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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
e) Explain the elements of this formula.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern