Prelim 2 & Solutions - this potential well(b Solve the...

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ECE 306 Preliminary Exam 2 Instructions: Attempt every problem. You may use a single sheet of notes to assist you on this exam. A calculator is not needed and not allowed during this exam. 1. Consider a hydrogen atom in a quantum mechanical state described as 0 , 2 , 3 | 1 , 1 , 2 | 0 , 0 , 1 | | c b a + + = ψ where the m l n , , | states represent the complete set of normalized eigenstates spanning the space and a,b,c are the relative amplitudes of each eigenstate. (a) Normalize Ψ in terms of a, b and c. (b) What is the probability that an energy measurement will yield the value E 2 (E 2 is the energy of the atom in the n = 2 state)? (c) Write down the time evolution of the wave function Ψ . (d) Determine the expectation value < L z > for this system. 2. Consider an infinite potential well in 3-dimensions. The potential is described as V = 0 for –a < x < a, -a < y < a, and –a < z < a. The potential V is infinite outside of this cubic region. (a) Write the complete time-independent Schroedinger equation for an electron inside
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Unformatted text preview: this potential well. (b) Solve the Schroedinger equation to determine the wave function, Ψ (x, y, z) = X(x)Y(y)Z(z) of this system. You do not need to normalize the wave function. (c) What are the energies of the 1 st and 2 nd states? (d) What are the degeneracies of the 1 st and 2 nd states? Neglect spin. 3. The Hamiltonian for the rotational energy of a system is given by 2 2 2 2 1 2 1 ) ( 2 1 z y x L I L L I H + + = where I n is the moment of inertia in the respective orientations of the system. (a) Write down the eigenfunctions for this system which satisfy H Ψ = E Ψ . What are the quantum numbers for this system (not numerical values, but tell me descriptively how many quantum numbers there will be and what they represent physically)? (b) What are the eigenenergies of this system? Write down the general expression in terms of the quantum numbers for the system....
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