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harmonic oscillator

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Title: harmonic oscillator


1
harmonic oscillator
In mathematics, you dont understand things. You
just get used to them.John von Neumann
2
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.
3
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!
4
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!
5
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
6
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.
7
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.
8
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.
9
Here are a couple of plots.
Wave Functions
10
Probability Densities
n 2
n 1
11
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!
12
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
13
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!
14
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.
15
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.
16
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!
17
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.
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