Evenly spaced energy level systems [duplicate]

I'm looking for examples of quantum mechanical systems which have evenly spaced energy levels. A couple of them are -

1. Quantum Harmonic Oscillator;
2. Inverted Quantum Harmonic Oscillator.

It appears to me that no more such systems are possible. Is that true? If not, what are the examples of that?

• I think I remember from the QM course that only harmonic potentials allow for evenly spaced energy levels. I do not know how to prove that for you, though May 7, 2021 at 13:33
• I guess some semi-trivial examples are spin-$s$ systems with Hamiltonians of the form $a + b S_z$, with $a$ and $b$ constants, or variations of it. May 7, 2021 at 13:43
• Possible duplicate: physics.stackexchange.com/q/132688/2451 May 7, 2021 at 16:00

Here's an example in Thanu Padmanabhan's wonderfully-titled book Sleeping Beauties in Theoretical Physics (the "Isochronous Curiosities" chapter). A potential of the form $$V(x) = ax^2 + \frac{b}{x^2},\tag{\star}\label{\star}$$ with $$a$$ and $$b$$ positive, has evenly-spaced energy levels.

To get a feel for why this might be, note that the Hamiltonian for a 3D harmonic oscillator, $$H = \frac{1}{2m}(p_x^2+p_y^2+p_z^2) + \frac{1}{2}m\omega^2(x^2+y^2+z^2),$$ has evenly-spaced energy levels $$E = \hbar\omega\left(n_x+n_y+n_z+\tfrac{3}{2}\right)\tag{\dagger}\label{\dagger}$$ because it is the sum of three independent, 1D harmonic oscillators. Going to polar coordinates, the radial Schrodinger equation is $$-\frac{\hbar^2}{2m}\frac{d^2u}{dr^2} + V_{\text{eff}}(r)u = Eu$$ where $$V_{\text{eff}}(r) = \frac{1}{2}m\omega^2r^2 + \frac{\hbar^2l(l+1)}{2mr^2}.$$ This is identical to the Schrodinger equation for a particle moving in one dimension in a potential of the form ($$\star$$) and has energy levels given by a subset of those described by ($$\dagger$$).

It turns out that this subset takes the form $$E = \hbar\omega(2n+l+3/2)$$ for $$n,l\geq 0$$.

It also turns out that potentials of the form ($$\star$$) have evenly-spaced energy levels, even if $$b$$ can't be written as $$\hbar^2l(l+1)/2m$$ for some integer $$l$$. To prove this, solve the Schrodinger equation for potential ($$\star$$) using a power series method.