- D And E Please 1 (219.24 KiB) Viewed 56 times
(d) and (e) please.
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answerhappygod
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(d) and (e) please.
(d) and (e) please.
3 (ax Suppose that we know the eigenstates, In), of a Hamiltonian, Ho, with eigenvalues e(n) and that the system is acted upon by an additional, time independent potential, \V, that we wish to treat perturbatively. By expanding the eigenvalues and eigenstates of H + V as a power series in 1: EN + XE + 1² EN + ... |N) nNĒM (11N1) + 12|N2) +...] with n=1 - \n)(n], show that EN = (n|V\n) = = = and Ēn|N1) = (elm) – PHPn)"? ,V\n). [4 marks] (b) Suppose that a hydrogen atom is placed in a magnetic field B = B2, where B is a constant and 2 is a unit vector in the z-direction. Given that the interaction between the electron and the magnetic field is Minneto B. I, where I is the angular momentum operator for the electron, what is the effect (to Thest order in the applied field) on the energy of the 2s and the three 2p states? 14 marks] (c) Explain why the formalism in part (a) breaks down if a subset of the excited states of the system is degenerate and briefly summarise the procedure used to resolve this problem. marks (d) Suppose that the hydrogen atom in its first excited state (n = 2) is placed in a constant electric field of magnitude E pointing along the z-axis, so that the atom feels a perturbing potential V = eEz, where -e is the charge of the electron and z its z-coordinate. Prove that [L2, 2] = 0 and hence prove that this potential does not mix states with different eigenvalues of Lz. [4 marks] (e) Given that the matrix element of the perturbing potential for the system described in part (d) above between the 2s state, (280), and the 2p-state with eigenvalue of Lz = 0, 2po), is: (250 V2po) = -3eEao, where a, is the Bohr radius of hydrogen, derive the eigenvalues and eigenvectors of V for this system. [4 marks]
3 (ax Suppose that we know the eigenstates, In), of a Hamiltonian, Ho, with eigenvalues e(n) and that the system is acted upon by an additional, time independent potential, \V, that we wish to treat perturbatively. By expanding the eigenvalues and eigenstates of H + V as a power series in 1: EN + XE + 1² EN + ... |N) nNĒM (11N1) + 12|N2) +...] with n=1 - \n)(n], show that EN = (n|V\n) = = = and Ēn|N1) = (elm) – PHPn)"? ,V\n). [4 marks] (b) Suppose that a hydrogen atom is placed in a magnetic field B = B2, where B is a constant and 2 is a unit vector in the z-direction. Given that the interaction between the electron and the magnetic field is Minneto B. I, where I is the angular momentum operator for the electron, what is the effect (to Thest order in the applied field) on the energy of the 2s and the three 2p states? 14 marks] (c) Explain why the formalism in part (a) breaks down if a subset of the excited states of the system is degenerate and briefly summarise the procedure used to resolve this problem. marks (d) Suppose that the hydrogen atom in its first excited state (n = 2) is placed in a constant electric field of magnitude E pointing along the z-axis, so that the atom feels a perturbing potential V = eEz, where -e is the charge of the electron and z its z-coordinate. Prove that [L2, 2] = 0 and hence prove that this potential does not mix states with different eigenvalues of Lz. [4 marks] (e) Given that the matrix element of the perturbing potential for the system described in part (d) above between the 2s state, (280), and the 2p-state with eigenvalue of Lz = 0, 2po), is: (250 V2po) = -3eEao, where a, is the Bohr radius of hydrogen, derive the eigenvalues and eigenvectors of V for this system. [4 marks]