Quantum Mechanics: Evaluating Operator Expressions

The problem asks us to evaluate two expressions involving quantum mechanical operators and states.

Part a: Evaluate when the transformation matrix from the basis to the basis is known, and .

Since is a Hermitian operator, its eigenstates form a complete orthonormal basis. The property means that are the eigenstates of with eigenvalues .
For a function of a Hermitian operator , if are the eigenstates of with eigenvalues , then .

We want to evaluate . We can insert a complete set of states in the basis:

So,

Since are eigenstates of , we have .
Therefore,

The terms and are elements of the transformation matrix between the and bases. Specifically, is the transformation matrix element from to , and is the transformation matrix element from to .
Let be the transformation matrix from the basis to the basis. Then .
And .
So,

This is the desired expression.

Part b: Using the continuum analogue of the result obtained in (a), evaluate .
Here, , where are operators. This implies that is the position operator magnitude.
The continuum analogue of the discrete sum is an integral .
The continuum analogue of the discrete states are the position eigenstates or momentum eigenstates .
In this problem, is a function of the position operator magnitude . The eigenstates of the position operator are , and .
We want to evaluate .
We can insert a complete set of position eigenstates:

So,

Since are eigenstates of , we have .
Therefore,

We know the transformation bracket between position and momentum eigenstates:

And

Substituting these into the expression:

Let . Then the integral is a 3D Fourier transform of .

Since depends only on the magnitude , we can perform the integral in spherical coordinates. Let be along the z-axis.
.
.

The integral over gives .
The integral over : Let , . When ; when .

So the integral becomes:

Let .

This is the simplified expression.

Final Answer:
Part a:
The expression for is given by inserting a complete set of eigenstates of :

Since , it follows that .
Therefore, .
Let be the transformation matrix element from the basis to the basis. Then and .

Part b:
To evaluate , we use the continuum analogue of the result from part (a) by inserting a complete set of position eigenstates :

Using the position-momentum transformation brackets:

Substituting these into the integral:

Let . Since depends only on the magnitude , we can perform the integral in spherical coordinates.

Substituting this back into the expression for :

The final answer is .