Quantum mechanics-projection operator in terms of momentum

[ke1be11]

Homework Statement

<x|p^2|x'>=?

Homework Equations

<x|p|x'>=-ih d/dx dirac delta(x-x')
I=Integral(dp|p><p|)

The Attempt at a Solution

<x|p^2|x'>=<x|p I p|x'>=<x|p Integral(dp)|p><p|p|x'>

I know that <x|p|x'> = -ih d/dx diracdelta(x-x') so does that mean that <p|p|x'>=-ih d/dx diracdelta(p-x') but that doesn't make sense because the x' and p are different quantities. Maybe just an explanation on the the bra and ket of p vs. the bra and ket of x? I really am lost here...

 

[Jhero]

I understand your confusion and I am happy to provide an explanation to help clarify things.

First, let's define some terms. In quantum mechanics, a "bra" is a vector in the dual space of the Hilbert space, denoted by <x|, and a "ket" is a vector in the Hilbert space, denoted by |x>. The <x| and |x> are called "bra-ket" notation.

Now, let's take a closer look at the equation <x|p^2|x'>. This is a matrix element, where <x| and |x'> are the bra and ket vectors, respectively, and p^2 is the operator. In general, the matrix element of an operator A can be written as <x|A|x'>. So, in this case, <x|p^2|x'> is the matrix element of the operator p^2.

Now, let's look at the equation <x|p|x'>=-ih d/dx dirac delta(x-x'). This is a well-known result in quantum mechanics, known as the momentum operator in position representation. It tells us that the matrix element of the momentum operator (p) is equal to -ih times the derivative of the dirac delta function with respect to x.

To answer your question, <p|p|x'> is not equal to -ih d/dx diracdelta(p-x'). This is because <p| is a bra vector in the momentum space, while |p> is a ket vector in the momentum space. These vectors represent different quantities, as you correctly pointed out. So, the equation <p|p|x'>=-ih d/dx diracdelta(p-x') does not make sense.

In conclusion, in the equation <x|p^2|x'>, the bra and ket vectors are in the position space, while in the equation <p|p|x'>, the bra and ket vectors are in the momentum space. They represent different quantities and cannot be equated. I hope this explanation helps clarify things for you.