# "Physically distinct" Hermitian operators with the same eigenspaces

I understand that Hermitian operators can be decomposed in terms of their eigenbasis: $$$$H = \sum_i\lambda_i|i\rangle\langle i|$$$$ where the $$\lambda_i$$ are all real. I've kept the summation index ambiguous since I am interested in both finite- and infinite-dimensional operators. My question is, does anyone know any examples of two "physically relevant" Hermitian operators (e.g meaningful observables, or Hamiltonians for some real-world systems) that have the same eigenspaces but not all the same eigenvalues? That is: $$$$H_1 = \sum_i\lambda_i|i\rangle\langle i|;\;\;\;H_2 = \sum_{i}\mu_i|i\rangle\langle i|$$$$ where $$\lambda_i\neq\mu_i$$ for at least one $$i$$. Ideally $$H_1$$ and $$H_2$$ are also "physically distinct," e.g not just two differently scaled versions of the same system. For example, one could assemble an operator $$H_{HO}'$$ with the same eigenstates as the harmonic oscillator Hamiltonian $$H_{HO}$$, but with $$H_{HO}'|n\rangle = n^2|n\rangle$$ or $$\frac{1}{n^2}|n\rangle$$ or something. $$H_{HO}'$$ then commutes with $$H_{HO}$$, but I feel that they correspond to very different physics.

I'd appreciate any examples of pairs of operators as described above, or a more general/abstract statement on such pairs of operators if anyone knows one.

• Jan 2 at 7:12
• Would $H_1=H$, $H_2=L^2$ and $H_3=L_z$, with - say - $\vert n\ell m\rangle$ hydrogen atom kets, fit the bill? Jan 2 at 8:56

Let $$A$$ be a self-adjoint operator with spectral decomposition $$A = \int_{\sigma_A} \lambda \ \mathrm dP^A(\lambda)$$ which reduces to your expression $$\sum_i \lambda_i |i\rangle\langle i|$$ in the case that the spectrum is purely discrete and all of the eigenspaces are 1D. Given some (Borel-measurable) function $$f(A)$$, we can define $$f(A) := \int_{\sigma_A} f(\lambda) \ \mathrm dP^A(\lambda) \rightsquigarrow \sum_i f(\lambda_i) |i\rangle\langle i|$$

$$A$$ and $$f(A)$$ satisfy the requirements you're looking for. Certainly one ubiquitous example would be the position operator $$X$$ and any arbitrary potential operator $$V(X)$$, with the caveat that these operators are a bit subtle because of their continuous spectra. The Hamiltonian operator $$H$$ and the time-evolution operator $$U=\exp[-iHt/\hbar]$$ are another example, in both finite and infinite dimensions.

This is intended as an extended comment: let $$A$$ and $$B$$ two hermitian matrices sharing a common eigenbasis $$(\phi_i)_i$$, such that each of the multiplicities of the eigenvalues ($$(\lambda_i)_i$$ for $$A$$ and $$(\mu_i)_i$$ for $$B$$) is one. Then not only do $$A$$ and $$B$$ commute: each of them is a polynomial in the other one.

Indeed, there are, from Lagrange interpolation, two polynomials $$P$$ and $$Q$$ such that for all $$i$$, $$P(\lambda_i) = \mu_i$$ and $$Q(\mu_i) = \lambda_i$$. Therefore, $$P(A) = B$$ and $$Q(B) = A$$.

So, both $$A$$ and $$B$$ can be seen as relabellings of each other, and in some loose sense, they can be thought as equivalent.

Of course, if the eigenvalues have a dimension, it may not be meaningful to feed them to nonmonomial polynomials, so at some point physics may consider $$A$$ and $$B$$ as inequivalent.