1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Show that the next eigenfunction has a zero between 2 zeros

  1. Dec 10, 2015 #1
    1. The problem statement, all variables and given/known data
    Screen Shot 2015-12-10 at 11.40.54 pm.png
    Screen Shot 2015-12-10 at 11.41.06 pm.png

    2. Relevant equations
    Wronskian theorem:
    Screen Shot 2015-12-11 at 12.00.50 am.png

    3. The attempt at a solution
    I've gotten the relationship given by the question but I do not know how to continue.

    Since ##\psi_n(a)=\psi_n(b)=0##,
    LHS ##=\psi_n'(b)\,\psi_{n+1}(b)-\psi_n'(a)\,\psi_{n+1}(a)##

    If LHS ##=0##, RHS ##=0##, then by the First Mean Value Theorem for Integrals, ##\psi_{n+1}(c)=0## for some ##c\in(a,b)##.

    But LHS is not necessarily ##0##.

    Attached below is the derivation of Wronskian theorem.
     

    Attached Files:

    Last edited: Dec 10, 2015
  2. jcsd
  3. Dec 15, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. Dec 16, 2015 #3

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    As a and b are consecutive zeros of ##\psi_n##, ##\psi_n## won't change sign between a and b.
    Assume, without loss of generality, that ##\psi_n## is positive between a and b.
    What does that tell you about the signs of ##\psi_n'(a)## and ##\psi_n'(b)##?

    Now assume that ##\psi_{n+1}## has no zeros between a and b. It follows that ##\psi_{n+1}## won't change sign between a and b.
    Now just check what the signs of the two expressions will be (remember that ##E_{n+1}>E_n##).
     
    Last edited: Dec 16, 2015
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted