## Desai Chapter 9 notes and problems.

Posted by peeterjoot on November 29, 2010

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# Motivation.

Chapter 9 notes for [1].

# Notes

# Problems

## Problem 2.

### Statement.

On the basis of the results already derived for the harmonic oscillator, determine the energy eigenvalues and the ground-state wavefunction for the truncatedoscillator

### Solution.

We require , so our solutions are limited to the truncated odd harmonic oscillator solutions. The normalization will be different since only the integration range is significant. Our energy eigenvalues are

And its wave function is

where is the first odd wavefunction for the non-truncated oscillator. Normalizing this we find , or

## Problem 3.

### Statement.

Show that for the harmonic oscillator in the state , the following uncertainty product holds.

### Solution.

I tried this first explicitly with the first two wave functions

For the state we find easily that

and this is zero since we are integrating an odd function over an even range (presuming that we take the principle value of the integral).

For the state this we have

Since each is a polynomial times a factor we have for all states .

The momentum expecation values for states and are also fairly simple to compute. We have

For the state our derivative is odd since a factor of is brought down, and we are again integrating an odd function over an even range. For the case our derivative is proportional to

Again, this is an even function, while is odd, so we have zero. Noting that we can express each in terms of Hankel functions

where is even and is odd, we note that this expectation value will always be zero since we will have an even times odd function in the integration kernel.

Knowing that the position and momentum expectation values are zero reduces this problem to the calculation of . Either of these expecation values are again not too hard to compute for . However, we now have to keep track of the proportionality constants. As expected this yields

These are respectively

However, these integrals were only straightforward (albeit tedious) to calculate because we had explicit representations for and . For the general wave function, what we have to work with is either the Hankel function representation of 3.7 or the derivative form

Expanding this explicitly for arbitrary isn’t going to be feasible. We can reduce the scope of the problem by trying to be lazy and see how some work can be avoided. One possible trick is noting that we can express the squared momentum expectation in terms of the Hamiltonian

So we can get away with only calculating , an exersize in integration by parts

The second term in this remaining integral is proportional to , which leaves us with

Our squared momentum expectation value is then

This completes the problem, and we are left with

## Problem 4.

### Statement.

Consider the following two-dimensional harmonic oscilator problem:

where are the coordinates of the particle. Use the separation of variables technique to obtain the energy eigenvalues. Discuss the degeneracy in the eigenvalues if .

### Solution.

Write . Substitute and dividing throughout by we have

Introduction of a pair of constants for each of the independent terms we have

For each of these equations we have a set of quantized eigenvalues and can write

The complete eigenstates are then

with total energy satisfying

A general state requires a double sum over the possible combinations of states , however if , we cannot distinguish between and based on the energy eigenvalues

In this case, we can write the wave function corresponding to a general state for the system as just . This reduction in the cardinality of this set of basis eigenstates is the degeneracy to be discussed.

## Problem 5,6.

### Statement.

Consider now a variation on Problem 4 in which we have a coupled oscillator with the potential given by

Obtain the energy eigenvalues by changing variables to such that the new potential is quadratic in , without the coupling term.

### Solution.

This has the look of a diagonalization problem so we write the potential in matrix form

The similarity transformation required is

Our change of variables is therefore

Our Laplacian should also remain diagonal under this orthonormal transformation, but we can verify this by expanding out the partials explicitly

Squaring and summing we have

Our transformed Hamiltonian operator is thus

So, provided , the energy eigenvalue equation is given by 3.26 with , and .

## Problem 7.

### Statement.

Consider two coupled harmonic oscillators in one dimension of natural length and spring constant connecting three particles located at , and . The corresponding Schr\”{o}dinger equation is given as

Obtain the energy eigenvalues using the matrix method.

### Solution.

Let’s start with an initial simplifying substutition to get rid of the factors of . Write

These were picked so that the differences in our quadratic terms involve only factors of

Schr\”{o}dinger’s equation is now

Putting our potential into matrix form, we have

This symmetric matrix, let’s call it M

has eigenvalues , with orthonormal eigenvectors

Writing

Writing , and , we see that the Laplacian has no mixed partial terms after transformation

Schr\”{o}dinger’s equation is then just

Or

Separation of variables provides us with one free particle wave equation, and two harmonic oscillator equations

We can borrow the Harmonic oscillator energy eigenvalues from problem 4 again with , and .

## Problem 8.

### Statement.

As a variation of Problem 7 assume that the middle particle at has a different mass . Reduce this problem to the form of Problem 7 by a scale change in and then use the matrix method to obtain the energy eigenvalues.

### Solution.

We write , and then Schr\”{o}dinger’s equation takes the form

With , we have

We find that this symmetric matrix has eigenvalues , and eigenvectors

The rest of the problem is now no different than the tail end of Problem 7, and we end up with , .

# References

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

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