Ladder operators for a general second-order linear differential equation

Question

We know that Schrödinger equation for a 1D harmonic oscillator

$$ \left( \hat{p}^2 + \frac{1}{2} m \omega^2 \hat{x}^2 \right) \psi(x) =\left( -\frac{\hbar^2}{2m}\frac{\mathrm{d}^2}{\mathrm{d}x^2} +\frac{1}{2} m \omega^2 x^2 \right) \psi(x) =E \, \psi(x) $$

can be solved with the help of ladder operators

$$ \hat{a} = \sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} + {\frac{\mathrm{i}}{m \omega}} \hat{p} \right) \quad\text{and}\quad \hat{a}^\dagger = \sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} - {\frac{\mathrm{i}}{m \omega}} \hat{p} \right). $$

Schrödinger equation for a hydrogen atom can also be solved by this way, although it is much more complicated.

Thus, for a general differential equation like

$$ y''(x) + P(x)\,y'(x) + Q(x)\,y(x) + R(x) = 0, $$

can we solve it with "ladder operators"?

Answer 1

It turns out that many of the differential equations of mathematical physics are closely related to expressions of Casimir invariants of some Lie groups expressed in some appropriate representation. In addition, the majority of special functions commonly found in mathematical physics are also simply related to the action of generators of Lie algebras on basis states.

As explicit examples, the angular part of the Laplacian in spherical coordinates is basically the $su(2)$ Casimir operator $\vec L\cdot \vec L$ operator in differential form; the 2nd order ODE in $\theta$ equation is nothing but the Legendre differential equation. Indeed there is a deep connection between special functions and the common differential equations of mathematical physics; details can be found in the textbooks of

• Talman [Talman, James D. Special functions: a group theoretic approach. (1968)],
• Vilenkin [Vilenkin, Naum Iakovlevich. Special functions and the theory of group representations. Vol. 22. American Mathematical Soc., 1978] or
• Miller [Miller, Willard. Lie theory and special functions (1968)].

If the underlying algebra (or group) is one of the semi-simple algebras, then the whole machinery of raising and lowering operators can be invoked, with these operators mapping directly to generalized ladder operators. Even if the algebra is not semi-simple (like, for instance, the Euclidean algebras) it is quite possible to define raising and lowering operators although some finesse is required and there are technical issues.

For instance, in the case of $e(2)$, one can define "weight states" $\vert m\rangle$ which are eigenstates of the rotation generator $\hat L_z$. The two generators of translation in the plane can be arranged as $p_+$ and $p_-$ so that $p_\pm \vert m\rangle\propto \vert m\pm 1\rangle$.

There is a priori no need for hermiticity. If the underlying algebra (or group) is compact, then the finite dimensional irreps are equivalent to unitary representations, so that $\Gamma(A_+)$ can be related to $\Gamma(A_-)$. This elegant paper gives examples of finite dimension, non-unitary but indecomposable representations of $e(2)$ for which $p_+$ and $p_-$ act as raising and lowering operators for which the matrix representation of $p_+$ is unrelated to that of $p_-$. Of course this is linked to the lack of unitarity in the representation.

(If you think indecomposable representations are out of bounds, check the last paper of Dirac before he passed away, called "The future of Atomic Physics", in which Dirac lobbies for increased attention to indecomposable representations. Indecomposable representations were investigated in the context of unstable particles in this paper Raczka, R. "A theory of relativistic unstable particles." Ann. d. IHP A 19 (1973): 341.

Answer 2

Hermitian operators, like the Hamiltonians you've mentioned, have the property of admitting real eigenvalues with orthogonal eigenfunctions. Ladder operators are then simply the operators that take you from one eigenfunction to a neighboring eigenfunction. So any hermitian operator will also admit ladder operators.

It is important to note that the general, second-order differential equation that you've mentioned may not be hermitian, and if the operator in question is not hermitian, there may not exist a complete set of eigenfunctions that span the range of solutions. However, decomposition onto an orthonormal basis set of functions and use of ladder operators can often be numerically useful even for non-hermitian operators.

On top of their many practical applications, there also exists deep connections between ladder operators and group theory, which I'll leave to the references.

Here's a few examples of ladder operators used outside of their normal scope:

Generalized Laguerre Functions, ${\bf su}(1,1)$

Spherical Harmonics, ${\bf so}(2,1)$ and ${\bf so}(3,2)$