# Why do wave functions have to be single-valued?

In some situations, like the wave function of an electron in a hydrogen atom, we must require that the wave function $\psi(r,\theta,\phi)$ to be single-valued, i.e. $\psi\propto e^{i2\pi m\phi}(m\in\mathbb{Z})$. But the result strongly relies on the symmetry of the system. Does there exist multi valued wave functions which do not violate the basic principles of quantum mechanics?

As far as I know, whenever you have a wavefunction defined in the "real world", it has to be single-valued, namely it has to come back to the same expression after a $2\pi$ rotation (in the real world). We see in real life that $2\pi$ rotations are irrelevant.

However this is not the case for wavefunctions defined in spaces which are not our real space, like spin space for example. Spinors change sign after a $2\pi$ rotation so in this sense they are not single-valued. I also encountered an interesting case when I was reading about geometric phases in molecular systems. Using Born-Oppenheimer approximation the total wavefunction is split into an electronic wavefunction and a nuclear one. Since only the total one is "in real space", it has to be single-valued, but the electronic and nuclear one separately do not have this restriction. They can both change sign after a cyclic evolution, with the condition that their product remains the same.

I hope it helps.

edit: by the way, I think the single-valuedness of real stuff is the reason why orbital angular momentum always takes integer values, even though the general theory of angular momentum allows half integer values also.

Hopefully it is not off-topic, but I think I can also provide a (sketch of) mathematical explanation for the difference between rotations in real space and in spin space. In real space we have a 3-dimensional matrix representation of the rotation group $SO(3)$, where the rotation with $2\pi$ corresponds to the identity matrix. The wavefunction is this case is single-valued.

When it comes to 1/2 spin, we start with a 2-dimensional representation of the Lie algebra of $SO(3)$, the Pauli matrices. However, it cannot be exponentiated to a representation of the group because the group is not simply-connected. It will be exponentiated to a representation of the universal covering group of $SO(3)$, namely the simply-connected group $SU(2)$. And indeed this leads to a projective representation of $SO(3)$, which is actually what we want (we deal with the space of states, not with the Hilbert space itself). Now, $SU(2)$ is a double –covering of $SO(3)$, roughly speaking $SU(2)$ is made out of two copies of $SO(3)$. Rotation with $2\pi$ is sent to minus the identity matrix, while rotation with $4\pi$ is sent to the identity matrix. Both are projected to the same element in $SO(3)$. The wavefunction in this case is not single-valued, but double-valued.

It’s a similar discussion with the Lorentz group and $SL(2,\mathbb{C})$. The Dirac spinors also change sign at $2\pi$ rotations. The book of B.Thaller "The Dirac equation" provides a rigorous discussion of what I tried to say here with covering groups and sign changes.

• Gosh Sylvia, two answers for the price of one! But polite cough, spin is real. See Hans Ohanian’s 1984 paper what is spin?. – John Duffield Apr 10 '18 at 16:15

It is so because in case it is multi-valued then it will lead to different probabilities of finding the Particle at given point of space at a given time which is not possible