17_580_hw_13 - Physics 580 Quantum Mechanics I Prof P M...

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Physics 580 Handout 17 29 November 2010 Quantum Mechanics I webusers.physics.illinois.edu/ goldbart/ Prof. P. M. Goldbart 3135 (& 2115) 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 ( ¯ h 2 2 m d 2 dx 2 + fx ) ψ ( x ) = ϵ ψ ( x ) . By introducing the shifted dimensionless length Y ( x/λ ) ( ϵ/fλ ), where λ is the characteristic length (¯ h 2 / 2 mf ) 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 ϵ n obey the quantisation condition Ai( ϵ n /fλ ) = 0 and that the (un-normalised) eigenfunctions are given by translations of the scaled Airy function: ψ n ( x ) = Ai ( x λ ϵ n ) . 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/k B T ) exp ( mgx/k B T ). 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 BE n n =1 e BE n Ai( X E n ) 2 0 dX Ai( X E n ) 2 , where X x/λ , B fλ/k B T and E n ϵ n /fλ , 1
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e) Explain the elements of this formula.
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