QMGreensite_merged

336 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA

Often,indealingwithproblemsinatomic,nuclear,orparticle physics, weare
interested onlyina finitesubspace ofHilbert space consistingof eigenstates of a
definitetotalangularmomentum;i.e. definitevalueofthequantumnumberj. For
example, the spin angular momentumof an electron corresponds to j =^12 , and
nothingcaneverchangethatvalue,shortofannihilatingtheelectron. Soitmakes
sensetorestrictconsiderationstothej=^12 subspaceofHilbertspace,spanned by
thetwoeigenstates

|j=

1

2

,m=

1

2

> and |j=

1

2

,m=−

1

2

> (21.193)

Inthissubspace,theangularmomentumoperatorsare

J^2 =

3

4

̄h^2

[

1 0

0 1

]

Jz=

1

2

̄h

[

1 0

0 − 1

]

J+ = ̄h

[

0 1

0 0

]

J− = ̄h

[

0 0

1 0

]

(21.194)

Usingeq. (21.192),wecanwritetheangularmomentumoperatorsJx, Jy, Jzinthe
j=^12 subspaceas

Jx=

̄h

2

σx Jy=

̄h

2

σy Jz=

̄h

2

σz (21.195)

wherethematrices

σx=

[

0 1

1 0

]

σy=

[

0 −i

i 0

]

σz=

[

1 0

0 − 1

]

(21.196)

areknownasthePauliSpinMatrices. Theywillbeusefulinourlaterdiscussion
ofelectronspin.

21.4 Canonical Quantization

Quantummechanicsas presentedinpreviouschaptershas beenformulatedinthe
x-representation.Letusreviewhowwegofromtheclassicalmechanicsofaparticle
movinginonedimension, tothecorrespondingquantumtheory: Thetransitionis
madeasfollows:

Classical Quantum

−−−−−−−−−− −−−−−−−−−−−

physicalstate (x,p) wavefunction ψ(x)

Observables O=O(x,p) Operators O ̃=O ̃( ̃x, ̃p)

∂tx=∂∂Hp, ∂tp=−∂∂Hx i ̄h∂tψ(x,t)=H ̃(x ̃,p ̃)ψ(x,t)

cc (21.197)