Magnetism 1

Raising and lowering operators
The origin of the spin degree of freedom: relativistic energy–momentum
relationship
2 2 2 2 4
E = c p + m c
1
2
−
2
⎛ v ⎞
p= γ ⋅ mv,
γ = ⎜1− 2 ⎟
⎝ c ⎠
∂
E →i, ∂t
p→
∇
i
correspondence principle
4-dimensional covariant form (treat time and space the same way)
P → iδµ
4-dimensional vector
µ
⎛ E ⎞
Pµ = ⎜γmv, ⎟
⎝ c ⎠
⎛ 1 ∂ ⎞
∂ µ = ⎜−∇, c γ t
⎟
⎝ ⎠
→ Klein–Gordon equation:
2 2 4
⎛ ⎞
1 ∂ mc
⎜∆−
− Ψ= 0
2 2 2 ⎟
⎝ c ∂t
⎠
This is a differential equation of the second kind in time and space
∂Ψ
→ the solution requires starting values for Ψ and
γ t
Conclusion: This cannot be!
Dirac’s idea: linearize the equation
2 2
E−c α 0
( ipi − βmc ⎛E c α )
jpj βmc
⎞
⎜
+ +
⎟
i ⎝ j
⎠
this agrees with the Klein–Gordon equation if
α α + α α = 2δij
2
∑ ∑ = i, j ε { x, yz , }
i j j i
αβ+ βa
= 0
i i
relativistic counterpart of
Schrödinger equation
β = 1
cannot be fulfilled with ordinary numbers. Need to consider certain 4x4 matrices.
⎛0 α = ⎜
⎝σ
σ ⎞ ⎛1 ⎟β =
1
⎜
⎠ ⎝0 0⎞ ⎛0 ⎟ with σx = ⎜
1⎠ ⎝1 1⎞ ⎛0 ⎟, σ y = ⎜
0⎠ ⎝i −i
⎞ ⎛1
⎟, σz
= ⎜
0 ⎠ ⎝0
0 ⎞
⎟
−1⎠
2
→ each solution to E∓c 0
( ∑αipi
∓ β mc = is thus also a solution to the Klein–
)
Gordon equation.
→ Dirac’s equation for rel. free particle
⎛ ∂ ⎞
⎜i−cαp−βmc⎟Ψ = 0
⎝ ∂t
⎠
i
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