Time-Dependent Superposition of Rigid Rotor Eigenstates

expansion coefficients

c

0,0

1.

c

0,0

0.

c

1,0

0.

c

1,1

0.

SubscriptBox[\(c\), \(2, \(-2\)\)]

0.

SubscriptBox[\(c\), \(2, \(-1\)\)]

0.

c

2,0

0.

c

2,1

0.

c

2,2

0.

time

0.

This Demonstration looks at a time-dependent superposition of the nine lowest quantum rigid-rotor eigenstates,

ψ(θ,ϕ)=

2

∑

J=0

J

∑

M=-J

c

J,M

-

E

J

t/ℏ



M

Y

J

(θ,ϕ)

, where

M

Y

J

(θ,ϕ)=

(2J+l)

4π

(J-M)!

(J+M)!

M

P

J

(cosθ)

Mϕ

e

,

M

P

J

are the associated Legendre polynomials, and

J

and

M

are the orbital and magnetic quantum numbers, respectively. The energy levels are

E

J

=J(J+1)

B

, where

B

≡

h

8

2

π

Ic

is the rotational constant. The figure shows the complex wavefunction

ψ(θ,ϕ)

, where the shape is its modulus and the coloring is according to its argument, the range

0

to

2π

corresponding to colors from red to magenta.