# The Handbook of Mathematics, Physics and Astronomy Data is provided

The Handbook of Mathematics, Physics and Astronomy Data is provided KEELE UNIVERSITY EXAMINATIONS, 2011/12 Level II Thursday 12th January 2012, 09.30 – 11:30 PHYSICS/ASTROPHYSICS PHY-20006 QUANTUM MECHANICS Candidates should attempt ALL of PART A and TWO questions from PART B. PART A yields 40% of the marks, PART B yields 60%. A sheet of useful formulae can be found on page 8. NOT TO BE REMOVED FROM THE EXAMINATION HALL PHY-20006 Page 1 of 8 PART A Answer all TEN questions A1 Give a physical interpretation of the wave function Ψ in terms of the observed position of the particle and explain how this leads to the concept of a normalised wavefunction. [4] A2 Explain the following concepts and give an example of a physical system that demonstrates each concept. • Wave-particle duality. • Quantisation of energy. [4] A3 Calculate the expectation value for the position, hxi, of a particle with the normalized wave function √ Ψ(x, t) = 2π e−πx e−iωt x > 0. You may use the following integral without proof in your answer. Z ∞ rk 0 exp(−αr) dr = k! αk+1 [4] A4 State three differences between the predictions of classical physics and the predictions of quantum mechanics for the properties of a [4] particle in a harmonic oscillator potential, V (x) = 12 kx2 . A5 Sketch the energy eigenfunction, ψ1 and ψ2 , for the ground state and first excited state of the finite square-well potential V (x) = PHY-20006 0 −V0 0 x < −a −a ≤ x ≤ a x>a Page 2 of 8 [4] A6 Calculate the expectation value hEi and uncertainty ∆E for the energy of a particle with the wavefunction 1 1 Ψ(x, t) = √ ψ1 (x)e−iE1 t/h̄ + √ ψ2 (x)e−iE2 t/h̄ , 2 2 where E1 = 1 eV and E2 = 2 eV. [4] A7 Estimate the natural line width in Hz for a transition from an energy level with a lifetime δt = 10−8 s. [4] A8 Explain why identical quantum particles in the same region of space are also indistinguishable. [4] A9 List all possible values of J, the magnitude of the total angular momentum, for an electron with orbital angular momentum quantum number ℓ = 2. Give your answers in units of h̄. [4] A10 Explain the origin of “fine-structure” in the spectrum of the hydrogen atom. [4] /Cont’d PHY-20006 Page 3 of 8 PART B B1 Answer TWO out of FOUR questions A particle of mass m is trapped in the following potential: V (x) = ∞ 0 ∞ x<0 0≤x≤a x>a (a) Show that the solutions of the time independent Schrödinger equation are 0 q ψn (x) = 2 a sin(nπx/a) n = 1, 2, 3, . . . , 0 x<0 0≤x≤a x>a and so derive an expression for the energy of the particle in terms of a and m. [20] (b) What is the expectation value for x in this case? Justify your answer. (Hint: No calculation required.) [3] (c) Write down the wavefunction, Ψ(x, t), for this particle in terms of a and m [3] (d) Discuss briefly whether the following two statements are consistent with each other. • The momentum of a particle with kinetic energy E and mass m is given by p2 = 2mE. • The expectation value of the momentum for the particle with the eigenfunction ψn (x) is hpi = 0. [4] /Cont’d PHY-20006 Page 4 of 8 B2 Consider a particle with the wave function Ψ(x, t) = ψ(x)e−iωt where ψ(x) = 0 A [1 + cos(πx)] 0 x ≤ −1 −1 < x < 1 x≥1 (a) Normalize this wavefunction. [8] (b) Show that the ground state has definite parity and state its value. [4] (c) Calculate the uncertainty in the observed position, ∆x. [8] (d) Show that ψ has the required mathematical properties for a valid wave function at the boundaries x = ±1. [5] (e) Discuss whether Ψ(x, t) can be a valid wave function for a particle in a harmonic oscillator potential if it is not one of the energy eigenfunctions, ψn (x). [5] You may use the following standards integrals without proof in your answers. Z [1 + cos(x)]2 dx = Z 1 −1 1 [6x + 8 sin(x) + sin(2x)] + C 4 x2 [1 + cos(πx)]2 dx = 1 − 15 2π 2 /Cont’d PHY-20006 Page 5 of 8 B3 Consider a particle with mass m and energy E incident on a barrier with height VB and width a, such that E < VB . VB E 0 a x (a) Show that the energy eigenfunction ψ1 = AI eikx + AR e−ikx , is a solution of the time-independent Schrödinger equation for the region x < 0 and hence derive an expression for k in terms of E. [10] (b) What is the physical interpretation of the quantity R = |AR |2 |AI |2 ? [3] (c) State an expression for the energy eigenfunction in the region x > a in terms of an amplitude AT . Explain your answer. [6] (d) Give one example of a physical process that demonstrates or 2 T| exploits the fact that T = |A 2 |AI | > 0. State the physical origin and approximate value of the barrier potential, VB , for your example. [6] (e) Discuss what happens to the value of T in the case that the mass m is very large. [5] /Cont’d PHY-20006 Page 6 of 8 B4 The wave functions for an electron in a simple model of the hydrogen atom have the form Ψn,ℓ,mℓ (r, θ, φ, t) = u(r) Yℓ,mℓ (θ, φ)e−iEt/h̄ . r The radial eigenfunction for an electron in a hydrogen atom in the 2p state is ! 1 r 2 −r/2a0 u(r) = √ , e 24a0 a0 where a0 is the Bohr radius. (a) State the physical quantity most closely associated with each of the quantum numbers n, ℓ and mℓ , and state the possible values for each quantum number for an electron in a 2p state. [6] (b) Calculate the expectation value, hri, for an electron in the 2p state. You may use the following standard integral without proof in your answer. Z ∞ rk 0 exp(−αr) dr = k! αk+1 [8] (c) With the aid of a sketch, explain why hri is different from the most probable observed value of r for the electron. [4] (d) With the aid of a labelled diagram, describe the main features of the Stern-Gerlach experiment. Explain how this experiment shows that the eigenfunctions Ψn,ℓ,mℓ do not give a complete description for the properties of an electron in a hydrogen atom. [12] /Cont’d PHY-20006 Page 7 of 8 Quantum Mechanics formulae Time independent Schrödinger equation d2 ψ 2m + 2 [E − V (x)] ψ = 0 dx2 h̄ PHY-20006 Page 8 of 8

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