Download PHY4604 Fall 2007 Problem Set 1 - Probability Distributions and Wave Functions and more Assignments Physics in PDF only on Docsity! PHY4604 Fall 2007 Problem Set 1 Department of Physics Page 1 of 3 PHY 4604 Problem Set #1 Due Friday September 7, 2007 (in class) (Total Points = 60, Late homework = 50%) Reading: Griffiths Chapter 1. Useful Math: )( 2 1 2 1 1 0 22 + + ∞ − Γ=∫ nnxan adxex , where Γ(x) is the gamma function and Γ(x+1) = xΓ(x). Γ(1) = Γ(2) = 1, Γ(n) = (n-1)! if n is a positive integer, and π=Γ )( 21 . Integration by parts: b a b a b a fggdx dx dfdx dx dgf +−= ∫∫ Problem 1 (12 points): Consider a room containing 14 people, whose ages are as follows: One person aged 14, One person aged 15, Three people aged 16, Two people aged 22, Two people aged 24, Five people aged 25. (a) (1 point) If you selected one person from the room, what is the probability that the person’s age would be 15? (b) (1 point) What is the most probable age? (c) (1 point) What is the median age? (d) (1 point) What is the average age? (e) (1 point) Let N(j) be the number of people with age j. Histogram N(j) versus j. (f) (2 points) Compute <j2> and <j>2. (g) (3 points) Determine Δj = j - <j> for each j, and compute the variance of the distribution using, σ2 = <(Δj)2>. What is the standard deviation, σ, of this distribution. (h) (2 points) Compute the standard deviation using, 22 ><−><= jjσ , and show that you get the same answer as in part (g). (Hint: see Griffiths section 1.3) Problem 2 (10 points): Consider the (Gaussian) wave function 2 2 )()( axAex −−= λ ψ , where A, a, and λ are positive real constants. The probability density is defined by )()()()( 2 xxxx ψψψρ ∗≡= . (a) (1 point) Find the value of A that normalizes this wavefunction such that 1)( =∫ +∞ ∞− dxxρ . PHY4604 Fall 2007 Problem Set 1 Department of Physics Page 2 of 3 (b) (2 points) Find <x> and <x2> for this wavefunction. (c) (2 point) Sketch the graph ρ(x) versus x. (d) (2 points) Find <px> and <px2> for this wavefunction. (e) (3 points) Compute Δx = σx and Δpx = xpσ . Is the product ΔxΔpx consistent with the uncertainty principle? (Hint: see Griffiths section 1.4, 1.5, and 1.6) Problem 3 (16 points): Consider a particle of mass m described by the wavefunction )/( 2),( itmxaAetx +−= hψ , where A and a are positive real constants and 1−=i . (a) (1 point) Find the value of A that normalizes this wavefunction such that 1)( =∫ +∞ ∞− dxxρ . (b) (2 points) For what potential energy function V(x) does Ψ satisfy Schrödinger’s equation? (c) (4 points) Calculate the expectation values of x, x2, px, and px2. (d) (3 points) Compute Δx and Δpx. Is the product ΔxΔpx consistent with the uncertainty principle? (e) (6 points) The kinetic energy, T, of the particle is defined by m pT x 2 2 = . Calculate the expectation value of the kinetic energy T. What is ΔT for this state? (Hint: see Griffiths section 1.4, 1.5, and 1.6) Problem 4 (10 points): Use Schrödinger’s equation to show that the time derivative of the expectation value of px is equal to the expectation value of minus the derivative of the potential, V(x). Namely, x V dt pd x ∂ ∂ −= >< . (Hint: take V(x) to be a real function, see Griffiths 1.5 and integrate by parts) Problem 5 (12 points): The time dependent Schrödinger equation is given by ),()(),( 2 ),( 2 22 txxV x tx mt txi ψψψ + ∂ ∂ −= ∂ ∂ h h . Suppose that the potential V(x) is complex and Ψ(x,t) is a (normalizable) wavefunction. (a) (2 points) Use Schrödinger’s equation to show that in general ),()(Im),(),( 2 txxV x txj t tx ρρ h=∂ ∂ + ∂ ∂ , where ρ(x,t) is the probability density defined by ),(),(),(),( 2 txtxtxtx ψψψρ ∗≡= , and j(x,t) is the probability current given by
