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I've been learning about quantum angular momentum, and I have a question about the relationship between quantum mechanical angular momentum wavefunctions with respect to different axes. I know that when constructing angular momentum eigenfunctions, we can only find simultaneous eigenfunctions of $L^2$ and one component of $\vec{L}$ (because the individual component of $\vec{L}$ don't commute with each other), so we choose $L_z$ for mathematical convenience.

Say we have a system that is in some superposition of these eigenfunctions which are simultaneous eigenfunctions of $L^2$ and $L_z$, and then we measure the angular momentum of the system with respect to some arbitrary axis. In order to use the Born probability rule, it will have to end up being the case that whatever superposition of the simultaneous eigenfunctions of $L^2$, and $L_z$ the system was in, this superposition will correspond to a superposition of states that are simultaneous eigenfunctions of $L^2$ and the component of $\vec{L}$ in the arbitrary direction we chose to make the measurement.

But this is not mathematically intuitive, so I just wanted to check whether I was correct: Is it the case that any superposition of simultaneous eigenfunctions of $L^2$ and $L_z$ can be expanded as a superposition of simultaneous eigenfunctions of $L^2$ and the component of $\vec{L}$ in any direction whatsoever? This would seem to be quite remarkable.

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Yes, you're right. One of the postulate of QM is that the set of eigenfunctions of a complete set of operators (where with complete I mean a set of physical quantity whose knowledge completely describe the state of the system) is a complete set of functions, where this time with complete I mean that every solution $\psi$ of the relative Shroedinger's equation is some linear combination of these. So if you consider the set of eigenfunctions $\psi_k$ of $L^2$ and $L_z$ then you can write: $$ \psi=\sum_{k=0}^{\infty}c_k\psi_k $$ Now, if $L^2$ and $L_z$ represents a complete set of operators, also $L^2$ and $L_s$ (where $L_s$ is the component of angular momentum in some other direction) does too. In fact the chose of direction depends only on the reference frame in which you work. So, also the simultaneous eigenfunctions of $L^2$ and $L_s$ constitute a complete set of functions in the space of Shroedinger equation's solutions and you can represent your wave function as a linear combination of them.

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