QM: More Spherical Harmonics

In summary, the system's wavefunction is proportional to sin^2p and the possible results of measurements of Lz and L^2 are -2 and +2. To find the probabilities, a linear combination of spherical harmonic basis functions can be used, with the hint given being -3sin^2θ + 2 = 3cos^2θ - 1. One of the spherical harmonics has no angle dependence, and Y(2,0) can be subtracted by an amount of Y(0,0) to remove the constant and normalize the wavefunction.
  • #1
rabbit44
32
0

Homework Statement


A system's wavefunction is proportional to sin^2p. What are the possible results of measurements of Lz and L^2? Give the probabilities of each possible outcome.

I'm using p for theta and q for phi.

Homework Equations


The Attempt at a Solution


So I believe that the value of L^2 is 6 (as l is 2), and the possible values of Lz are -2 and +2. But I can't find the probabilities. I tried:

|psi> = a|2,2> + b|2,-2>

where the kets are |l,m> and the spherical harmonice are Y(l,m)

<2,2|psi> = int dpdq <2,2|p,q><p,q|psi>
<2,2|psi> = int dpdq Y(2,2)* [aY(2,2) + bY(2,-2)]

And then inserted the spherical harmonic expressions in, but I just get a=a, so I guess this isn't the way.

I also thought about using the ladder operators, applying L+|psi> = 2a|2, -1> but I don't think this is helpful.

Any help wouldbe much appreciated!

EDIT: Another thing I tried was saying Lz|l,m>=m|l,m>. So I thought if I had a matrix representation of Lz, I could maybe have

Lz (a*, b*) = 2(a*, -b*)

Where ( , ) is a column matrix. But I don't know the matrix representation of Lz or what basis it should be in. Please help!
 
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  • #2
the wavefunction has NO phi dependence?
 
  • #3
malawi_glenn said:
the wavefunction has NO phi dependence?

Yeah that was my first reaction. But I don't think it is possible to have no phi dependence and be proportional to sin^2, so I assumed that it meant it was proportional to sin^2 as well as some function of phi. I may be wrong though?
 
  • #4
The spherical harmonics are basis functions for angular functions, so one should be able to find a linear combination of them to build up sin^2 theta.

Now if I give you this hint, you should be able to do it.

[tex]-3\sin ^2 \theta +2 = 3\cos ^2 \theta - 1[/tex]

Also recall that one of the S. harm's does not depend on any angle at all.
 
  • #5
malawi_glenn said:
The spherical harmonics are basis functions for angular functions, so one should be able to find a linear combination of them to build up sin^2 theta.

Now if I give you this hint, you should be able to do it.

[tex]-3\sin ^2 \theta +2 = 3\cos ^2 \theta - 1[/tex]

Also recall that one of the S. harm's does not depend on any angle at all.

Ahhhh I forgot about that one.

I see now, I just need Y(2,0) and then need to subtract an amount of Y(0,0) to get rid of the constant, and normalise.

Thank you!
 
  • #6
rabbit44 said:
Ahhhh I forgot about that one.

I see now, I just need Y(2,0) and then need to subtract an amount of Y(0,0) to get rid of the constant, and normalise.

Thank you!

correct observation :-)

Good luck
 

1. What are spherical harmonics?

Spherical harmonics are a mathematical tool used to represent the angular dependence of solutions to partial differential equations on a sphere. They are commonly used in quantum mechanics to represent the angular part of wavefunctions.

2. How are spherical harmonics used in quantum mechanics?

Spherical harmonics are used in quantum mechanics to describe the angular part of wavefunctions, which represent the probability of finding a particle in a specific location and state. They are also used to calculate the energy levels and angular momentum of particles in a spherical potential.

3. What is the difference between spherical harmonics and regular harmonics?

Spherical harmonics are similar to regular harmonics in that they are both mathematical functions that represent oscillations. However, spherical harmonics are specifically used to represent the angular dependence of solutions on a sphere, while regular harmonics can represent oscillations in any direction.

4. How are spherical harmonics related to the quantum numbers of an atom?

The quantum numbers of an atom, specifically the azimuthal quantum number (l) and magnetic quantum number (m), determine the shape and orientation of the orbital. These quantum numbers correspond to the angular momentum and angular part of the wavefunction, which are described by spherical harmonics.

5. Can spherical harmonics be used to describe other physical systems besides atoms?

Yes, spherical harmonics can be used to describe the angular dependence of solutions to partial differential equations on any spherical system, not just atoms. They have applications in other fields such as electromagnetism, fluid dynamics, and astrophysics.

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