Hamiltonian, hisenberg's eqn of motion etc

[subny]

Homework Statement

A particle of mass m and charge q is subject to a uniform electrostatic
eld ~E
.
(a) Write down the Hamiltonian of the particle in this system
(Hint: consider the potential energy of an electric dipole);
(b) Find the Heisenberg equation of motion for the expectation value of the position operator ~r, i.e. find the expression for d<r>/dt .
(c) Find now an expression for d^2<r>/dt^2
(d) Rearrange the last expression to show that this is Newton's
second law of motion.

Homework Equations

The Attempt at a Solution

how to get the potential given the field - once the potential is known the hamiltonian can be solved.

for part b, do we find [r,H] ??

 

[TSny]

Welcome to PF!

Think about how to write an expression for the force (vector) on the charge. Then think about how to use the force to get the potential energy.

Or, you can follow the hint if you are already familiar with the potential energy of a dipole in an electric field.

 

[subny]

F=qE

how to get the PE from this ??

 

[TSny]

Do you remember how you get the potential energy of a spring, $U = \frac{1}{2}kx^2$, from the force of a spring, $F=-kx$?

 

[subny]

no i do not.

but for the sum i used

v= q.integration (Edr)

 

[TSny]

v= q.integration (Edr)

OK, the negative integral of the force gives the potential energy for a conservative force. Thus, the potential energy at a point $p$ is

$V(p) = -\int_{r.p.}^p q\textbf{E} \cdot \,\mathrm{d}\textbf{s}$

Here, the integral is a line integral along any path connecting the reference point (r.p.) to point $p$. The reference point is the point where you choose $V = 0$, say the origin of your coordinate system. The integrand contains the dot product between $\textbf{E}$ and an infinitesimal displacement $\mathrm{d}\textbf{s}$ along the path. Since the field is uniform, you can carry out the integration.

 

[subny]

yes thanks i could do it afterall