Homework Assignment 6, due 3 March 2006 Problem 1 - HMC Physics

Problem 1 – Townsend 6.1(a) Use induction to show thatHomework Assignment 6, due 3 March 2006[ˆx n , ˆp x ] = inˆx n−1 (1)(b) By expanding F (x) in a Taylor series, show that [F (ˆx), ˆp x ] = i ∂F∂x (ˆx)(c) For the one-dimensional Hamiltonian Ĥ = ˆp2 xd〈px〉2m+ V (ˆx) show thatdt= 〈 − dVdxProblem 2 – Townsend 6.4(a) Show for a free particle of mass m initially in the stateψ(x) = 〈x|ψ〉 =1√√ πae −x2 /2a 2〉thatψ(x, t) = 〈x|ψ(t)〉 =1√√ π[a + (it/ma)]e −x2 /2a 2 [1+(it/ma 2 )]and therefore∆x = √ a ( ) t 2√1 + 2 ma 2(b) What is ∆p x at time t? Suggestion: Use the momentum-space wave function toevaluate ∆p x .Problem 3 – Townsend 6.5Consider a wave packet defined by⎧⎪⎨ 0 p < −P/2〈p|ψ〉 = N −P/2 < p < P/2⎪⎩0 p > P/2(a) Determine a value for N such that 〈ψ|ψ〉 = 1 using the momentum-space wave functiondirectly.(b) Determine 〈x|ψ〉 = ψ(x).(c) Sketch 〈p|ψ〉 and 〈x|ψ〉. Use reasonable estimates of ∆p x from the form of 〈p|ψ〉 and∆x from the form of 〈x|ψ〉 to estimate the product ∆x∆p x . Check that your result isindependent of the value of P . Note: Simply estimate rather than actually calculatethe uncertainties.(d) Now calculate the uncertainties properly. Can you reconcile the contradiction?Problem 4 – Townsend 6.17 Calculate the reflection and transmission coefficients forscattering from the potential energy barrier2m 2 V (x) = λ b δ(x)Note the discussion in the preceding problem on the boundary conditions.Homework Assignment 6 3 March 2006

3 March 2006 Homework Assignment 6Problem 5 – Townsend 6.21 Electrons in a metal are bound by a potential that may beapproximated by a finite square well. Electrons fill up the energy levels of this well up to anenergy called the Fermi energy, as indicated Fig. 6.16a. The difference between the Fermienergy and the top of the well is the work function W of the metal. Photons with energiesexceeding the work function can eject electrons from the metal—the photoelectric effect.Another way to pull out electrons is through application of an external uniform electric fieldE, which alters the potential energy as shown in Fig. 6.16b. Show that the transmissioncoefficient for electrons at the Fermi energy is given by()T ≈ exp − 4√ 2mW 3/23e|E|How would you expect the field-emission current to vary with the applied voltage?Peter N. Saeta 2 Physics 116