Hello all. I’m researching rotational motion with a nearly harmonic potential using the basis of the particle on a ring eigenstates e(adsbygoogle = window.adsbygoogle || []).push({}); ^{(n*i*theta)}defined from theta=0 to theta=2*pi.

The total systems wave functions (eigenfunctions of the full Hamiltonian (KE+PE)) are then linear combinations of the particle on a ring eigenstates.

The thing that has me stumped is this: As the potential is symmetric around pi, the wave functions are necessarily either symmetric or anti-symmetric around pi. If you describe an anti-symmetric wave function with these types of states however, you’ll see that the real part cancels out and you’re left with all of the amplitude due to the imaginary part alone.

That is, for a given value of n, the coefficient in front of e^{(n*i*theta)}is either equal to that in front of the e^{(-n*i*theta)}term (then the i*sin(theta) part cancels because sin is an odd function) or is equal to -1*that coefficient (and the cosine part cancels because itis an even function).

I was under the impression that bound state wave functions MUST be real. However, I don’t see how anti-symmetric wave functions can exist in this system if they’re NOT purely imaginary and I don’t know of any reason to exclude the anti-symmetric states apart from that.

By “I was under the impression that…” I mean that it’s something that I’ve heard on the street and never really thought about. Can someone either help me out with how I’m looking at this incorrectly or tell me why “all bound states must be purely real” is a hard fast rule of quantum mechanics and I should exclude the asymmetric states from my analysis. This seems strange because asymmetric states of the harmonic oscillator aren't excluded and this system is basically the same but with a weird basis.

Any help would be much appreciated. Thanks in advance.

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# Purely imaginary bound state wave functions?

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