Karlstads Universitet Fysik Tentamen i Kvantfysik I [ VT 2018, FYGB07] Datum: 2018-06-07 Tid: 14.00 19.00 Lärare: Jürgen Fuchs c/o Andrea Muntean Tel: 0547002047 / 0706448585 Total poäng: 50 Godkänd / 3: 25 Väl godkänd: 37.5 4: 33.5 5: 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 2018 1 2018-06-21
Del 1 FYGB07 Tentamen 2018 2 2018-06-21
Problem 1A Basics The following three observations cannot be explained by classical physics, but can be understood with the help of quantum mechanics: Radiation exhibiting particle-like behavior. Matter exhibiting wave-like behavior. Spin. For at least two of these, describe an experiment in which the effect can be observed and summarize the main ideas and concepts of quantum mechanics that are needed to explain the observation. Problem 1B Basics Write down both the time-dependent and the time-independent Schrödinger equation. Describe a method by which the time-independent Schrödinger equation can be derived from the time-dependent one. Problem 1C Basics Give the commutator between the position operator x and the momentum operator p x. Compute this commutator explicitly both in the position space representation and in the momentum space representation of the operators. FYGB07 Tentamen 2018 3 2018-06-21
Problem 1D Basics Which of the following statements about the degeneracy of energy eigenvalues of bound states in a central potential V = V(r) are correct? All energy eigenvalues are degenerate. All energy eigenvalues are non-degenerate. The ground state energy is non-degenerate. An energy eigenvalue is degenerate if there is at least one corresponding energy eigenstate that is at the same time an eigenstate of L 2 with angular momentum quantum number l > 0. Problem 1E Basics Consider a particle of definite energy moving in the one-dimensional potential V(x) = 13x 2 +e x +e x. What is the expectation value x of the position of the particle? What is the expectation value p x of the momentum of the particle? What can be said qualitatively about the expectation values x 2 and p 2 x of the squares of these operators? FYGB07 Tentamen 2018 4 2018-06-21
Problem 1F Basics 2 p. What are the possible eigenvalues of the z-component J z of the total angular momentum operator J? What are the possible eigenvalues of the operator J 2? What are the possible eigenvalues of the operator J x? 2 p. Can one measure the value of the two operators J x and J 2 simultaneously with arbitrary accuracy? Can one measure the value of the two operators J x and J z simultaneously with arbitrary accuracy? Problem 1G Basics 3 p. Assume that at time t=0 the state vector of a system is given by 3 p. Ψ(t=0) = 1 2 ψ 1 1 2 3 ψ2, where ψ 1 and ψ 2 are normalized eigenvectors of a hermitian operator B corresponding to distinct non-degenerate eigenvalues b 1 and b 2, respectively. What are the possible results when a measurement of the dynamical variable that is described by the operator B is performed in this system? What is the probability of obtaining the value b 2 in a measurement of B? What is the expectation value of B in the state Ψ(t=0)? Assume further that the operator B 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 2018 5 2018-06-21
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Problem 2 Operators 7 p. a Show thatthe expectation valueof theoperator zp z, with z the z-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. b Let A and B be linear operators and α and β complex numbers. 2 p. Show that (αa+βb) = α A +β B and (AB) = B A, where A and B are the hermitian conjugates of A and B, respectively, and the star denotes complex conjugation. c Let A be a linear operator. What condition must A satisfy in order that the operator e ia is a unitary operator? d The wave function of a free particle can be described as a superposition 2 p. of plane waves e i k r, and also as a superposition of spherical waves j l (kr)y lm (θ,φ). To which observables are the basis functions e i k r, respectively the basis functions j l (kr)y lm (θ,φ), eigenfunctions? What are the corresponding eigenvalues and what are their degeneracies? FYGB07 Tentamen 2018 7 2018-06-21
