Practice Question for Exam 2 - Introduction Quantum mechanics 1 | PHY 4604, Exams of Physics

Download Practice Question for Exam 2 - Introduction Quantum mechanics 1 | PHY 4604 and more Exams Physics in PDF only on Docsity! PHY 4604 Fall 2008 – Practice Questions for Exam 2 (With Solutions) 1. A particle of mass m moves in one dimension under the potential V (x) = { ∞ for x < 0, V0aδ(x− a) for x > 0, where V0 and a are positive quantities. (a) Write down the general form of a stationary-state wave function of energy E > 0 in the regions x < 0, 0 < x < a, and x > a. Solution: The general form of the stationary-state wave function is ψ(x) =   0 for x < 0, A1e ikx +B1e −ikx for 0 < x < a, A2e ikx +B2e −ikx for x > a, where k = + √ 2mE/~. (b) By applying the appropriate boundary conditions, express the stationary-state wave function from (a) in terms of just one unknown amplitude. Solution: We must impose continuity of ψ at x = 0, where V (x) jumps to infinity: A1 +B1 = 0 ⇒ A1 = −B1 ⇒ ψ(x) = S sin kx for 0 < x < a. Given this, it simplifies matters slightly to rewrite ψ(x) = C cos kx+D sin kx for x > a. Then we must impose continuity of ψ at x = a: C cos ka+D sin ka = S sin ka D + C cot ka = S. (1) Finally, we must enforce ∆ψ′(a) = (2mV0a/~2)ψ(a): −kC sin ka+ kD cos ka = kS cos ka+ 2mV0a ~2 S sin ka D − C tan ka = S(1 + α tan ka), (2) where α = 2mV0a/(~ 2k). The solution to Eqs. (1) and (2) is C = −Sα sin2 ka, D = S(1 + α sin ka cos ka). Then, using sin θ cosφ− cos θ sinφ = sin(θ − φ), ψ(x) =   0 for x < 0, S sin kx for 0 < x < a, S [sin kx+ α sin ka sin k(x− a)] for x > a, where k = + √ 2mE/~ and α = 2mV0a/(~ 2k). (c) What is the reflection coefficient for a particle approaching from the far right? Solution: The probability flux j is obviously zero in the region x < 0. Since j must be spatially uniform in a stationary state, this means that in the region x > a, the incident and reflected fluxes must cancel: jI + jR = 0. This in turn means that the reflection coefficient R = −jR/jI = 1. 2. A quantum mechanical system is described by a two-dimensional vector space spanned by orthonormal basis vectors |1〉 and |2〉. The Hamiltonian for this system is Ĥ =  (−4|1〉〈1| + 4|2〉〈2| + 3|1〉〈2| + 3|2〉〈1|) , where  > 0. We will also consider the operator Λ̂ = λ0(|1〉〈2| + |2〉〈1|). (a) Provide the matrix representations of Ĥ and Λ̂ in the basis {|1〉, |2〉}. Solution: In the basis {|1〉, |2〉}, Ωmn = 〈m|Ω̂|n〉 for m,n = 1, 2. Ĥ ↔  (−4 3 3 4 ) , Λ̂ ↔ λ0 ( 0 1 1 0 ) . (b) Find the eigenvalues (E1 and E2, with E1 < E2) and normalized eigenkets (|E1〉 and |E2〉) of Ĥ in the basis {|1〉, |2〉}. The eigenvalues En satisfy 0 = det(Ĥ − EnÎ) = (−4− En)(4− En) − (3)2 = E2n − 252. ⇒ E1 = −5, E2 = 5. E1 = −5:( 1 3 3 9 )( a b ) = ( 0 0 ) ⇒ ( a b ) = 1√ 10 ( 3 −1 ) or |E1〉 = (3|1〉 − |2〉) / √ 10. Orthogonality implies that |E2〉 = (|1〉 + 3|2〉) / √ 10. (c) Find the matrix representation in the basis {|1〉, |2〉} of the propagator ˆU(τ) defined by Û(τ)|Ψ(t)〉 = |Ψ(t+ τ)〉 for any |Ψ(t)〉. Solution: As shown in Homework 4, Question 1, the propagator for a finite- dimensional system is Û(τ) = ∑ n e−iEnt/~|En〉〈En|,