a) Consider an electron in a hydrogen atom. Describe the time evolution of a hermitian operator...

 

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a) Consider an electron in a hydrogen atom. Describe the time evolution of a hermitian operator Ô (with no explicit time dependence), corresponding to a particular observable for this quantum system. Further, state the time evolution of its eigenkets (base kets). Please give the relevant expressions in Schrödinger picture and Heisenberg picture. b) Now consider that the electron, as mentioned above, is subject to a (weak) constant external electric field E = E3. Comment on the time evolution of (L₁), (3), (12), where the expectation values are evaluated in the ground state (R10(r)YO (0, 6)) of the unperturbed hydrogen atom. Which of these expectation values are conserved? Please explain, if applicable, using symmetry argument. Consider the spin part of the electron is in a (normalized) quantum state ) = a 1/2, 1/2) + 31/2, -1/2), where {1/2, 1/2), [1/2, -1/2)} denote the eigenstates of $3. Evaluate (a Sla) for i = {1,2}. How does the expectation value (a|S2|a) transform if we perform a spatial rotation by an angle 0 about the third axis (3)? Show by explicit calculation. d) Show that (o.p) (o.p) = p2, where o, denotes the i-th Pauli matrix. a) Consider an electron in a hydrogen atom. Describe the time evolution of a hermitian operator Ô (with no explicit time dependence), corresponding to a particular observable for this quantum system. Further, state the time evolution of its eigenkets (base kets). Please give the relevant expressions in Schrödinger picture and Heisenberg picture. b) Now consider that the electron, as mentioned above, is subject to a (weak) constant external electric field E = E3. Comment on the time evolution of (L₁), (3), (12), where the expectation values are evaluated in the ground state (R10(r)YO (0, 6)) of the unperturbed hydrogen atom. Which of these expectation values are conserved? Please explain, if applicable, using symmetry argument. Consider the spin part of the electron is in a (normalized) quantum state ) = a 1/2, 1/2) + 31/2, -1/2), where {1/2, 1/2), [1/2, -1/2)} denote the eigenstates of $3. Evaluate (a Sla) for i = {1,2}. How does the expectation value (a|S2|a) transform if we perform a spatial rotation by an angle 0 about the third axis (3)? Show by explicit calculation. d) Show that (o.p) (o.p) = p2, where o, denotes the i-th Pauli matrix.

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a In the Schrdinger picture the time evolution of a hermitian operator with no explicit time depende... View the full answer