soll - Physics 741 Graduate Quantum Mechanics 1 Solution Set L 1[10 In class we found the even parity bound states for H= P2 V(Q 2m where 0 V(Q = V0 if

soll - Physics 741 Graduate Quantum Mechanics 1 Solution...

This preview shows page 1 - 2 out of 3 pages.

Physics 741 – Graduate Quantum Mechanics 1 Solution Set L 1. [10] In class we found the even parity bound states for ( ) 2 , 2 P H V Q m = + where ( ) 0 0 if if Q a V Q V Q a < = > Repeat the analysis for the odd parity bound states. We know from class notes eq. (7.60) that in the three regions, the solution takes the form ( ) ( ) ( ) ( ) ( ) cos sin x I II x III x Ae x B kx C kx x De β β ψ ψ ψ = = + = We are interested in the odd parity bound states. Since parity relates regions I and III to each other, and region II to itself, this implies ( ) ( ) I III x x ψ ψ = and ( ) ( ) II II x x ψ ψ = . We therefore have A = - D and B = 0. We now wish to match boundary conditions. We will choose to do so at x = a, where we follow exactly the notation of (7.62) in the notes to yield ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin exp cos exp II III II III a a C ka D a a a kC ka D a ψ ψ β ψ ψ β β = = = = − Dividing the second equation by the first, we find ( ) cot k ka β = − Using equations (7.57) and (7.59), it is easy to see that 2 2 2 2 2 0 0 2 2 k mV mV k β β + = = = = Plugging this in and dividing by - k , we find ( ) 0 2 2 2 cot 1 mV ka k = = This is equation (7.66). Solutions of this equation can then be plugged into (7.65) to get
Image of page 1
Image of page 2

You've reached the end of your free preview.

Want to read all 3 pages?

  • Spring '04
  • Unknow
  • Physics, mechanics, Eigenvalue, eigenvector and eigenspace, Hilbert space, odd parity

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes