Understanding the Significance of Kets and Vectors in the Schrodinger Equation

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In summary, the ket |\alpha \rangle is a complex vector with the elements given by the dot product <Pn/S>. The ket |\bold{r} \rangle is just a vector in your vector space.
  • #1
Dathascome
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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?
 
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  • #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.
 
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  • #3
No it's not referring to a unit vector, just the position vector, normalized or not.
 
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  • #4
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].
 
  • #5
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.
 
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  • #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?
 
  • #7
Is this an understandable mistake or is there something that I'm truly missing?

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>
 
  • #8
Right...my mistake...I'm at work and wasn't being careful :biggrin:

As for my probelm though...?
 
  • #9
How many <Pk|S>'s are there? :-p

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!)
 
  • #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.
 

Related to Understanding the Significance of Kets and Vectors in the Schrodinger Equation

1. What are kets and vectors?

Kets and vectors are mathematical objects used in quantum mechanics to represent states and observables, respectively.

2. What is the difference between a ket and a vector?

The main difference is the notation used. Kets are typically represented by the symbol | > and are used to denote quantum states, while vectors are usually written using column notation and can represent any type of mathematical object.

3. How are kets and vectors used in quantum mechanics?

Kets are used to represent quantum states, which can describe the properties of a physical system such as position, momentum, or energy. Vectors, on the other hand, are used to represent observables, which are physical quantities that can be measured.

4. Can kets and vectors be manipulated mathematically?

Yes, kets and vectors can be manipulated using mathematical operations such as addition, multiplication, and normalization. This allows us to perform calculations and make predictions about the behavior of quantum systems.

5. Are kets and vectors the same as matrices?

No, kets and vectors are not the same as matrices. While they can both be represented using matrices, kets and vectors have specific properties and meanings in quantum mechanics that make them distinct from matrices.

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