is, in the transverse gauge they do commute
2 2
⎛ e ⎞ 2 e
e 2
→⎜ ∇+ A⎟ =−∆+ 2 A∇+
A 2
⎝ i c ⎠
i c c
2 2
− e 1 e 2
2 Z⋅e ⇒ H = ∆+ A⋅∇+ A − 2
2me i cme 2me
c r
1
transverse gauge ⇒ A (
= H× r)
2
2 2
−
e 2
⇒ (
)
e
H r (
H r)
Ze
H = ∆+ × ⋅∇+ × −
2
2me i mec 8mc
r
cyclic property (
H× r) ⋅∇= (
r×∇) ⋅H
recognize that r× p = l
op. of angular momentum
r× ∇ = l
i
with the definition
e⋅
= µ B
2m
e
and p
i ∇→
µ = 9.274⋅ 10
B
2 nd term:
µ L⋅H
B
“Bohr magneton”
−24
2
Am
paramagnetic term for orbital magnetism
3 rd term:
assume H zˆ
( 2
) 2 2 2
H× r = H ( x + y ) diamagnetic term
2 2
−
c Ze
= ∆+ B ⋅ + + −
2
2m 8me
r
→ ˆ z
2 2 2
H µ H L H ( x y )
paramagnetic term
diamagnetic term
- 4 -
2