3. Consider the motion of the electron of a Hydrogen atom. whose Hamiltonian is H = p²/(2m) - e²/f, where e is the charg

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3 Consider The Motion Of The Electron Of A Hydrogen Atom Whose Hamiltonian Is H P 2m E F Where E Is The Charg 1 (371.1 KiB) Viewed 15 times

3 Consider The Motion Of The Electron Of A Hydrogen Atom Whose Hamiltonian Is H P 2m E F Where E Is The Charg 2 (237.73 KiB) Viewed 15 times

3. Consider the motion of the electron of a Hydrogen atom. whose Hamiltonian is H = p²/(2m) - e²/f, where e is the charge of the electron, mis its mass, and of the opera- tor representing the distance from the origin. The eigenstates of ♬ are labelled as n.l.m). where n = 1, 2,... is the principal quantum number, while (+1) and hmm are the eigenvalues of Land L. respectively, with L the angular momentum operator. (a) The particle is prepared in the eigenstate of II with n = 2, 6 – 1 and m-1. Using the integral for ^c-¹dr = kl, which you can assume without proof, calculate the variance of F. [6] (b) The angular part of the wavefunction of the state with 7 = 2,0 = 1 and m = 1 is given by the spherical harmonic Yi1(0,6). Using Eq. (Syn-3) in the Synoptic page, show that Y₁ (0,0) = -√ sin fe 3 8T (c) The atom is now subjected to an external potential with y the position operator along the y axis of the reference frame. Such external potential can be considered perturbative with respect to the Hamiltonian H. Use perturbation theory to find the first-order correction in the perturbative parameter & to the energy of the state considered in part (a). You may use, without proof, the integrals 21 16 do sin² o = -. do de sin ²015 [8] (d) Assume now that the electron is prepared in the state (0))=√(₂ (|n = 2, € – 1, m = 1) − \n = 2, ( = 1, m – −1)) and is subjected to an external magnetic field directed along the z axis of the reference frame. Under these conditions, you can assume without proof that the Hamiltonian of the system thus becomes HI₁ =  +wL., where is a parameter depending on the amplitude of the external magnetic field. 1. Show that the time evolved state (t)) = e-it/h|u(0)) reads |v(t)) = ( (e¯²2, 1, 1) – e¹|2,1, -1)) /v2. [3] ii. Calculate the expectation value of Lover (4)). e Nove
Spherical harmonics 21 + 1 (1 - \m)! pm (cos 0)eimo, 4T (1+m)! (Syn-3) (m 20) are the associated Legendre functions, and P(x) Yim (0,0) = (−1)(m+|m|)/2 where Pm (cos 0) 1 sind Picos) (d cos 0) 2(x²-1)² are the Legendre polynomials.