What Am I Missing in My Third Year QM Assignment Solution?

[karlhoffman_76]

See the picture below for details. I have included my solution thus far, but it doesn't feel right and I'm just wondering what I am missing. Any assistance would be greatly appreciated, cheers.

 

[blue_leaf77]

What about analyzing which powers of x,y, and z appear in the wavefunction and compare it with the spherical harmonics?
Comment to your second equation in your solution: $\psi(\mathbf{r})$ is supposed to be superposition of several eigenfunctions of $L_z$, means you can't write the equation such that where you can associate the wavefunction with one value of m.

 

[karlhoffman_76]

What about analyzing which powers of x,y, and z appear in the wavefunction and compare it with the spherical harmonics?
Comment to your second equation in your solution: $\psi(\mathbf{r})$ is supposed to be superposition of several eigenfunctions of $L_z$, means you can't write the equation such that where you can associate the wavefunction with one value of m.

Hey blue_leaf77, thanks for your reply. What do you mean by analyzing powers of x, y and z that appear in the wave function?

 

[blue_leaf77]

The purpose of the question is to check which eigenfunctions of $L^2$ and $L_z$ may appear when the original wavefunction is expanded as a sum of those eigenfunctions, which are the spherical harmonics $Y_{lm}$. Now if you remember the characteristic dependency of $Y_{lm}$ w.r.t to the three coordinates x,y, and z is governed by the indices $l$ and $m$. Using this fact you can get the first rough idea of which $l$ and $m$ are allowed to appear because they conform with the dependency of the original wavefunction to three coordinates. First of all, do you know how the spherical harmonics looks like and its dependency on x,y,z?

For example, $Y_{2,-2} \propto (x-iy)^2$, by comparing this with the manner x,y, and z appear in the original wavefunction, do you think $Y_{2,-2}$ will appear in the expansion?

 

[karlhoffman_76]

The purpose of the question is to check which eigenfunctions of $L^2$ and $L_z$ may appear when the original wavefunction is expanded as a sum of those eigenfunctions, which are the spherical harmonics $Y_{lm}$. Now if you remember the characteristic dependency of $Y_{lm}$ w.r.t to the three coordinates x,y, and z is governed by the indices $l$ and $m$. Using this fact you can get the first rough idea of which $l$ and $m$ are allowed to appear because they conform with the dependency of the original wavefunction to three coordinates. First of all, do you know how the spherical harmonics looks like and its dependency on x,y,z?

For example, $Y_{2,-2} \propto (x-iy)^2$, by comparing this with the manner x,y, and z appear in the original wavefunction, do you think $Y_{2,-2}$ will appear in the expansion?

I think I see where this is going now. Given that the final form of the expansion of $(x-iy)^2$ does not match the spherical harmonic (in terms of its Cartesian representation) of the wave function in question I would NOT expect to see $Y_{2,-2}$ in this particular wave function. So after reading your comments and looking into it a bit further I have a new solution. First of all, looking up a table of spherical harmonics I was able to ascertain that,

$x = \sqrt{\frac{2\pi}{3}}r(Y_{1,-1}-Y_{1,1})$

$y = \sqrt{\frac{2\pi}{3}}ir(Y_{1,1}+Y_{1,-1})$

$z = \sqrt{\frac{4\pi}{3}}rY_{1,0}$

and hence,

$\psi(\vec{r}) = (x+y+z)f(r) = \sqrt{\frac{2\pi}{3}}r[(Y_{1,-1}-Y_{1,1})+i(Y_{1,1}+Y_{1,-1})+\sqrt{2}Y_{1,0}]f(r)$

and so the allowed values of $l$ are $l = 1$ only and the allowed values of $m$ are $m = -1,0,1$. Thus the allowed values of $\hat{L}_{z}$ are $\hat{L}_{z} = -\hbar,0,\hbar$ and the allowed values of $\hat{L}^2$ are $\hat{L}^2 = 2\hbar^2$ only.

 

[blue_leaf77]

yes, that's what I think the correct one.

 

[karlhoffman_76]

yes, that's what I think the correct one.

Thanks so much blue_leaf77!