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Quantum motion of a charged particle in a magnetic field
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[QUOTE="EightBells, post: 6334469, member: 621301"] [B]Homework Statement:[/B] 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. [B]Relevant Equations:[/B] 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! [/QUOTE]
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Quantum motion of a charged particle in a magnetic field
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