is given by
~j= ̄h
2 iμ
[ψ∗∇~ψ−(∇~ψ∗)ψ+
2 ie
̄hc
Aψ~ ∗ψ].
Remember the flux satisfies the equations∂(ψ
∗ψ)
∂t +
∇~~j= 0.
- Consider the problem of a charged particle in an external magnetic fieldB~ = (0, 0 ,B) with
the gauge chosen so thatA~= (−yB, 0 ,0). What are the constants of the motion? Go as far as
you can in solving the equations of motion and obtain the energy spectrum. Can you explain
why the same problem in the gaugesA~= (−yB/ 2 ,xB/ 2 ,0) andA~= (0,xB,0) can represent
the same physical situation? Why do the solutions look so different?
- Calculate the top left 4×4 corner of the matrix representation ofx^4 for the harmonic oscillator.
Use the energy eigenstates as the basis states.
- The Hamiltonian for an electron in a electromagnetic field can be written asH = 21 m[~p+
e
c
A~(~r,t)]^2 −eφ(~r,t) + e ̄h
2 mc~σ·
B(~~r,t). Show that this can be written as the Pauli Hamiltonian
H=
1
2 m
(
~σ·[~p+
e
c
A~(~r,t)]
) 2
−eφ(~r,t).
20.7 Sample Test Problems
- A charged particle is in an external magnetic field. The vector potential is given byA=
(−yB, 0 ,0). What are the constants of the motion? Prove that these are constants by evalu-
ating their commutator with the Hamiltonian.
- A charged particle is in an external magnetic field. The vector potential is given byA=
(0,xB,0). What are the constants of the motion? Prove that these are constants by evaluating
their commutator with the Hamiltonian.
- Gauge symmetry was noticed in electromagnetism before the advent of Quantum Mechanics.
What is the symmetry transformation for the wave function of an electron from which the
gauge symmetry for EM can be derived?
