Question

In classical mechanics, you should have seen a Hamiltonian of a particle of charge $q$ in an electromagnetic field \begin{equation} H = \frac{1}{2m} \left[p - \frac{q}{c}A(r)\right]^2 + q\phi(r) \end{equation} where $A(r)$ and $\phi(r)$ are the vector and electric potentials, and $r = (x, y, z)$. To \textquotedblleft quantize\textquotedblright this problem, we replace the momentum and position with operators: \begin{equation} \hat{H} = \frac{1}{2m} \left[\hat{p} - \frac{q}{c}A(\hat{r})\right]^2 + q\phi(\hat{r}). \end{equation} A. Derive the Heisenberg equations of motion for $\hat{r}^H(t)$ and $\hat{p}^H(t)$. B. Eliminate $\hat{p}^H(t)$ from your equations and obtain a second-order-in-time differential equation on $\hat{r}^H(t)$ that should resemble the Newton's law equation: $m\ddot{r}(t) = $ $\frac{q}{c}\dot{r}(t) \times B + qE$ [Hint: While the described procedure works for arbitrary coordinate-dependent EM fields, let us simplify by considering spatially-uniform fields $B = \text{const}$ and $E = \text{const}$. Thus let us use a specific electromagnetic gauge $\phi(r) = -E \cdot r$ and $A(r) = \frac{1}{2}[B \times r].$]

          In classical mechanics, you should have seen a Hamiltonian of a particle of charge $q$ in an
electromagnetic field
\begin{equation}
H = \frac{1}{2m} \left[p - \frac{q}{c}A(r)\right]^2 + q\phi(r)
\end{equation}
where $A(r)$ and $\phi(r)$ are the vector and electric potentials, and $r = (x, y, z)$. To \textquotedblleft quantize\textquotedblright
this problem, we replace the momentum and position with operators:
\begin{equation}
\hat{H} = \frac{1}{2m} \left[\hat{p} - \frac{q}{c}A(\hat{r})\right]^2 + q\phi(\hat{r}).
\end{equation}
A. Derive the Heisenberg equations of motion for $\hat{r}^H(t)$ and $\hat{p}^H(t)$.
B. Eliminate $\hat{p}^H(t)$ from your equations and obtain a second-order-in-time differential
equation on $\hat{r}^H(t)$ that should resemble the Newton's law equation: $m\ddot{r}(t) = $
$\frac{q}{c}\dot{r}(t) \times B + qE$
[Hint: While the described procedure works for arbitrary coordinate-dependent EM fields,
let us simplify by considering spatially-uniform fields $B = \text{const}$ and $E = \text{const}$. Thus let
us use a specific electromagnetic gauge $\phi(r) = -E \cdot r$ and $A(r) = \frac{1}{2}[B \times r].$]

In classical mechanics, you should have seen a Hamiltonian of a particle of charge q in an
electromagnetic field

H = (1)/(2m)[p - (q)/(c)A(r)]^2 + qϕ(r)

where A(r) and ϕ(r) are the vector and electric potentials, and r = (x, y, z). To “quantize”this problem, we replace the momentum and position with operators:

Ĥ = (1)/(2m)[p̂ - (q)/(c)A(r̂)]^2 + qϕ(r̂).

A. Derive the Heisenberg equations of motion for r̂^H(t) and p̂^H(t).
B. Eliminate p̂^H(t) from your equations and obtain a second-order-in-time differential
equation on r̂^H(t) that should resemble the Newton's law equation: mr̈(t) =
(q)/(c)ṙ(t) × B + qE
[Hint: While the described procedure works for arbitrary coordinate-dependent EM fields,
let us simplify by considering spatially-uniform fields B = const and E = const. Thus let
us use a specific electromagnetic gauge ϕ(r) = -E · r and A(r) = (1)/(2)[B × r].]

Added by Colin L.

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