1. Not finding help here? Sign up for a free 30min 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!

Definition needed

  1. Jun 17, 2011 #1
    1. The problem statement, all variables and given/known data
    What is the meaning of the phrase "up to a scale" as applied to eigenvectors in quantum mechanics.


    2. Relevant equations
    None

    3. The attempt at a solution
    N/A did an extensive search of the web and my texts. No joy.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 17, 2011 #2

    Mark44

    Staff: Mentor

    I'll guess that this means that two eigenvectors point in the same or opposite direction. IOW, each one is a scalar multiple of the other.
     
  4. Jun 17, 2011 #3
    I guess I wasn't specific enough in my question. Sorry. The definition I would like is as the phrase applies to the following statement:

    ie [itex]\Lambda|\omega_i>[/itex] is an eigenvector of [itex]\Omega[/itex] with eigenvalue [itex]\omega_i [/itex]. Since the vector is unique [itex]\underline{up\ to\ a\ scale}[/itex],

    [itex]\Lambda|\omega_i > = \lambda_i | \omega_i >[/itex]
     
    Last edited: Jun 17, 2011
  5. Jun 17, 2011 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, that is exactly what Mark44 was referring to. If v is an eigenvector of linear transformation, A, with eigenvector [itex]\lambda[/itex], then [itex]Av= \lambda v[/itex]. If u is any "scalar multiple" of v, u= sv for some scalar, s, then, since A is linear, [itex]Au= A= (sv)= s(Av)= s(\lambda v)= \lambda(sv)= \lambda u[/itex] so that u is also an eigenvector with eigenvalue [itex]\lambda[/itex]. That is, the eigenvector is unique "up to a scalar multiple" which is, I presume, what this physics text means by "up to scale". (You might want to recheck the exact wording. "Up to a scale" doesn't seem grammatically correct.)
     
  6. Jun 17, 2011 #5
    This is from the proof of theorem 13 page 44 of Principles of Quantum Mechanics- 2nd edition by R Shanker. The up to scale is a direct quote.

    Thanks all for the help.

    Gary R.
     
  7. Jun 18, 2011 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    But what you said before was "up to a scale" which is not a direct quote.

     
  8. Jun 18, 2011 #7
    I rechecked the text. The actual statement is "up to a scale". I goofed in the second message. Sorry

    Gary R.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook