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Why are wavefunctions represented as eigenvectors?

  1. Oct 26, 2013 #1
    Hi I was learning about eigenvectors, inner products, Dirac notation etc. But I don't get why wave functions are represented as eigenvectors?
     
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  3. Oct 26, 2013 #2

    Simon Bridge

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    Because it makes the math easier.

    Technically - the wave-function is a state-vector.
    We like to resolve them into eigenvectors (also called eigenfunctions) for the same reason we like to resolve a velocity vector into components: it makes the math easier.

    All wave-functions are vectors because they transform as vectors - so "function" and "vector" mean the same thing.
     
  4. Oct 26, 2013 #3
    could you elaborate with an example please?
     
  5. Oct 26, 2013 #4

    Vanadium 50

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    Do you even know what an eigenvector is? The reason I ask is that four hours ago you asked "Hey all, does anyone a great place to learn linear algebra online? Thanks for any help. " It took me more than four hours to learn linear algebra.
     
  6. Oct 27, 2013 #5

    Simon Bridge

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    I'm with Vanadium 50.
    There are plenty of examples in standard texts in linear algebra - you should have seen them already.
     
  7. Oct 27, 2013 #6
    I only learnt eigenvectors, eigenvalues, dirac notation and inner products. I already knew how to work with matrices.I just haven't seen how to use them in qm.

    I'm motivated. Science is my life. I don't spend my time playing games like my other friends.
     
    Last edited by a moderator: Oct 31, 2013
  8. Oct 27, 2013 #7

    Simon Bridge

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    Do you know what a vector space is?
    Have you seen that functions are vectors in their own right?
    Or are you thinking that a vector is a column (or row) of numbers?

    See also:
    https://ece.uwaterloo.ca/~ece204/howtos/functions/

    All the rest is just different ways of writing vectors down.
     
    Last edited: Oct 27, 2013
  9. Oct 27, 2013 #8
    Thanks, I didn't know that vectors=functions in a way.
     
  10. Oct 27, 2013 #9

    Simon Bridge

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    Yep - after grokking that, it remains only to figure which representation of vectors is the one you want to deal with. Some of the cooler ways can look a lot like magic.

    This gets very useful when you have to deal with much more complicated systems where writing down the functions normally can occupy a whole page - it can also obscure the relationships that you are interested in.
     
  11. Oct 27, 2013 #10
    Thanks again, is this the basis for the linear algebra formulation of QM?
     
  12. Oct 27, 2013 #11

    Simon Bridge

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    Um - it is a demonstration that there is no useful distinction to be made between the wavefunction and state-vector formulation. The formulation is just notation.
     
  13. Oct 31, 2013 #12
    We have a linear algebra formulation of QM because Schrödinger's equation is linear. That is, if A and B are solutions, then c1*A + c2*B is a solution.
     
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