Cardiff University Introductory Quantum Mechanics Exam Paper 2022-2023 PDF

PX2132 CARDIFF UNIVERSITY EXAMINATION PAPER Academic year: 2022-23 Examination Period: Autumn Examination Paper Number: PX2132 Examination Paper Title: Introductory Quantum Mechanics Duration: 2 hours Do not turn this page over until instructed to do so by the Senior Invigi- lator Structure of Examination Paper: There are eight pages. There are five questions in total. This examination paper is divided into two sections. There are no appendices. The maximum mark for the examination paper is 80 (60% of the module mark). The mark obtainable for a question or part of a question is shown in brackets along- side the question. Students to be provided with: 1 answer book Mathematical Formulae and Physical Constants Handbook Instructions to students: Answer ALL questions from section A, and TWO questions from section B. If addi- tional section B questions are attempted be clear which should be marked, otherwise only the first two answers as they appear on your paper will be marked. Students are advised to spend an equal amount of time on each section. Calculators which have been pre-programmed and calculators with an alphabetic keyboard are not permitted in any examination. The use of translation dictionaries between English and Welsh or another language, bearing an appropriate departmental stamp, is permitted in this examination. 1 PX2132 Section A Answer all questions 1. Scattering and Tunnelling. A particle is incident from the left (x = −∞) on a potential step defined by ( 0, x < 0 (region I) V (x) = V0 , x ≥ 0 (region II). (i) Sketch the potential. [2 marks] (ii) State the time independent Schrödinger equation (TISE) in terms of V (x). [1 mark] (iii) Assuming the energy E > V0 explain why the solutions to the TISE take the forms φI (x) = exp (ikx) + r exp (−ikx) φII (x) = t exp (ik 0 x). [3 marks] (iv) Using the TISE, find expressions for k and k 0 in terms of E and V0. [3 marks] (v) State the two boundary conditions obeyed at the step. [2 marks] (vi) The probability current density is given by ∂φ∗ i~ ∗ ∂φ j (x) = φ −φ. 2m ∂x ∂x Find the probability current densities for the reflected and transmitted waves. [4 marks] (vii) Find the probability of transmission in terms of k, k 0 , and the transmission amplitude t. [3 marks] (viii) Thinking of the ‘velocity’ of the particle as the ratio of its momentum to its mass, or otherwise, explain physically why the probability for transmission is not simply the square modulus of the transmission amplitude. [2 marks] 2 PX2132 2. Bound States. Consider the TISE ~2 d2 φn (x) − + V (x) φn (x) = En φn (x) 2m dx2 for the infinite potential well defined by ( 0, 0≤x≤L V (x) = ∞, otherwise. (i) Sketch the potential and the lowest three energy eigenfunctions. [3 marks] (ii) Find the eigenvalues En and normalized eigenfunctions φn (x). [7 marks] (iii) A particle is found to have energy En. Find the probability, Pn (α), for the particle to exist in the region 0 ≤ x ≤ αL where 0 ≤ α ≤ 1. [3 marks] (iv) State the three probabilities Pn (α = 0), Pn (α = 1/2), and Pn (α = 1), and explain why they must take these values. [3 marks] (v) The particle is in the first excited state E2. Sketch the probability Pn=2 (α) as a function of α marking on the points found in (iv), and the values of α at which the maxima and minima of the gradient dPn=2 (α) /dα occur. [4 marks] 3 Turn over PX2132 Section B Answer two questions. 