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pset3 - t What is the probability that a measurement of the...

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Spring Term 2003 22.02 Introduction to APPLIED NUCLEAR PHYSICS Problem Set #3 1. A particle of mass m 0 is just being bound by a one-dimensional potential well of width 2 a and depth V 0 . What is the minimum value of: V 0 6 = 0 for an ODD parity eigenfunction? (Hint: This problem can be solved with the simple de Broglie wave concept, but this must be justi fi ed. Otherwise, the more tedious way of using boundary conditions at x = a and x = a must be used. In either case ask yourself what the condition of “just being bound” implies.) X V(x) X=a X= -a -V_o 2. Libo ff , problems 5.12, 5.19, 5.25
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3. Suppose system of Question 1 from Problem Set #2 is prepared at t = 0, with wave- functions, ψ = a 1 ψ 1 + a 3 ψ 3 . Evolve the system according to the Schr¨ o dinger equation to derive ψ ( x, t ) and compute the probability that a measurement of the particle’s position fi nds the particle between x = L 2 and x = L 2 + x , at time
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Unformatted text preview: t . What is the probability that a measurement of the energy of the system f nds: a) E = E 1 , b) E = E 2 , and c) E = E 3 , where the eigenstate energies are E n = ¯ h 2 π 2 2 mL 2 n 2 ? 4. For a quantum particle moving in a 3D potential, V ( x ) = V ( x, y, z ), which directions, if any, of the particle’s momentum are conserved if the potential takes the following form: (a) V = ax + by 2 + cz 3 , where the constants, a , b , and c are all positive. (b) V = a arctan( x/L ) (c) V = b exp( − y 2 /d 2 ) 5. For which systems characterized by the potentials listed below, do the energy and parity observables have common eigenfunctions? (a) V = e 2 /r ; where r = √ x 2 + y 2 + z 2 . (b) V = ax + by 2 + cz 3 (c) V = axy 6. Libo f , problems 9.3, 9.25(a), 9.30, 9.34, 9.35....
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