Problem 3 One-dimensional problems 8 p. The wave function Ψ(x, t) of a particle of definite energy E in some one-dimensional potential V(x) is given by Ψ(x,t) = Cx e γx2 iet/ with some constants C and γ > 0. a Does this wave function Ψ(x,t) describe a bound state or a scattering state? Motivate your answer! b Determine the constant C such that the wave function Ψ(x, t) is normalized. 2 p. Hint: One possibility to compute the relevant integral I is as follows: reduce it to a simpler integral Ĩ by writing it as I = d dγ Ĩ. The integral Ĩ, in turn, can befound by computing its square Ĩ2 with the help of polar coordinates. c Making use of the fact that the function Ψ(x, t) satisfies the time-dependent 3 p. Schrödinger equation, find the potential V(x). d Basedonthe so obtainedresult for V(x), determine theenergy eigenvalue E. e Give an argument showing that Ψ is not the ground state of the system, but an excited state. Determine the difference between the eigenvalue E of Ψ and the ground state energy E 0. FYGB07 Tentamen 2018 8 2018-06-21
Problem 4 One-dimensional problems 8 p. At some given time t the state Ψ(t ) of a one-dimensional system is given by a spatial wave function ψ(x)=ψ(x,t ) of the form C for a w x a+w, ψ(x) = C for a w x a+w, 0 else. with C a complex constant. a Determine the possible value(s) of the number C for which the wave function ψ(x) is normalized. b Compute the expectation values of the operators x and x 2 in the state 2 p. Ψ(t ). c Determine the wave function φ(p x ) in momentum space and compute the 3 p. expectation value of the momentum operator p x. d Discuss qualitatively how the probability density in momentum space and the expectation value of p x change if ψ(x) is modified to C for a w x a+w, ˆψ(x) = e iα C for a w x a+w, 0 else. for some real number α. e A possible realization of the wave functions ψ(x) or ˆψ(x) is as an approximate description of the wave function along the line on which the two slits of a double-slit experiment are located. Discuss the implications of the result of part d for the interference pattern observed in such a double-slit situation. FYGB07 Tentamen 2018 9 2018-06-21
Problem 5 Three-dimensional problems 8 p. At some given time t the (unnormalized) wave function of an electron in the hydrogen atom is given by ψ( r) = 3ψ 1,0,0 ( r) 3ψ 3,2,1 ( r)+2ψ 4,2,0 ( r), where ψ n,l,m are the standard energy eigenfunctions of the hydrogen atom. a For each possible combination of the quantum numbers n, l, m, give the 2 p. probability of finding the electron at time t in the state ψ n,l,m. b Determine the expectation value of the following operators in the state ψ: 4 p. the square L 2 of the angular momentum; the component L z of the angular momentum; the component L x of the angular momentum; the energy, expressed as a multiple of the ground state energy E 0,0,0. c What are the possible outcomes of a measurement of the component L x of the angular momentum if the system is in the state ψ? d Is the state ψ an eigenstate of the parity operator? Motivate your answer. FYGB07 Tentamen 2018 10 2018-06-21
Problem 6 Angular momentum and spin 9 p. For a system that is in an eigenstate of J 2 corresponding to the value j=1 of the angular momentum quantum number j, the angular momentum operators J z and J ± =J x ±ij y can be described in terms of matrices as follows: J z = 1 0 0 0 0 0 0 0 1 0 2 0, J + = 0 0 2, J = 0 0 0 0 0 0 2 0 0. 0 2 0 We denote by χ 1,1, (z) χ (z) 1,0 and χ (z) 1, 1 the eigenvectors of J z in this matrix representation that correspond to the eigenvalues +, 0 and, respectively. a Determine the expectation value J y of the operator J y in the J z -eigenstate 2 p. χ (z) 1, 1. b Find the eigenvectors χ 1,1, (y) χ (y) 1,0 and χ (y) 1, 1 of J y in the given matrix 4 p. representation. Hint: It may help to remember that i 2 = 1. c Consider the following experiment. A beam of atoms, all of which are in the eigenstate with j=1 and with eigenvalue +, of J z, passes through a region with a magnetic field that is inhomogeneous in y-direction, whereby the beam gets split into three parts. (This is the situation that is studied in Stern- Gerlach-type experiments.) Of the three resulting beams, only the one that corresponds to the eigenvalue 0 of the operator J y is allowed to travel further. This beam then passes through a region with a magnetic field that is inhomogeneous in z-direction, whereby also this beam splits up. 3 p. Determine the intensity of the beams that come out of the second inhomogeneous magnetic field as compared to the intensity of the original incoming beam. Hint: To solve this part one should use the result of part b. But qualitatively the question can be answered even without having obtained that result. When correct, such qualitative arguments will also give points. FYGB07 Tentamen 2018 11 2018-06-21