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**[SOLVED] Eigenstates and Angular Momentum**

**1. Homework Statement**

At a given instant, a rigid rotor is in the state:

[tex]\Psi(\theta,\phi)=\sqrt{\frac{3}{4\pi}}Sin(\phi) Sin(\theta)[/tex]

If the z component of the orbital angular momentum is measured, what are the possible values of [tex]<\hat{L_{z}}>[/tex], and with what probability will they occur?

**2. Homework Equations**

These are the equations that I think are relevant:

[tex]\hat{L_z}=\frac{\hbar}{i}\frac{\partial}{\partial \theta}[/tex]

[tex]L_z = m_l \hbar[/tex]

[tex]|m_l| \leq l[/tex]

[tex]<\hat{L_z}>=\int_0^{2\pi} \Psi^*(\theta, \phi)\left(\frac{\hbar}{i}\frac{\partial}{\partial \theta}\right)\Psi(\theta, \phi) d\phi[/tex]

[tex]Y_1^1=-\sqrt{\frac{3}{8\pi}}Sin(\theta)e^{i\theta}[/tex]

[tex]Y_1^{-1}=\sqrt{\frac{3}{8\pi}}Sin(\theta)e^{-i\theta}[/tex]

...and also the exponential identities for Sine and Cosine.

**3. The Attempt at a Solution**

I suppose my question really has to do with the nature of the wavefunctions defined by Spherical Harmonics. Since the solution to The Schrodinger Equation in spherical coordinates has solutions that correspond to Spherical Harmonics (i.e., [tex]Y_l^{m_l}[/tex] corresponds to [tex]\Psi(\theta,\phi)[/tex] above), it seems like we should

*only*be able to get solutions (ie wavestates) that are in this form! And while [tex]\Psi(\theta,\phi)[/tex] is close to both [tex]Y_1^1[/tex] and [tex]Y_1^{-1}[/tex], it isn't the same. Furthermore, it looks like it

*could*be a superposition of the two, but it's not!

So, really, I need to find [tex]l[/tex] and [tex]m_l[/tex]. Since our [tex]\phi[/tex] dependence in our wavefunction has a coefficient of 1 (i.e. [tex]Sin(\phi)[/tex] corresponds to [tex]e^{im_l\phi}[/tex] where [tex]m_l=1[/tex]). So, [tex]l[/tex] can have the possible values of [tex]+1[/tex] or [tex]-1[/tex]. But, like I stated in the above paragraph, neither [tex]Y_1^1[/tex] or [tex]Y_1^{-1}[/tex] correspond to our wavefunction! So what the hell is [tex]m_l[/tex] and [tex]l[/tex]???