Probability of finding a state with azimuthal orbital quantum number

[IHateMayonnaise]

STATEMENT:

A system is known to be in the normalized state described by the wave function

$$\psi(\theta,\phi)=\frac{1}{\sqrt{30}}(5Y_4^3+Y_6^3-2Y_6^0)$$

What is the probaility of finding the system in a state with azmithual orbital quantum number m=3?WHAT I HAVE:

We know that:

$$P=\sum_i|<Y_i^3|\psi(\theta,\phi)>|^2$$

So, simply find the dot product of $$Y_i^3$$ and $$\psi(\theta,\phi)$$. If we resolve this into the three components, this translates to:

$$Y_i^3 \dot \psi(\theta,\phi) = (y_1^3+y_2^3+y_3^3)\dot(5Y_4^3+Y_6^3-2Y_6^0)=(y_1^3)(5Y_4^3)+(y_2^3)(Y_6^3)-(y_3^3)(2Y_6^0)$$

How can I solve this dot product without resolving the components into the solutions to the spherical harmonics? Am I missing anything, any thoughts? Thanks yall

IHateMayonnaise

EDIT: either PF changed the format for LaTeX or I turned stupid. My first LaTeX statement, within tex brackets is: \psi(\theta,\phi)=\frac{1}{\sqrt{30}}(5Y_4^3+Y_6^3-2Y_6^0). Hmmm. The following is a test of theta:

$$\theta$$

which is (replacing []s with ()'s):

(tex)\theta(/tex)

And here is a copy/paste of the LaTeX code from another thread that IS showing up (in another thread, obviously not this one):

[tex]
\alpha\beta\chi\delta\epsilon\phi\varphi\gamma
[tex]

Which for whatever reason copies withOUT the / in the tex bracket, but still does not render. Now I've got two problems..

 

[G01]

It seems there are some latex problems since the move to the new server. Greg is working on remedying it as quickly as possible, I'm sure.

In regards to your physics question:

You do not need to find any dot products here.

You need to find the total probability for the particle to have m=3, and you know the state of the particle is the some of a finite number of eigenstates. This is a direct example of the expansion postulate.

HINT: What do the coefficients of the eigenstates in the sum tell you?

P.S. This is from a practice Physics GRE, correct?