130_notes.dvi

17 3D Symmetric HO in Spherical Coordinates*

We have already solved the problem of a 3D harmonic oscillator by separation of variables in Carte-
sian coordinates (See section 13.2). It is instructive tosolve the same problem in spherical
coordinatesand compare the results. The potential is

V(r) =

1

2

μω^2 r^2.

Our radial equation is
(
d^2
dr^2

+

2

r

d

dr

)

REℓ(r) +

2 μ

̄h^2

(

E−V(r)−

ℓ(ℓ+ 1) ̄h^2

2 μr^2

)

REℓ(r) = 0

d^2 R

dr^2

+

2

r

dR

dr

−

μ^2 ω^2

̄h^2

r^2 R−

ℓ(ℓ+ 1)

r^2

R+

2 μE

̄h^2

R = 0

Write the equation in terms of the dimensionless variable

y =

r

ρ

.

ρ =

√

̄h

μω

r = ρy

d

dr

=

dy

dr

d

dy

=

1

ρ

d

dy

d^2

dr^2

=

1

ρ^2

d

dy^2

Plugging these into the radial equation, we get

1

ρ^2

d^2 R

dy^2

+

1

ρ^2

2

y

dR

dy

−

1

ρ^4

ρ^2 y^2 R−

1

ρ^2

ℓ(ℓ+ 1)

y^2

R+

2 μE

̄h^2

R = 0

d^2 R

dy^2

+

2

y

dR

dy

−y^2 R−

ℓ(ℓ+ 1)

y^2

R+

2 E

̄hω

R = 0.

Now find the behavior for largey.

d^2 R

dy^2

−y^2 R= 0

R≈e−y

(^2) / 2
Also, find the behavior for smally.
d^2 R
dy^2

+

2

y

dR

dy

−

ℓ(ℓ+ 1)

y^2

R= 0

R≈ys

s(s−1)ys−^2 + 2sys−^2 =ℓ(ℓ+ 1)ys−^2

s(s+ 1) =ℓ(ℓ+ 1)

R≈yℓ