QMGreensite_merged

10.3. THEPARTICLEINASQUARE 161

where

V(x,y) = v(x)+v(y)

v(x) =

{

0 0 <x<L

∞ otherwise

(10.54)

ThevariablesintheHamiltonianareseparable:

H ̃ = H ̃x+H ̃y

H ̃x = −h ̄

2

2 m

∂^2

∂x^2

+v(x)

H ̃y = −h ̄

2

2 m

∂^2

∂y^2

+v(y) (10.55)

sothemethodofseparationofvariables

φ(x,y)=ψ(x)φ(y) (10.56)

canbeappliedtosolvethetime-independentSchrodingerequation

H ̃φ=Eφ (10.57)

Weget
φ(y)H ̃xψ(x)+ψ(x)H ̃yφ(y)=Eψ(x)φ(y) (10.58)

anddividingbyφ(x,y)onbothsides

H ̃xψ(x)

ψ(x)

+

H ̃yφ(y)

φ(y)

=E (10.59)

Sincethefirstratioisonlyafunctionofx,andthesecondisonlyafunctionofy,the
onlywaythatthesumofthetworatioscanbeaconstant,forallvaluesofxandy,
isifeachratioisaconstant,i.e.

H ̃xψn(x)=Enψn(x)

H ̃yφm(y)=Emφm(y) (10.60)

Inthiswaythetwo-dimensionalequationhasbeenreducedtotwoone-dimensional
equations,eachofwhichisidenticaltotheSchrodingerequationforaparticleina
tube,andwhichhavethesamesolutions:

ψn(x) =

√

2

L

sin[

nπx

L

] En=n^2

̄h^2 π^2

2 mL^2

φm(y) =

√

2

L

sin[

mπy

L

] Em=m^2

̄h^2 π^2

2 mL^2

(10.61)