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Kets, vectors.

  1. Aug 18, 2005 #1
    This might be a really stupid question but what is the signifigance of having a something like this

    /r >

    Which is supposed to be an r vector inside of a ket.

    In the book I'm reading they use this notation when talking about the shrodinger equation, where r is the positon vector.

    I'm sort of confused because at some point a big deal is made of making a distinction between kets/bras and the vectors (arrows) we all know and love. So the arrows are a certain subset of vecotrs in a vector space (R^3), and there's a relation between kets and vectors but I don't understand what it means to have a vector, like the position vector inside as a ket. Maybe I'm just thinking too much and making something out of nothing?
     
    Last edited: Aug 18, 2005
  2. jcsd
  3. Aug 18, 2005 #2
    If what your talking about is like a little ^ right above the vector, it means that the vector is a unit vector. The elementary property of a unit vector is that its magnitude is 1 (or unity) thus the name "unit vector". I should also mention that unit vectors, themselves, are dimensionless. The physical quantity (or the scalar) that is associated with the unit vector, gives the vector its dimensions.
     
    Last edited: Aug 18, 2005
  4. Aug 18, 2005 #3
    No it's not refering to a unit vector, just the position vector, normalized or not.
     
    Last edited: Aug 18, 2005
  5. Aug 18, 2005 #4

    Gokul43201

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    A ket [itex]| \alpha \rangle [/itex] is nothing but a complex vector, or an element of the n-dimensional complex vector space [itex]\mathbb {C}^n[/itex].
     
  6. Aug 19, 2005 #5

    Galileo

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    The ket [itex]|\bold{r} \rangle[/itex] is just a vector in your vector space. A ket IS a vector and a vector is a ket, same thing. But keep in mind they live in the state space or Hilbert space). Whatever you put inside the [itex]| \rangle[/itex] is just a label to distinguish between different kets. The label [itex]\bold {r}[/itex] is chosen because [itex]| \bold{r} \rangle[/itex] is the eigenket (or eigenvector) of the position operator [itex]\vec R=\hat X \vec i + \hat Y \vec j +\hat Z \vec k[/itex] corresponding to the eigenvalue [itex]\bold r[/itex] (yes, that's a vector, because [itex]\vec R[/itex] is a vector-operator. This vector DOES live in 3D geometric space).

    If it helps, try to think of the 1D analog. The position operator [itex]\hat X[/itex] has eigenkets [itex]|x \rangle[/itex] corresponding to the eigenvalue x.
     
    Last edited: Aug 19, 2005
  7. Aug 20, 2005 #6
    So then I have position vectors as the elements/components of the ket then right?

    I think that the thing that confused me is that in the book I'm reading they talk about the matrix representation of kets/bras, where in they define the ket , /S>, as a nX1 column vector with it's elements/component given by the dot product <Pn/S> where /Pn> is the nth basis ket of /S>. Specifically I thought that this dot product should give me a scalar and not a vector, so it seemed sort of odd to me to have a ket with it's elements given by a vector, because I didn't see how you could get a vector out of that dot product.

    Is this an understandable mistake or is there something that I'm truly missing?
     
  8. Aug 20, 2005 #7

    Hurkyl

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    You've given "n" two different meanings.

    The coordinate representation of |S> in the ordered basis {|Pk>} (k = 1 .. n) is the nx1 matrix whose k-th element is <Pk|S>
     
  9. Aug 20, 2005 #8
    Right...my mistake...I'm at work and wasn't being careful :biggrin:

    As for my probelm though...?
     
  10. Aug 20, 2005 #9

    Hurkyl

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    How many <Pk|S>'s are there? :tongue2:

    BTW, they're components of the coordinate representation of the ket, not the ket itself. (The phrase "components of the ket" might not even make sense!)
     
  11. Aug 20, 2005 #10
    Sometimes I find these little details to be the hardest part of learning this sort of stuff. I know that they are important and saying the wrong thing can lead to thinking the wrong thing, but the terminology and phrasing is hard to get down sometimes.
     
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