Tentamen i Kvantfysik I

Transcription

1 Karlstads Universitet Fysik Tentamen i Kvantfysik I [ VT 2017, FYGB07] Datum: Tid: Lärare: Jürgen Fuchs c/o Krister Svensson Tel: Total poäng: 50 Godkänd / 3: 25 Väl godkänd: : : 42 Tentan består av 2 delar som inlämnas separat: Del 1: 10 p. Del 2: 40 p. Hjälpmedel: Del 1 & 2: Ordbok engelska svenska Del 2 (efter del 1 har inlämnats) dessutom: Ett handskrivet A4 ark med valfritt innehåll (skrivet på ena sidan, ej maskinskriven eller maskinkopierad) inlämnas tillsammans med tentan Physics Handbook Mathematics Handbook Endast en uppgift per sida. Svaren måste vara väl motiverade. FYGB07 Tentamen

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3 Problem 1A Basics Give two examples of experimental observations which cannot be explained by classical physics, but which can be understood with the help of quantum mechanics. Describe briefly the main ideas and concepts of quantum mechanics that are needed to explain these observations. Problem 1B Basics Out of the seven main postulates of quantum mechanics, formulate: one postulate that is only concerned with state vectors; one postulate that is only concerned with observables; one postulate that combines state vectors and observables. Problem 1C Basics What are the possible energy eigenvalues for a particle in a one-dimensional harmonic oscillator potential? Are the corresponding eigenstates bound states or scattering states? Explain how the answer to this question is related to the shape of the potential. FYGB07 Tentamen

4 Problem 1D Basics Can one measure the position and the momentum of a particle simultaneously with arbitrary accuracy? If your answer is no, then describe the relation between the uncertainty x in (the x-component of) the position and the uncertainty p x in (the x- component of) the momentum. Problem 1E Basics Describe qualitatively the similarities and differences between the classical and quantum behavior of a particle travelling in the presence of a one-dimensional potential barrier. Hint: It can be convenient to mention the energy dependence of the reflection coefficient R and the transmission coefficient T. Problem 1F Basics Define the following concepts: (a) a linear operator; (b) the hermitian conjugate (or adjoint) A of a linear operator A; (c) a hermitian operator. Give an example of an operator that is not linear, and an example of a linear operator that is not hermitian. FYGB07 Tentamen

5 Problem 1G Basics Give an example for each of the following types of one-dimensional potentials: Potentials for which all physical solutions are scattering states. Potentials that have both scattering and bound states as physical solutions. Potentials for which all physical solutions are bound states. In the latter case, also give a formula for the bound states energies, at least qualitatively. Problem 1H Basics 3 p. Assume that at time t=0 the state vector of a system is given by 3 p. Ψ(t=0) = 1+i 2 ψ 1 1 i 2 ψ 2, where ψ 1 and ψ 2 are normalized eigenvectors of a hermitian operator A corresponding to distinct non-degenerate eigenvalues a 1 and a 2, respectively. What are the possible results when a measurement of the dynamical variable that is described by the operator A is performed in this system? What is the probability of obtaining the value a 1 in a measurement of A? What is the expectation value of A in the state Ψ(t=0)? Assume further that the operator A commutes with the Hamilton operator H of the system. What does the state vector Ψ(t) then look like at arbitrary times t? FYGB07 Tentamen

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7 Problem 2 Operators 7 p. a 2 p. Show that the momentum operator p = i is a hermitian operator. b Showthattheexpectationvalueoftheoperator zp z, with z thez-component 2 p. of the position operator and p z the z-component of the momentum operator, satisfies (in any state) zp z ( zp z ) = i, where the star denotes complex conjugation. c 3 p. Assume that A and B are linear operators. Show that also any function F(A) as well as the commutator [A, B] are linear operators. (The function F is assumed to have a convergent power series expansion.) Assume further that A and B are hermitian operators. Determine the behavior of the commutator [A, B] under hermitian conjugation. Show that the operator e ia as well as the product e ia e ib are unitary operators. FYGB07 Tentamen

8 Problem 3 One-dimensional problems 8 p. Consider a particle of mass m moving in a one-dimensional potential of the form for x<0, V(x) = 0 for 0<x<b, for x>b, with b a positive parameter. a Find the energy eigenvalues and the normalized energy eigenfunctions of the 4 p. system. b Assume that at some time t the wave function of the particle is given by 3 p. { cx(b x) for 0 x b, ψ(x) = 0 else, with c a non-zero parameter. Compute the probability that a measurement of the energy at time t yields the ground state energy. c Determine the probability that a measurement of the energy at time t yields a value E in the interval 0 < E < 4π2 2 mb 2. FYGB07 Tentamen

9 Problem 4 One-dimensional problems 8 p. Consider the time-independent Schrödinger equation for a particle in a one-dimensional potential well, i.e. for the potential { 0 for x < l or x > l, V(x) = V 0 for x < l, with V 0 > 0. a For which range of the energy E are there scattering solutions, and for which range can there exist bound states? b Obtain the general solution to the time-independent Schrödinger equation 5 p. with energy eigenvalue E = 3V 0, and specialize it to the situation that there is no incoming wave from the right. c For the solution with E = 3V 0 and without incoming wave from the right, 2 p. compute the reflection coefficient R and the transmission coefficient T. FYGB07 Tentamen

10 Problem 5 Three-dimensional problems 8 p. At some given time t the wave function of an electron in the hydrogen atom is in spherical coordinates given by ψ(r,θ,φ) = 6πA re r/aµ, where a µ is the (scaled) Bohr radius and A is a constant. a 3 p. Determine the value of A such that ψ is normalized to 1. b Argue that ψ is not an energy eigenfunction. 2 p. c Compute the probability to find the system in the ground state ψ 1,0,0 of the 3 p. hydrogen atom when measuring the energy. Hint: 1. For obtaining this probability, you can make use of the integral ρ 5/2 e ρ dρ = 15 π In case you do not know the explicit form of ψ 1,0,0 : One can reconstruct it by combining the following properties that the ground state wave function must have: its behavior for large and small values of r (which you should remember), that it is spherically symmetric, and that it is normalized. FYGB07 Tentamen

11 Problem 6 Angular momentum and spin 9 p. Consider a particle of spin 1/2. Denote by S x, S y and S z the Cartesian components of the spin operator. a With the help of the Pauli matrices, find the eigenvalues a ± of the operator 5 p. A = S x +S y, and determine the corresponding eigenvectors χ (A) ±. b Assuming that the particle is in the eigenstate χ (A) + corresponding to the 2 p. larger eigenvalue a +, compute the probability of obtaining the value 1 in 2 a measurement of the operator S z. c Compute the expectation value of the operator S x in the state χ (A) +. 2 p. FYGB07 Tentamen