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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?

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    $\begingroup$ 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 $\endgroup$ – Davide Dal Bosco May 7 at 13:33
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    $\begingroup$ 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. $\endgroup$ – secavara May 7 at 13:43
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    $\begingroup$ Possible duplicate: physics.stackexchange.com/q/132688/2451 $\endgroup$ – Qmechanic May 7 at 16:00
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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.

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