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harmonic oscillator

In mathematics, you dont understand things. You

just get used to them.John von Neumann

5.11 Harmonic Oscillator

Recall from math how functions can be written in

the form of a Maclaurins series (a Taylor series

about the origin)

If F represents a restoring force (a force that

pulls the system back to the origin) then F(0)

0.

For small displacements x, all the higher order

terms (involving x2, x3, etc.) are small, so

The sign enters because F is a restoring force,

so the derivative is negative.

So in the limit of small displacements, any

restoring force obeys Hookes Law

If any restoring force obeys Hookes Law, it must

be worth studying!

Classically, a harmonic oscillator is subject to

Hooke's law.

Newton's second law says F ma. Therefore

Another differential equation to solve!

The solution to this differential equation is of

the form

where the frequency of oscillation is f, and

Recall from your first semester (mechanics)

physics course, that the harmonic oscillator

potential is

So what?

Many systems are described by harmonic

oscillators. We had better see what quantum

mechanics has to say about them!

A truly classic example is the swinging bowling

ball demo.

Before we continue, lets think about harmonic

oscillators

Classically, all energies are allowed. What will

QM say?

Only quantized energies?

Classically, an energy of zero is allowed. What

will QM say?

Nonzero, like particle in box?

Classically, the oscillator can't exist in a

state in "forbidden" regions. For example, a

pendulum oscillating with an amplitude A cannot

have a displacement greater than A.

Could there be a nonzero probability of finding

the system in "forbidden" regions. I wonder what

that means for our swinging bowling ball

Now, let's solve Schrödinger's equation for the

harmonic oscillator potential.

all you do is plug in the correct potential and

turn the math crank

Why ? instead of ??

If we let

then Schrödinger's equation becomes

Solutions to this equation must satisfy all the

requirements we have previously discussed, and ?

must be normalized.

The solution is not particularly difficult, but

is not really worth a day's lecture. Instead, I

will quote the results.

The equation can be solved only for particular

values of ?, namely ?2n1 where n 0, 1, 2, 3,

...

For those values of ?, the wave function has the

form

A normal human would say this looks nasty, but a

math-ematician would say it is simple. Just a

bunch of numbers, an exponential function, and

the Hermite Polynomials Hn. Polynomials are

simple. H0(y) 1, H1(y) 2y, and other

polynomials are given in Table 5.2 of Beiser.

More important, we find that the wave equation is

solvable only for certain values of E (remember,

? 2E/hf 2n1), given by

The energies of the quantum mechanical harmonic

oscillator are quantized in steps of hf, and the

zero point energy is E0 ½hf.

Here is a Mathcad document illustrating QM

harmonic oscillator energy levels, probabilities,

and expectation values.

Because of the scaling we did in re-writing

Schrödingers equation, it is difficult to

identify the classically forbidden regions in the

graphs in the Mathcad document. See Figures 5.12

and 5.13, page 191 of Beiser, for an illustration

of how the amount of wavefunction tails in the

forbidden region shrinks as n increases.

Here are a couple of plots.

Wave Functions

Probability Densities

n 2

n 1

Things you ought to study in relation to

harmonic oscillators

Figure 5.13, to see how the quantum mechanical

harmonic oscillator "reduces" to the classical

harmonic oscillator in limit of large quantum

numbers.

Figure 5.11, to see how the different potentials

for different systems lead to different energy

levels (we will do the hydrogen atom, figure

5.11a, in the next chapter).

Example 5.7, page 192, expectation values.

Like, before exam 2!

The BIG PICTURE.

It took us forever to get through chapter 5.

What are some big ideas?

Wave functions probability densities

normalization expectation values good and

bad wave functions calculating probabilities.

Particle in box how to solve the SE energy

levels quantization expectation values

effect of box length calculating probabilities.

Particle in well how to solve the SE energy

levels quantization expectation values

effect of well length effect of well height

calculating probabilities compare and contrast

with infinite well classically forbidden

regions.

possible and not possible

Tunneling (how to solve the SE) transmission

probability reflection probability effect of

particle mass and energy on tunneling probability

effect of barrier height on tunneling

probability.

We didnt discuss applications, but there are

many. Scanning tunneling microscope. Quantum

effects as integrated circuits shrink towards the

quantum world! Your life will be impacted by

quantum effects in a big way!

Harmonic Oscillator (how to solve the SE)

energy levels zero point energy quantization

expectation values classically forbidden

regions classical limit.

This is not guaranteed to be an all-inclusive

list!

Optional information on applications (not for

test)

Readable introduction to quantum computing at

Caltech, April 2000.

A single photon is incident on a beamsplitter,

which is a mirror that reflects with a 50

probability and transmits with a 50 probability.

Half the photons of a beam of light reach A, and

half reach B.

Classically, half the photons should be detected

at A, half at B.

Experimentally, all of the photons reach A.

The photon wave travels both paths at once, and

interference at the second beamsplitter gives

this result.

The ability of a quantum system to contain

information about many states simultaeously is

(within the limits of my simple-minded

understanding) the basis of quantum computing.

In December 2001, IBM researchers built a quantum

computer in this test tube and used to find the

two prime factors of 15.

We still have a ways to go before making a real

computer. Youre not too late to get into the

field.

Howstuffworks is always a good place to go for

easy-to-understand explanations.

Todays (10-15-03) special bonus feature was how

quicksand works. The quantum computer feature

also leads you to teleporting photons, and an

explanation of how teleportation will work.

The only catch is, the original object being

teleported has to be destroyed. Better hope

there is no power outage if that object is you!

As semiconductor device sizes become smaller, and

we approach devices which confine single

electrons in a quantum well, quantum effects

become dominant.

This is bad if you are trying to design

conventional semiconductor devices. They dont

behave like you want them to.

This is good if you know your physics and can

take advantage of quantum mechanics.

Quantum dots arent single-electron devices, but

they are getting there.

You can actually buy quantum dot products today.