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