Normalization Factor - Introduction to Quantum Mechanics - Exam, Exams of Quantum Mechanics

Download Normalization Factor - Introduction to Quantum Mechanics - Exam and more Exams Quantum Mechanics in PDF only on Docsity! 1 2005 Ph.D. Qualifier, Quantum Mechanics DO ONLY 3 OUT OF 4 OF THE QUESTIONS Problem 1. A particle of mass m in an infinitely deep square well extending between x = 0 and x = L has the wavefunction Ψ(x, t) = A [ sin (πx L ) e−iE1t/~ − 3 4 sin ( 3πx L ) e−iE3t/~ ] , (1) where A is a normalization factor and En = n 2h2/8mL2. (a) Calculate an expression for the probability density |Ψ(x, t)|2, within the well at t = 0. (b) Calculate the explicit time-dependent term in the probability density for t 6= 0. (c) In terms of m, L, and h, what is the repetition period T of the complete probability density? Problem 2. Let us consider the spherical harmonics with l = 1. (a) Determine the eigenvalues for aLz, where a is a constant. (b) Determine the matrix for Lx for the basis set |lm〉 with l = 1 using the fact that L±|lm〉 = ~ √ (l ∓m)(l ±m + 1)|l, m± 1〉. (2) and that the step operators are given by L± = Lx ± iLy. (c) Determine the eigenvalues of aLx for the states with l = 1. (d) Determine the matrix for L2 from the matrices for L+, L−, and Lz. Problem 3. Consider a two-dimensional harmonic oscillator H0 = ~ωxa † xax + ~ωya † yay (3) with ~ωx  ~ωy. The number of excited states is given by N = nx + ny, where a†x|nx〉 = √ nx + 1|nx + 1〉 and a†y|ny〉 = √ ny + 1|ny + 1〉 (a) Express the normalized state with N = 2 with the lowest energy in terms of the step operators and the vacuum state |0〉, i.e. the state with no oscillators excited. The system is now perturbed by H1 = K(a † xay + a † yax). (4) (b) Calculate for the state found in (a): the correction in energy up to first order. (c) Express the correction in energy up to second order. (d) Give the lowest-order correction to the wavefunction. NOTE: The correction term to the wavefunction is given by |ψ1n〉 = ∑ m6=n 〈ψ0m|H1|ψ0n〉 E0n − E0m |ψ0m〉 (5)