# Expanding wavefunctions in terms of $L_z$ eigenkets

The angular momentum operator along the $$z$$-axis $$L_z$$ satisfies the secular equation $$L_z|m,l\rangle = \hbar m |m,l\rangle,$$ where $$l$$ is the corresponding (integer-valued) eigenvalue of the simultaneously diagonalizable operator $$|\mathbf L|^2$$ and $$m$$ is an integer between $$-l$$ and $$+l$$. Dropping the number $$l$$ from the state, we can write in angular coordinates the associated wavefunction $$\chi_m(\varphi):=\langle \varphi|m\rangle =\frac{1}{\sqrt{2\pi}}e^{im\varphi}.$$ Consider the (angular) position eigenket $$|\varphi\rangle$$. Since the eigenvectors of $$L_z$$ form a complete set of the Hilbert space, we should be able to write $$|\varphi\rangle=\sum_m \langle m|\varphi\rangle|m\rangle=\sum_m \chi_m^*(\varphi)|m\rangle,$$ or more generally any wavefunction $$\Psi(\varphi)$$ as $$\Psi(\varphi)=\langle \varphi|\Psi\rangle=\sum_m \langle\varphi|m\rangle\langle m|\Psi\rangle=\sum_m c_m\chi_m(\varphi)=\frac{1}{\sqrt{2\pi}}\sum_m c_m e^{im\varphi}$$ which is just a Fourier expansion with coefficients given by $$c_m=\int d\varphi\langle m|\varphi\rangle\langle \varphi|\Psi\rangle=\int d\varphi \chi_m^*(\varphi)\Psi(\varphi).$$ My question, coming back to the original eigenket $$|m,l\rangle$$, is the following: on all of these sums indexed by $$m$$, does $$m$$ vary freely in $$\mathbb Z$$ or is it bound by the quantum numer $$l$$?

• Let me get this straight: you are looking at even l spherical harmonics for θ =0? That is, are you truly considering SU(2), of just U(1), as your bottom manipulations assume? Apr 30, 2021 at 23:06
• @CosmasZachos I'm basically considering just the $\varphi$-dependant part of $Y_{lm}(\theta,\varphi)$. If you will, the circle rather than the sphere... Apr 30, 2021 at 23:55
• Indeed, for the circle you have only U(1) and unlimited range m s. May 1, 2021 at 0:56
• Thank you @CosmasZachos. I have to admit that I am not very familiar with group theoretic language. Could you elaborate a little on that? May 1, 2021 at 14:16
• No, not really. You have to study up on properties of spherical harmonics and why $|m|\leq l$ when the m s are not limited in a circle. May 1, 2021 at 14:33