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Practise-Final-exam-June - Why h is the Planck constant and...

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110A Physical Chemistry: Quantum Mechanics Practise Final Exam; December 1 st , 2010 Problem # 1 (45 pts) Compute the radius of the sphere that encloses a 50% probability of finding a hydrogen 1s electron; repeat the calculation for a 90% probability. ( 15 pts) Prove that the average value of the position operator in the 1s and 2s states of hydrogen is 3a 0 /2Z and 6a 0 /Z, respectively. a 0 is the Bohr radius and Z the nuclear charge. (15 pts) Consider the angular momentum operator operators L 2 , L x , L y and L z and give their commutation rules. Which physical principle determines these commutation rules? Can one observe simultaneously values of L 2 , L x , L y and L z ? Do L 2 and L z have common eigenfunctions? (15 pts) Problem # 2 (11 pts) Use a trial function of the form exp (- β x 2 ) to calculate the ground state energy of a harmonic oscillator in one dimension. β is a variational parameter ( β > 0). Compare your results with the exact energy h ν /2. Is the agreement good?
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Unformatted text preview: Why? [ h is the Planck constant and ν the frequency ]. (21 pts) Problem # 3 (24 pts) Identify the unperturbed Hamiltonian H (0) , perturbation H (1) , ground state of unperturbed Hamiltonian ψ (0) and ground state energy of the unperturbed Hamiltonian, E (0) ,for the following problems: 1) A 1D oscillator governed by the potential v(x) = (k/2) x 2 + (g/6)x 3 + (b/24)x 4 (12 pts) 2) A particle constrained to move in one dimension in the region [0,a] with the potential V(x) = 0 for x within 0 and a/2 and V(x) = b for x within a/2 and a. (12 pts) 3) Problem # 4 (10 pts) The term symbol of a certain atom is 3 P : what are the spin, angular and total angular momentum of the atom? Does the atom have a closed shell configuration? (Explain why). Do you expect the state 1 P 0 of the same atom to have higher or lower energy? (Explain why). Hydrogen Atom Radial Schroedinger Equation:...
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