- #1

EightBells

- 11

- 1

- Homework Statement
- Find eigenenergies for a charged particle in magnetic ﬁeld B and in a parabolic central potential, ##U(r) = \frac {kr^2} 2##. The particle has mass m and charge q.

- Relevant Equations
- Hamiltonian for charged particle in magnetic field ##H = \frac 1 {2m} \left( p- \frac {qA} {c^2} \right)^2 +q \phi##

Once I know the Hamiltonian, I know to take the determinant ##\left| \vec H-\lambda \vec I \right| = 0 ## and solve for ##\lambda## which are the eigenvalues/eigenenergies.

My problem is, I'm unsure how to formulate the Hamiltonian. Is my potential ##U(r)## my scalar field ##\phi##? I've seen equations relating B and E fields to the scalar and vector fields, ##\vec B = \nabla {} \times \vec A## and ##\vec E = -\nabla \phi - \frac 1 c \frac {d \vec A} {dt}##, but I'm not sure how to incorporate them or if I even need to.

Do I need an equation for the vector potential ##\vec A##? What is the momentum ##p## in this case?

Thanks!

My problem is, I'm unsure how to formulate the Hamiltonian. Is my potential ##U(r)## my scalar field ##\phi##? I've seen equations relating B and E fields to the scalar and vector fields, ##\vec B = \nabla {} \times \vec A## and ##\vec E = -\nabla \phi - \frac 1 c \frac {d \vec A} {dt}##, but I'm not sure how to incorporate them or if I even need to.

Do I need an equation for the vector potential ##\vec A##? What is the momentum ##p## in this case?

Thanks!