Total problems: 5 B = V, Ay = B Problem 5 (25 points). A particle of charge q moves in a uniform magnetic field whose di

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Total Problems 5 B V Ay B Problem 5 25 Points A Particle Of Charge Q Moves In A Uniform Magnetic Field Whose Di 1 (108.95 KiB) Viewed 17 times

Please answer parts d, e and f
Total problems: 5 B = V, Ay = B Problem 5 (25 points). A particle of charge q moves in a uniform magnetic field whose direction is in the z-direction (B = BZº). Let A be the vector potential. (a) Define the mechanical momentum lĩ = p - qĂ, where p is the canonical momentum. Evaluate [IL, Ily]. (Answer: iħqB) (b) To represent the magnetic field, the vector potential can be chosen as Ax 2x, A, = 0. Why? Ex (c) Write down the Hamiltonian of the particle in terms of ILY, Ily, and pz using the choice in (b). (d) Justify why the eigenvalues of the Hamiltonian can be written as the sum of the eigenvalues of the pz term(s) and the terms containing IIx and Ily. (e) Figure out the eigenvalue of the pz term(s) in (c), assuming ħk is the continuous eigenvalue of the p:operator. (f) By comparing the terms containing IIx and Ily in the Hamiltonian written in (c) and the result obtained in (a) with those of the 1-D harmonic oscillator problem, show that the eigenvalues of the terms containing IIx and IIy are qB lh P (nt 2 3 mc where n = 0, 1, 2, ....