3. Quantum superposition. (i) Show that the probability density ρ (x) = |ψn (x, t)|2 is time independent for all energy eigenfunctions. [1 mark] (ii) Now consider the superposition χ (x, t) = αψn (x, t) + βψm (x, t) where α and β are complex numbers, n 6= m, and the corresponding energy eigenvalues obey En 6= Em. Given that ψn and ψm are correctly normalized, find a condition on α and β such that χ is correctly normalized. [4 marks] (iii) For the special case that α and β and all ψn (x, 0) are purely real, show that the probability density ρχ (x) = |χ (x, t)|2 is now time dependent, and find an expression for its period. [4 marks] (iv) Explain why any function f (x) matching the same boundary conditions as the energy eigenstates ψn (x, t) can be written as ∞ X f (x) = fn ψn (x, t = 0) n=1 where it is assumed the ground state has n = 1. [1 mark] (v) Show that Z ∞ fn = ψn∗ (x, t = 0) f (x) dx. −∞ [4 marks] 4 PX2132 (vi) Consider the infinite potential well ( 0, −L/2 ≤ x ≤ L/2 V (x) = ∞, otherwise which has energy eigenstates  q 2 nπx  L cos L , n odd ψn (x, t = 0) = q 2 nπx  L sin L , n even for integer n > 0. The following correctly normalized state is prepared at time t = 0: ( √ 12L−3/2 (L/2 + x) , −L/2 ≤ x ≤ 0 f (x) = √ 12L−3/2 (L/2 − x) , 0 ≤ x ≤ L/2. Sketch f (x), indicating the values of maxima and minima within the well. [2 marks] (vii) The energy of the particle described by f (x) is measured. Find the probablility that either of the two lowest energies is found. [4 marks] 5 Turn over PX2132 4. Finite dimensional Hilbert spaces. (i) Prove that Hermitian matrices have real eigenvalues. [4 marks] (ii) Consider the spin matrices ~ 0 1 ~ 0 −i ~ 1 0 Ŝx = , Ŝy = , Ŝz =. 2 1 0 2 i 0 2 0 −1 State three properties which make them appropriate as a choice of operators to encode spin-1/2. [3 marks] (iii) In this basis, explain why the states corresponding to spin-up and spin- down along x must then take the form 1 1 | ↑x i = α , | ↓x i = β 1 −1 where α and β are (possibly complex) constants. [2 marks] (iv) Find α and β, explaining your resoning. [3 marks] (v) Express 1 | ↑z i = 0 as a linear combination of | ↑x i and | ↓x i. [3 marks] (vi) A Stern Gerlach apparatus is used to establish that a spin-1/2 silver atom has spin up along z with probability 1. The same atom then passes through a second Stern Gerlach apparatus oriented so as to perform a measurement of the spin along x. Using your previous results, calculate the probability that the outcome is spin down along x. [3 marks] (vii) Rather than record the outcome of the second measurement, the two possible routes are redirected so as to rejoin one another, and the resulting route passes through a third Stern Gerlach apparatus so as to measure the spin along z. Calculate the probability that the outcome is spin down along z. [2 marks] 6 PX2132 5. The quantum harmonic oscillator. The Hamiltonian for the quantum harmonic oscillator is p̂2 1 Ĥ = + mω 2 x̂2. 2m 2 (i) Defining the lowering operator r mω i â = x̂ + p̂ 2~ mω and the raising operator r † mω i â = x̂ − p̂ 2~ mω show that the Hamiltonian can be rewritten as † 1 Ĥ = ~ω â â + Î 2 where Î is the identity operator. You will need to use the canonical commuta- tion relation: [x̂, p̂] = i~Î. [4 marks] (ii) Using the canonical commutation relation, show that † â, â = Î. [3 marks] (iii) The TISE is given by Ĥ|ni = En |ni. By acting on both sides with â, show that † 1 ~ω â â + Î (â|ni) = (En − ~ω) (â|ni). 2 [3 marks] (iv) Hence explain why â is called the lowering operator. [2 marks] 7 Turn over PX2132 (v) The ground state of the harmonic oscillator, denoted by |0i, obeys â|0i = 0. In the position basis, we can denote the ground state wavefunction hx|0i = φ0 (x). Find the normalised state φ0 (x). [Hint]: you will need to rewrite the lowering operator in the position basis to obtain a first-order ordinary differential equation which you must solve. You may wish to use the fact that Z ∞ r 2 π exp −αx dx =. −∞ α [4 marks] (vi) Find the ground state wavefunction in the momentum basis: hp|0i = φ̃0 (p). [4 marks] 8X

Cardiff University Introductory Quantum Mechanics Exam Paper 2022-2023 PDF

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