Problem Set 3 for Physics 324: Harmonic Oscillator and Quantum States, Assignments of Quantum Mechanics

Download Problem Set 3 for Physics 324: Harmonic Oscillator and Quantum States and more Assignments Quantum Mechanics in PDF only on Docsity! Physics 324 Problem Set # 3 Due 10/27/03 1. A harmonic oscillator consists of a 1 g mass on a spring. Its oscillation frequency is 2 Hz and its amplitude of oscillation is 1 cm. Approximately, what is the maginitude of the energy quantum number associated with this classical oscillator? 2. From Eqns. 2.48 and 2.54 in your textbook, we find that the normalized wavefunction for the lowest energy state of a harmonic oscillator potential can be written as: ψ0(x, t) = (mω πh̄ )1/4 e−mωx 2/2h̄ e−iωt/2 (a) Using Eqns. 2.51 and 2.54 in your textbook, find an expression for the normalized wavefunction of the first excited state of the harmonic oscillator, ψ1(x, t). (b) What are the expectation values of position and momentum, 〈x〉 and 〈p〉, for the two states ψ0(x, t) and ψ1(x, t) ? (You may use symmetry arguments to reduce the work required.) (c) Compute 〈x2〉 and 〈p2〉 for state ψ0(x, t). (d) Use the results from parts (b) and (c) to verify the uncertainty principle for state ψ0(x, t). (e) Using the results that 〈KE〉 = 〈p2/2m〉 and 〈V (x)〉 = 〈mω2x2/2〉, compute 〈KE+V (x)〉 for state ψ0(x, t). Is the result what you expect? 3. An alternate way of expressing the energy eigenstates (solutions to the time independent Schrodinger equation) for the harmonic oscillator potential comes from the analytic approach, summarized in Eqn. 2.69 in your text: ψn(x) = (mω πh̄ )1/4 1√ (2n n!) Hn(ζ)e−ζ 2/2 where ζ = √ mω h̄ x and Hn(ζ) are Hermite polynomials which satisfy the following recursion relation: Hn+1(ζ) = 2ζHn(ζ) − 2nHn−1(ζ) (a) Using the results that H0(ζ) = 1 and H1(ζ) = 2ζ, use the recursion relation to compute Hn(ζ) for n = 2 through n = 6. (b) A particle in a harmonic oscillator potential is prepared in the initial state: ψ(x, 0) = A [ ψ0(x) + 2ψ1(x) + ψ2(x) ] where ψn(x) is the normalized solution to the time independent Schrodinger equation for the harmonic oscillator potential for the nth energy level. Without explicitly evaluating any integrals, find A. (c) Without explicitly evaluating any integrals, find the expectation value for the energy, 〈Ĥ〉, for the state ψ(x, t) in part (b). 1