QMGreensite_merged

17.4. DEGENERATEPERTURBATIONTHEORY 277

17.4.2 Example-TheTwo-DimensionalHarmonicOscillator

TheHamiltonianforaharmonicoscillatorintwodimensionsis

H 0 =−

̄h^2

2 M

[

∂^2

∂x^2

+

∂^2

∂y^2

]

+

1

2

k(x^2 +y^2 ) (17.80)

Thissystemissymmetricwithrespectto rotations(aroundthez-axis),reflections
(x→−xandy →−y),andinterchangex →y, y→ x. Its nothard toverify
thatthese symmetriesdo notall commutewitheachother,andthereforeatleast
someoftheenergyeigenvaluesmustbedegenerate.Infact,itiseasytosolveforthe
eigenvaluesandeigenstatesofH 0 bythemethodofseparationofvariables.Writing

H 0 =h[x]+h[y] (17.81)

whereh[x]andh[y]areone-dimensionalharmonicoscillatorHamiltoniansinthex
andycoordinatesrespectively,and

φ(0)mn(x,y)=φm(x)φn(y) (17.82)

weendupwiththeone-dimensionalharmonicoscillatoreigenvalueequations

h[x]φm(x) = Emφm(x)

h[y]φn(y) = Enφn(y) (17.83)

withthetotalenergyeigenvalueofφ(0)mn(x,y)being

Enm(0)=Em+En (17.84)

Theeigenvalueequations(17.83)wehavesolvedlongagousingraising/loweringop-
erators. Inthiscase,weshouldintroduceseparateraising/loweringoperatorsinthe
xandycoordinates,i.e.

a =

1

√

2 ̄h

(

√

Mωx+i

px

√

Mω

)

b =

1

√

2 ̄h

(

√

Mωy+i

py

√

Mω

)

(17.85)

withcommutatorrelations
[a,a†]=[b,b†]= 1 (17.86)

and
[a,b]=[a,b†]=[a†,b]=[a†,b†]= 0 (17.87)

Intermsoftheseoperators,thetwodimensionalharmonicoscillatorissimply

H 0 =h ̄ω(a†a+b†b+1) (17.88)