Quantal and Classical Exponential Atmosphere - Notes for Homework 13 | PHYS 580, Assignments of Quantum Mechanics

Download Quantal and Classical Exponential Atmosphere - Notes for Homework 13 | PHYS 580 and more Assignments Quantum Mechanics in PDF only on Docsity! Physics 580 Handout 17 1 December 2008 Quantum Mechanics I http://w3.physics.uiuc.edu/∼goldbart Homework 13 Prof. P. M. Goldbart 2115 & 3135 ESB University of Illinois 1) Quantal and classical exponential atmosphere: Consider a particle of mass m moving in one spatial dimension, confined to the region x > 0 and subject to the linear potential V (x) = fx, where f is a positive constant. If, e.g., the potential were gravitational then f would be mg and the particle would be an atmospheric particle. a) Sketch the potential. b) The energy eigenproblem reads ( − h̄ 2 2m d2 dx2 + fx ) ψ(x) = ² ψ(x). By introducing the shifted dimensionless length Y ≡ (x/λ) − (²/fλ), where λ is the characteristic length (h̄2/2mf)1/3, show that the eigenproblem reduces to the dimen- sionless form Ψ′′(Y )− Y Ψ(Y ) = 0, i.e., to Airy’s equation. c) By imposing appropriate boundary conditions, establish that the energy eigenvalues ²n obey the quantisation condition Ai(−²n/fλ) = 0 and that the (un-normalised) eigenfunctions are given by translations of the scaled Airy function: ψn(x) = Ai ( x λ − ²n fλ ) . Give the first three eigenvalues and sketch the first three eigenfuntions. Note: You may wish to consult a standard reference, such as Abramowitz and Stegun. d) Now apply the Bohr-Sommerfeld quantisation scheme to motion in the potential V (x). Compute the lowest three eigenvalues it gives, and compare them with their exact values. Compare the Bohr-Sommerfeld eigenvalue spectrum at large n with the result you obtain for the spectrum using the asymptotic properties of the Airy function. Recall the problem of classical statistical mechanics in which we consider a system of non- interacting particles that constitute an isothermal atmosphere. In that setting, we ask the question: What is the probability density p(x) for finding a particle to be at height x, given the gravitational potential V (x) = mgx? We find the exponential atmosphere result: p(x) = (mg/kBT ) exp (−mgx/kBT ). Let us see how we can recover this result, starting with quantal rather than classical motion—a problem posed by my colleague Prof. Paul Debevec. According to the canonical ensemble of quantum statistical mechanics, p(x) = 1 λ P(X,B) ≡ 1 λ ∑∞ n=1 e −BEn ∞∑ n=1 e−BEn Ai(X − En)2∫∞ 0 dX ′ Ai(X ′ − En)2 , where X ≡ x/λ, B ≡ fλ/kBT and En ≡ ²n/fλ, 1 e) Explain the elements of this formula. Now let us try to understand the classical limit of this formula. The essential idea is to neglect tunnelling into the classically forbidden region. This means that, for a given value of x, states n only contribute to p(x) if ²n > fx. States lower in energy come with quantal probability densities that involve Airy functions evaluated at positive arguments, and these are small, only being nonzero by virtue of tunnelling. Thus we have P(X, B) ≈ 1∑∞ n=1 e −BEn ∞∑ n>ν(X) e−BEn Ai(X − En)2∫ En 0 dX ′ Ai(X ′ − En)2 , where Eν(X) ≈ X. Generally, for larger values of x fewer states contribute. To evaluate P we approximate the sum over states (i.e. over n) by an integral over n (characteristically, as discreteness/quantisation is an essentially quantal effect). To do the resulting integral we exchange continuous n for continuous E = (3π/2)2/3n2/3, suggested by the asymptotic spectrum at large n. f) Show that this leads to P(X, B) ≈ 1∫∞ 0 dE √ E e−BE ∫ ∞ X dE √ E e−BE Ai(X − E)2 ∫ E 0 dX ′ Ai(X ′ − E)2 . g) The asymptotic form of the Airy function at large negative argument reads Ai(−z) ≈ 1√ π z1/4 sin ( ζ + π 4 ) , where ζ ≡ (2/3)z3/2. Show that averaging over a few periods gives for the quantal probability factor Ai(X − E)2 ≈ 1 2π 1√ E −X and for its normalisation integral ∫ E 0 dX Ai(X − E)2 ≈ √ E π . h) Put the pieces togther to obtain the classical result P(X,B) ≈ ∫ ∞ X dE 1√ E −X e −BE 2 ∫ ∞ 0 dE √ E e−BE = B e−BX . Hence, we find the exponential atmosphere formula: p(x) = (f/kBT ) exp (−fx/kBT ). 2