Download Quantum Mechanics Exam and more Exams Physics in PDF only on Docsity! PHY4604 Fall 2006 Exam 1 Department of Physics Page 1 of 5 PHY 4604 Exam 1 (Total Points = 100) Problem 1 (20 points): Circle true or false for following (2 point each). (a) (True or False) One of the “breakthroughs” that lead to quantum mechanics was the idea of associating differential operators with the dynamical variables. (b) (True or False) Solutions of Schrödinger’s equation of the form )()(),( txtx φψ=Ψ correspond to states with definite energy E. (c) (True or False) Solutions of Schrödinger’s equation of the form )()(),( txtx φψ=Ψ correspond to states in which the probability density 2|),(|),( txtx Ψ=ρ is independent of time. (d) (True or False) The wave function Ψ(x,t) must vanish in a region of infinite potential. (e) (True or False) It is possible for a free particle to have a definite energy. (f) (True or False) In quantum mechanics particles can enter the “classically forbidden” region where V0 > E (i.e. KE < 0). (g) (True or False) The operator AopA↑op is hermitian. (h) (True or False) If Aop and Bop are hermitian then AopBop is also hermitian. (i) (True or False) The commutator operator [(px)op,(x2)op] is equal to hi2− . (j) (True or False) In position-space the commutator operator [(px)op,sin(kx)] is equal to )cos(kxkih− . PHY4604 Fall 2006 Exam 1 Department of Physics Page 2 of 5 Problem 2 (30 points): Consider an electron with mass me confined within an infinite square well defined by V(x) = 0 for 0 < x < L, V(x) = +∞ otherwise. (a) (2 points) Using Schrödinger’s equation calculate the allowed stationary state eigenfunctions ψn(x), where the complete wavefunctions are given by h/)(),( tiEnn nextx −=Ψ ψ , and normalize the eigenfunctions so that the probability of finding the electron somewhere in the box is one. (b) (2 points) Show that the wavefunctions Ψn(x,t) correspond to states with definite energy (i.e. show that ΔE = 0). (c) (2 points) Calculate the allowed energy levels, En, of the system. Express your answer in terms of the Compton wavelength of the electron, )/( cmee hD = , and the rest mass energy of the electron, 2cme . What is the numerical value of the ground state energy (in MeV) for the case eL D= ? (Note that mec2 = 0.511 MeV.) (d) (2 points) Show that the states, )(xnψ , form an orthonormal set. Namely, show that mnnm dxxx δψψ =∫ +∞ ∞− ∗ )()( . (e) (3 points) Calculate <x> for nth state stationary state. (f) (3 points) Calculate <px> for nth state stationary state. (g) (5 points) Suppose the electron in this infinite square well has a wave function at t = 0 which is an equal mixture of the first two stationary states as follows [ ])()()0,( 21 xxAx ψψ +=Ψ . What is the normalization A? If you measure the energy of this particle, what are the possible values you might get, and what is the probability of getting each of them? What is the expectation value of the energy for this state (i.e. average energy)? (h) (5 points) Suppose as in part (g) the electron in this infinite square well has a wave function at t = 0 which is an equal mixture of the first two stationary states as follows [ ])()()0,( 21 xxAx ψψ +=Ψ . What is the probability density 2|),(|),( txtx Ψ=ρ for this state. Does it depend on time? What is <x> and <px> for this state. Do they depend on time? (i) (6 points) Suppose the electron in this infinite square well has a wave function at t = 0 which is given by )/3cos()/sin(2)0,( LxLx L x ππ=Ψ . If you measure the energy of this particle, what are the possible values you might get, and what is the probability of getting each of them? What is the expectation value of the energy for this state (i.e. average energy)? V = +infinity V = +infinity Infinite Square Well 0 L x
