• Support PF! Buy your school textbooks, materials and every day products Here!

Angular momentum of a particle in a spherically symmetric potential

  • #1
1,444
2

Homework Statement


A particle in a spherically symmetric potential is in a state described by the wavepacked

[tex] \psi (x,y,z) = C (xy+yz+zx)e^{-alpha r^2} [/tex]

What is the probability that a measurement of the square of the angular mometum yields zero?
What is the probability that it yields [tex] 6\hbar^2 [/itex]?
If the value of l is found to be 2. what are the relative probabilities of m=-2,-1,0,1,2

2. The attempt at a solution

i think the first part is simply aking to calculate [itex] <L^2>[/itex]

but the carteisna coords are throwing me off... Should i convert to spherical polars?? Till now whenever the angular momentum L^2 and Lz were required, they were gotten using
[tex] \hat{L^2} \psi_{nlm_{l}} = l(l+1) \psi_{nlm_{l}} [/tex]

really from the spherical harmonics... however conversion to spherical polars doesnt yield any familiar spherical harmonic either.

can it written in a way that yields familiar spherical harmonics, however??
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
Yes. Convert to spherical coordinates. You won't necessarily get a spherical harmonic but you can decompose it into spherical harmonics in the usual way you split a wavefunction relative to an orthonormal basis.
 

Related Threads on Angular momentum of a particle in a spherically symmetric potential

Replies
3
Views
5K
  • Last Post
Replies
1
Views
1K
Replies
3
Views
768
Replies
5
Views
1K
  • Last Post
Replies
1
Views
1K
Replies
3
Views
1K
Replies
1
Views
4K
Replies
5
Views
1K
Replies
1
Views
5K
Replies
1
Views
886
Top