A strange wave function of the Hydrogen atom

[omegax241]

I am trying to solve the following exercise.

In a H atom the electron is in the state described by the wave function in spherical coordinates:

$$\psi (r, \theta, \phi) = e^{i \phi}e^{-(r/a)^2(1- \mu\ cos^2\ \theta)}$$

With $a$ and $\mu$ positive real parameters. Tell what are the possible values of the measurement of $L_z$, what is the probability that a measurement of $L^2$ gives the value $6 \hbar^2$, and then what is the minimum value of an energy measurement.

First I've observed that $m=1$, so the only possible measurement for $L_z$ is $\hbar$. (I'm pretty confident with this reasoning, but not so much.)For the other questions my idea was to rewrite the wavefunction as a linear combination of stationary states of the form $\phi_{n,l,1}$, but the problem is the square on the $r$ part, and the fact that the angular part $\theta$ is exponentiated.
In fact I've even started to ask myself is if necessary at all to find this decomposition to solve the problem, but nothing comes to mind.
Thank you for listening, every bit of help is appreciated.

 

[PeroK]

I am trying to solve the following exercise.

In a H atom the electron is in the state described by the wave function in spherical coordinates:

$$\psi (r, \theta, \phi) = e^{i \phi}e^{-(r/a)^2(1- \mu\ cos^2\ \theta)}$$

With $a$ and $\mu$ positive real parameters. Tell what are the possible values of the measurement of $L_z$, what is the probability that a measurement of $L^2$ gives the value $6 \hbar^2$, and then what is the minimum value of an energy measurement.

First I've observed that $m=1$, so the only possible measurement for $L_z$ is $\hbar$. (I'm pretty confident with this reasoning, but not so much.)For the other questions my idea was to rewrite the wavefunction as a linear combination of stationary states of the form $\phi_{n,l,1}$, but the problem is the square on the $r$ part, and the fact that the angular part $\theta$ is exponentiated.
In fact I've even started to ask myself is if necessary at all to find this decomposition to solve the problem, but nothing comes to mind.
Thank you for listening, every bit of help is appreciated.

To find the coefficient of a particular eigenfunction you do not need to completely decompose the wavefunction. Do you know how to do that?

 

[PeroK]

Thank you for listening, every bit of help is appreciated.

I've had a look at this. I must admit I don't see how to tackle it. That wave function looks horrible. Sorry, I've no ideas either.

 

[omegax241]

To find the coefficient of a particular eigenfunction you do not need to completely decompose the wavefunction. Do you know how to do that?

Yes, I should evaluate the product:
$$\langle \phi_{n, l ,m} | \psi \rangle$$

But suppose I want to tackle the second question, if a value of $6 \hbar$ is found for $L^2$ this means that $l = 2$, and then $m = \pm 1 ; \pm 2 ; 0$. So I should evaluate the coefficents​

$$\langle \phi_{n, 2, \pm 1} | \psi \rangle$$
$$\langle \phi_{n, 2, \pm 2} | \psi \rangle$$
$$\langle \phi_{n, 2, 0} | \psi \rangle$$

But how can I find a definite number with this n dependence ?

 

[PeroK]

Yes, I should evaluate the product:
$$\langle \phi_{n, l ,m} | \psi \rangle$$

But suppose I want to tackle the second question, if a value of $6 \hbar$ is found for $L^2$ this means that $l = 2$, and then $m = \pm 1 ; \pm 2 ; 0$. So I should evaluate the coefficents​

$$\langle \phi_{n, 2, \pm 1} | \psi \rangle$$
$$\langle \phi_{n, 2, \pm 2} | \psi \rangle$$
$$\langle \phi_{n, 2, 0} | \psi \rangle$$

But how can I find a definite number with this n dependence ?

You know that $m = 1$. It's the variable $n$ that's the problem.

 

[hutchphd]

Doesn't parity tell you that an eigenstate of l=2 must be even? The wave function provided is odd.