Problem Set 2 | Introduction to Quantum Mechanics | PHY 4604, Assignments of Physics

Download Problem Set 2 | Introduction to Quantum Mechanics | PHY 4604 and more Assignments Physics in PDF only on Docsity! PHY4604 Fall 2007 Problem Set 2 Department of Physics Page 1 of 3 PHY 4604 Problem Set #2 Due Wednesday September 19, 2007 (in class) (Total Points = 110, Late homework = 50%) Reading: Griffiths Chapter 2 (sections 2.1, 2.2, and 2.3). Useful Math: 4 2cos2sin 8 1 46 cos 8 2cos 4 2sin 4 cos 4 2sin 2 cos sincoscos 4 2cos2sin 8 1 46 sin 8 2cos 4 2sin 4 sin 4 2sin 2 sin cossinsin 2 sincossin 23 22 2 2 2 23 22 2 2 2 2 xxxxxxdxx xxxxxdxx xxxdx xxxxdxx xxxxxxdxx xxxxxdxx xxxdx xxxxdxx xxdxx +⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −+= ++= += += −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −−= −−= −= −= = ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ Problem 1 (10 points): The time-dependent Schrödinger’s equation is ),()(),( 2 ),( 2 22 txxV x tx mt txi Ψ+ ∂ Ψ∂ −= ∂ Ψ∂ h h , where we take the potential energy, V, to be only a function of x (not of time). We look for solutions of the form )()(),( txtx Φ=Ψ ψ . The time-dependent equation separates into two equations (separation constant E). )()()()( 2 2 22 xExxV xd xd m ψψψ =+− h and )()( tE dt tdi Φ=Φh . Thus, h/)( iEtet −=Φ . (a) (3 points) For normalizable solutions, show that the separation constant E must be real. (Hint: see Griffiths problem 2.1) (b) (3 points) Show that E corresponds to the total energy and that )()(),( txtx Φ=Ψ ψ corresponds to a state with definite energy (i.e. ΔE = 0). (c) (4 points) Show that the time-independent wave function, )(xψ , can always be taken to be a real function. (Hint: take V to be real and see Griffith’s problem 2.1) mn mn ppp nxdxmx nxdxmx p δπ δπ π πξ ξ π π 2 coscos 2 sinsin 87 1 5 1 3 11 6 )2( 4 1 3 1 2 11)( 0 0 2 222 2 = = =++++ = ++++= ∫ ∫ L L PHY4604 Fall 2007 Problem Set 2 Department of Physics Page 2 of 3 Problem 2 (50 points): Consider an infinite square well defined by V(x) = 0 for 0 < x < a, and V(x) = ∞ otherwise. (a) (4 points) Show that the stationary states of Schrödinger’s equation are given by h/)(),( tiEnn nextx −=Ψ ψ with )/sin(2)( axn a xn πψ = and 2 222 2ma nEn hπ = , and n is a positive integer. (Hint: see Griffiths section 2.2) (b) (2 points) Show that the states, )(xnψ , form an orthonormal set. Namely, show that mnnm dxxx δψψ =∫ +∞ ∞− ∗ )()( . (Hint: see Griffiths section 2.2) (c) (8 points) Calculate <x>, <x2>, <px>, and <px2> for the nth stationary state. (d) (6 points) Compute Δx = σx and Δpx = xpσ for the n th stationary state. Is the product ΔxΔpx consistent with the uncertainty principle? Which state comes closest to the uncertainty limit? (e) (8 points) Suppose that particle in this infinite square well has an initial wave function which is an even mixture of the first two stationary states: [ ])()()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? (Hint: see Griffiths problem 2.5) (f) (10 points) Suppose that particle in this infinite square well has the initial wave function ⎩ ⎨ ⎧ − =Ψ )( )0,( xaA Ax x axa ax ≤≤ ≤≤ 2/ 2/0 Sketch )0,(xΨ and determine the normalization A. Find ),( txΨ . What is the probability that a measurement of the energy of this state will yield the ground state energy E1? What is the expectation value of the energy for this state ),( txΨ ? (g) (12 points) Suppose a particle of mass m in this infinite square is in the ground state (i.e. n = 1). Now suppose that the well suddenly expands to twice its original size (the right wall moves from a to 2a) leaving the wave function (momentarily) undisturbed. If the energy of the particle is now measured, what is the most probable value and what is the probability of getting this value? Problem 3 (50 points): Consider the one dimensional harmonic oscillator with potential energy 2 2 1)( kxxV = . The Hamiltonian is given by ( )22 2 )( 2 1 2 xmp m V m p H x x ω+=+= where mk /=ω . We express the Hamiltonian in terms of the operators (px)op and (x)op, where x ip opx ∂ ∂ −= h)( and xx op =)( and 2 2 22 )()()( x ppp opxopxopx ∂ ∂ −== h .