Determining the probability of measuring a particular eigenvalue for angular momentum given the angular wave function

Question

Suppose I know the normalised angular wave function of a particle that is defined as $\psi(\theta,\phi)$

How would I use this to determine the probability of measuring a particular observable such as the probability of measuring a particular angular momentum.

Say that $L^2 = 2\hbar^2$ this would correspond to an angular momentum of $l=1$.

But how would I know the probability for measuring this.

I know that $$\int\int_{}\ |\psi(\theta,\phi)|^2 \sin(\theta) d\theta d\phi$$ would give me the probability of finding a particle in a particular position but am unsure how to change this to determine the probability of measuring $L^2 = 2\hbar^2$ or an angular momentum of $l=1$.

Answer 1

Assuming that the measurement is ideal, the act of measurement leaves the initial state in an eigenstate of the measured observable compatible with the result. So the probability of measuring a particular eigenvalue is best interpreted as the transition probability from the state prior to the measurement to an eigenstate of that particular eigenvalue (“an eigenstate” because the eigenspace could have degeneracy $>1$). For an observable $A$ and an initial state $\vert\psi\rangle$ you get, if $\langle\psi\vert\psi\rangle =1$, $$P_\psi^A(a_n)=\sum_{r=1}^{d(a_n)}\vert\langle\psi\vert a_n,r\rangle\vert^2$$ where $\vert a_n,1\rangle,\dots,\vert a_n, d(a_n)\rangle$ are the eigenstates of $A$ of eigenvalue $a_n$, with $d(a_n)$ the dimension of the eigenspace.

In the case in question, the operator $A$ would be the angular momentum $L$ with its eigenstates, which for the angular part are the spherical armonics $Y_\ell^m$. So the probability of measuring $\mathbf{L}^2=2\hbar^2$ is the sum of the transition probability from $\psi$ to $Y_1^1$ plus those from $\psi$ to $Y_1^{-1}$ and from $\psi$ to $Y_1^{0}$.