## Notes and problems for Desai chapter III.

Posted by peeterjoot on October 9, 2010

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# Notes.

Chapter III notes and problems for [1].

FIXME:

Some puzzling stuff in the interaction section and superposition of time-dependent states sections. Work through those here.

# Problems

## Problem 1. Virial Theorem.

### Statement.

With the assumption that is independent of time, and

show that

### Solution.

I floundered with this a bit, but found the required hint in physicsforums. We can start with the Hamiltonian time derivative relation

So, with the assumption that is independent of time, and the use of a stationary state for the expectation calculation we have

The exercise now becomes one of evaluating the remaining commutators. For the Laplacian commutator we have

For the potential commutator we have

Putting all the factors back in, we get

which is the desired result.

Followup: why assume is independent of time?

## Problem 2. Application of virial theorem.

Calculate with .

## Problem 3. Heisenberg Position operator representation.

### Part I.

Express as an operator for .

With

We want to expand

We to evaluate to proceed. Using , we have

This gives us

Or

### Part II.

Express as an operator for with .

In retrospect, for the first part of this problem, it would have been better to use the series expansion for this exponential sandwich

Or, in explicit form

Doing so, we’d find for the first commutator

so that the series has only the first two terms, and we’d obtain the same result. That seems like a logical approach to try here too. For the first commutator, we get the same result since .

Employing

I find

The triple commutator gets no prettier, and I get

Putting all the pieces together this gives

If there is a closed form for this it isn’t obvious to me. Would a fixed lower degree potential function shed any more light on this. How about the Harmonic oscillator Hamiltonian

… this one works out nicely since there’s an even-odd alternation.

Get

I’d not expect such a tidy result for an arbitrary potential.

## Problem 4. Feynman-Hellman relation.

For continuously parametrized eigenstate, eigenvalue and Hamiltonian , and respectively, we can relate the derivatives

Left multiplication by gives

which provides the desired identity

## Problem 5.

### Description.

With eigenstates and , of with eigenvalues and , respectively, and

and , determine in terms of and .

### Solution.

## Problem 6.

### Description.

Consider a Coulomb like potential with angular momentum . If the eigenfunction is

determine , , and the energy eigenvalue in terms of , and .

### Solution.

We can start with the normalization constant by integrating

To go further, we need the Hamiltonian. Note that we can write the Laplacian with the angular momentum operator factored out using

With zero for the angular momentum operator , and switching to spherical coordinates, we have

We can now write the Hamiltonian for the zero angular momentum case

With application of this Hamiltonian to the eigenfunction we have

In particular for we have

Collecting all the results we have

## Problem 7.

### Description.

A particle in a uniform field . Show that the expectation value of the position operator satisfies

### Solution.

This follows from Ehrehfest’s theorem once we formulate the force , in terms of a potential . That potential is

The Hamiltonian is therefore

Ehrehfest’s theorem gives us

or

Putting all the last bits together, and summing over the directions we have

## Problem 8.

### Description.

For Hamiltonian eigenstates , , , obtain the matrix element in terms of the matrix element of .

### Solution.

I was able to get most of what was asked for here, with a small exception. I started with the matrix element for , which is

Next, computing the matrix element for we have

Except for the part of this expression, the problem as stated is complete. The relationship 2.25 is no help for with , so I see no choice but to leave that small part of the expansion in terms of .

## Problem 9.

### Description.

Operator has eigenstates , with a unitary change of basis operation . Determine in terms of , and the operator and its eigenvalues for which are eigenstates.

### Solution.

Consider for motivation the matrix element of in terms of . We will also let . We then have

We also have

So it appears that the operator has the orthonormality relation required. In terms of action on the basis , let’s see how it behaves. We have

So we see that the operators and have common eigenvalues.

## Problem 10.

### Description.

With , and , show that

### Solution.

We also have

Or, for ,

This gives

## Problem 11. commutator of angular momentum with Hamiltonian.

Show that , where .

This follows by considering , and . Let

so that

We now need to consider the commutators of the operators with and .

Let’s start with . In particular

So our commutator with is

Since , all terms cancel out, and the problem is reduced to showing that

Now assume that has a series representation

We’d like to consider the action of on this function

Thus as expected, implying .

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

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