2. Consider an electron (charge -e) moving in a uniform magnetic field along z-axis (B = B2). We choose a particular gau

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2 Consider An Electron Charge E Moving In A Uniform Magnetic Field Along Z Axis B B2 We Choose A Particular Gau 1 (79.04 KiB) Viewed 22 times

2. Consider an electron (charge -e) moving in a uniform magnetic field along z-axis (B = B2). We choose a particular gauge so that the vector potential is A = -Byx and the scalar potential is zero. The Hamiltonian in this case is 2 1 e 2 1 eB 1 H = - 2 m (F + ² A) ³² = 2₂mm (P² - ² ² 4 ) ² + 2 ² (²+²) - Px 2m с 2m Let =p+ Ã be the kinetical momentum. (You might use w = eB whenever it simplifies the expressions.) mc (a) Calculate the commutators (i) [T, Ty], (ii) [Tx, T₂] and (iii) [Ty, T₂]. (b) Calculate the time derivative of (Tr)t, (Ty)t and (₂) t. (c) Solve the equations you have obtained above and express (T), (Ty) and (₂) in terms of t and their initial values at t = 0. [Hint: it might be easier to work with (TT) ±i(Ty).] (d) Calculate the time derivatives of (x), and (y). (e) Integrate the results in part (d).