The Position Operator

Thursday, November 29, 2018

The Position Operator

Position

In classical mechanics, we can describe a particle's position as a function of time. In quantum mechanics, weird things start to happen and so we describe position using a complex vector space.

As we move to more complicated systems, we have to add more dimensions to our vector space. For position, a particle can appear anywhere on a line and so we need an infinite dimension vector space. Functions are an infinite dimension vector space and so we use wave functions to describe particles. Wave functions work the same way as vectors since you can inner product them, add them together etc..

So for position we can use complex wave functions to describe a particle. ψ(x)

ψ(x) => |ψ(x)>

(In this post we will always be referring to wave functions as functions of position (x) and so I will just write ψ for ease.)

The operator for position (X) acts on ψ so that the new vector is ψ times x.

X|ψ> = |xψ>

So if this is actually the operator for position then it follows that it must be Hermitian?

Remember for an operator to be hermitian then <A|H|A> is always real

<ψ|X|ψ> is always real?

= <ψ|x|ψ> = ∫ψ(x)ψ(x)* dx

and this is always real and so X is definitely hermitian.

This means it must correspond to some kind of observable (we guessed at the start that the observable is position).

If we look at the eigenvectors and eigenvalues of X then maybe we can understand why this operator is position.

X|ψ> = λ|ψ>

means that functions of position that satisfy this equation are eigenvectors of X (where λ are the eigenvalues of X).

So we need to find a function that when we multiply by x, we get λ times the function.

xψ = λψ

ψ(x - λ) = 0

This means either ψ = 0 or x = λ.
So if x ≠ λ, ψ = 0

and if ψ ≠ 0, x = λ

So that means ψ is 0 everywhere except when x = λ.

Remember

Postulate 4

States for which the observable have a definite value are the eigenvectors of H.
The value for this eigenvector is the corresponding eigenvalues.

This makes sense. If position has a definite value then we would expect the wave function of the particle to have a value at one place.
For each possible value of λ, there is one eigenvector corresponding to the particle being somewhere else.

This function is know as the Dirac delta function.

The width of the tall spike is d then the height is 1/d. To look at the spike we look at small values of d so that it is tall and thin. Defining it like this means that the area under the function is 1.

This was a short introduction to the position operator in quantum mechanics and in the next post, we will look at the momentum operator.

Thanks for reading. If you enjoyed this post or any of my others, follow and subscribe to my blog. Feel free to discuss anything related to this post or ask questions in the comments below.