Velocity operator for a Hamiltonian that satisfies the generalized Schrodinger equation

Question

The velocity operator is defined as $\mathbf{v}=i[H,\mathbf{r}]$ for the Hamiltonian $H$ satisfying $H\psi=\epsilon \psi$. This can be obtained from the Ehrenfest theorem. I'm wondering if $\mathbf{v}=i[H,\mathbf{r}]$ still holds for $H$ satisfying the generalized Schrodinger equation $H\psi=\epsilon S\psi$, where $S$ is the overlap operator?

Answer 1

Consider a time-independent operator $\mathbf{r}$, its expectation value, and the time derivative of the expectation value (by applying the product rule for derivatives):

\begin{align} \tag{1} \langle \mathbf{r}\rangle &\equiv \int \psi^* \mathbf{r}\psi \,\,\, dr\\ \frac{\rm{d}}{\textrm{d}t}\langle \mathbf{r}\rangle &= \int \left(\frac{\rm{d}}{\textrm{d}t}\psi^*\right) \mathbf{r}\psi \,dr + \int \psi^* \mathbf{r}\left(\frac{\rm{d}}{\textrm{d}t}\psi\right)\,dr\tag{2}. \end{align}

Now let's plug in the Schrödinger equation (with $\hbar=1$ since that's what you had) and its conjugate:

\begin{align}\tag{3} \frac{\rm{d}}{\textrm{d}t}\psi &= -\textrm{i}H\psi \\ \frac{\rm{d}}{\textrm{d}t}\psi^* &= \textrm{i}H\psi^*. \tag{4}\\ \end{align}

The result is:

\begin{align} \frac{\rm{d}}{\textrm{d}t}\langle \mathbf{r}\rangle &= \int \left(\textrm{i}H\psi^* \right) \mathbf{r}\psi \,dr + \int \psi^* \mathbf{r}\left(-\textrm{i}H\psi \right)\,dr\tag{5} \\ \langle \mathbf{v}\rangle &\equiv\textrm{i} \left[H ,\langle \mathbf{r}\rangle \right].\tag{6} \end{align}

I never explicitly mentioned anyhting about eigenvalues and never used $H\psi = \epsilon\psi$ or $H\psi = \epsilon S \psi$: I just used:

• the definition of an expectation value $\langle \cdot \rangle$,
• the product rule for derivatives,
• the Schrödinger equation (see the derivation of the Schrödinger equation in Section 2.2 of this PDF to see why this is the equation of motion for $\psi$ if we want its norm to be preserved as Born's rule relates wavefunction norms to probabilities: i.e. unitary evolution), so we have used:
  • Born's rule,
  • the law that probabilities must sum up to 1,
• the definitions:
  • $\langle \mathbf{v}\rangle \equiv \frac{\rm{d}}{\textrm{d}t}\langle \mathbf{r}\rangle$,
  • $[A,B] \equiv AB-BA$,
  • $\textrm{i}^2=-1.$

On page 20 of the same aforementioned PDF you can see how the Heisenberg equation of motion is derived, if you want to do something similar without expectation values.