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Homework Help: 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
    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

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    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.
     
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