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Unique to within a constant factor?

  1. Sep 21, 2007 #1

    cks

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    unique to within a constant factor???

    we always listen to the phrase in quantum mechanics textbook that says this eigenvector is unique to within a constant factor.

    What does it actually mean?

    unique means there is one and only one??
    constant factor: a constant 2 or 3 or... maybe??

    I don't really catch the physical picture of it, I just hope for some mathematical examples that can illustrate this. Thank you
     
  2. jcsd
  3. Sep 21, 2007 #2

    CompuChip

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    Well, suppose that [tex]\left| a \rangle[/tex] is an eigenvector to an operator  with eigenvalue [itex]a[/itex], that is:
    [tex]\hat A \left| a \rangle = a \left| a \rangle[/tex].
    Then of course, since scalars commute with any operator,
    [tex]\hat A \left( \lambda \left| a \rangle \right) =
    \lambda \left( \hat A \left| a \rangle \right) = \lambda \left( a \left| a \rangle \right) =
    a \left( \lambda \left| a \rangle \right)[/tex].
    So as you see, [tex]\lambda \left| a \rangle[/tex] is also an eigenvector for the same eigenvalue, for any (complex) number lambda. But other than that, the eigenvector is really unique, that is, if
    [tex]\hat A \left| a' \rangle = a \left| a' \rangle[/tex]
    then |a'> cannot be anything different than a multiple of |a>.

    This is what is meant by "unique up to a constant factor", which is enough since in QM, the pre-factor doesn't carry any physical meaning anyway (we normally just use it for convenience, e.g. to normalize eigenkets).
     
    Last edited: Sep 21, 2007
  4. Sep 21, 2007 #3

    malawi_glenn

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    There is a complex (global-)phase: exp(i*phi), cos' we deal with complex numbers. And as convention we often we often taken phi = 0, so exp(i phi) = 1.

    Is that what you looked for? You can see an example of this in Sakurai "modern QM" chapter 1 in discussion of Sx and Sy and their eigen-vectors.
     
  5. Sep 21, 2007 #4

    cks

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    Thank you CompuChip,

    I understand now. Really appreciate that.
     
  6. Sep 22, 2007 #5

    dextercioby

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    quote: >>the pre-factor doesn't carry any physical meaning anyway (we normally just use it for convenience, e.g. to normalize eigenkets).<<

    Actually this phase freedom of vectors is of crucial importance when it comes to describing symmetries in QM.
     
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