Solving Angular Momentum in QM: Find Probability of L^2 Values

In summary, the problem is to find the probability of measuring a particular eigenvalue of the L^2 wavefunction of a particle on a sphere. The solution is to find the spherical harmonics eigenfunctions, and then calculate the probability of measuring a particular one.
  • #1
Izbitzer
5
0
Hello,

I'm trying to solve a problem dealing with finding the probability of measuring certain values of [tex]L^2[/tex] for a particle.

The particle is on a sphere and is in the state [tex]\Psi (\theta , \phi) = Ne^{\\cos{\theta }}[/tex].

I don't know quite how to start, I guess I have to decompose the wave function in eigenfunctions for [tex]L^2[/tex], and then find the corresponding eigenvalues, and form that find the probability of measuring that particular eigenvalue, but like I said, I don't really know where to start.

Does anybody have any pointers?

Thanks!
 
Last edited:
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  • #2
Nope, you have to do you wrote. That is try to write your wavefunction as a linear combination of angular momentum eigenstates.

Daniel.
 
  • #3
Thanks for the reply!

I've worked some more, and I think I've made some progress :smile:

The eigenfunctions of [tex]\^{L}^2[/tex] are spherical harmonics, but since the wave function is independent of [tex]\phi[/tex], m = 0, i.e [tex]\Psi(\theta,\phi) = \sum_{l} c_{l,0}Y_l^0 [/tex].
The eigenvalues are [tex]l(l+1)\hbar[/tex]. So I just see which value of [tex]l[/tex] corresponds with the values I'm supposed to find the probability of. The probability is calculated with: [tex]|c_{l}|^2 = |<\Psi|Y_l^0>|^2[/tex]. This is where I'm stuck at the moment. The integrals are really complicated and I can't find them in any books, is there any easier way to calculate this?

Thanks!
 
Last edited:
  • #4
Izbitzer said:
Thanks for the reply!

I've worked some more, and I think I've made some progress :smile:

The eigenfunctions of [tex]\^{L}^2[/tex] are spherical harmonics, but since the wave function is independent of [tex]\phi[/tex], m = 0, i.e [tex]\Psi(\theta,\phi) = \sum_{l} c_{l,0}Y_l^0 [/tex].
The eigenvalues are [tex]l(l+1)\hbar[/tex]. So I just see which value of [tex]l[/tex] corresponds with the values I'm supposed to find the probability of. The probability is calculated with: [tex]|c_{l}|^2 = |<\Psi|Y_l^0>|^2[/tex]. This is where I'm stuck at the moment. The integrals are really complicated and I can't find them in any books, is there any easier way to calculate this?

Thanks!
A suggestion: I don't recall the exact form of the[itex] Y^l_0 [/itex], but aren't they simple to write in terms of [itex] (cos \theta)^n [/itex] ? Then you simply have to Taylor expand [itex] e^{cos \theta)} [/itex]. The coefficients will simply be the usual 1/n!. If you have a closed form expression for the [itex] Y^l_0 (cos (\theta) [/itex], then you can find the coeeficients c_l without doing a single integral.

Patrick
 
  • #5
nrqed said:
A suggestion: I don't recall the exact form of the[itex] Y^l_0 [/itex], but aren't they simple to write in terms of [itex] (cos \theta)^n [/itex] ? Then you simply have to Taylor expand [itex] e^{cos \theta)} [/itex]. The coefficients will simply be the usual 1/n!. If you have a closed form expression for the [itex] Y^l_0 (cos (\theta) [/itex], then you can find the coeeficients c_l without doing a single integral.

Patrick

That probably would have worked, but I managed to solve the damn integrals before I read your reply :smile: All it took was some (a lot of) integration by parts, and it turned out ok. Now I can finally rest. :smile:

Thanks for taking the time guys!
 

Related to Solving Angular Momentum in QM: Find Probability of L^2 Values

1. What is angular momentum in quantum mechanics?

Angular momentum in quantum mechanics is a fundamental property of particles that describes the rotational motion of a particle around an axis. It is represented by the operator L and has both magnitude and direction.

2. How is angular momentum solved in quantum mechanics?

Angular momentum is solved in quantum mechanics using the angular momentum operator L and its corresponding wave function. The wave function represents the probability of finding a particle with a specific angular momentum value.

3. How do you find the probability of L^2 values in quantum mechanics?

To find the probability of L^2 values in quantum mechanics, you first need to calculate the eigenvalues and eigenvectors of the angular momentum operator L. The square of the magnitude of the eigenvector corresponds to the probability of the particle having that specific L^2 value.

4. What is the significance of solving angular momentum in quantum mechanics?

Solving angular momentum in quantum mechanics is significant because it allows us to understand the behavior of particles in terms of their rotational motion. It also helps us to accurately predict and describe the properties of atoms and molecules.

5. Can angular momentum in quantum mechanics be measured?

Yes, angular momentum in quantum mechanics can be measured using various experimental techniques such as spectroscopy. By measuring the angular momentum of particles, we can gain a better understanding of their physical properties and behavior.

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