harmonic oscillator
“In mathematics, you don’t 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
Maclaurin’s series (a Taylor series about the origin):
2
2
3
3



dF
x
d
F
x
d
F
 
F(x) = F(0) + x   +  2  +  3  + . . .
 dx 0 2!  dx 0 3!  dx 0
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
 dF 
F(x)  x   = - k x .
 dx 0
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 Hooke’s Law:
F(x) = - k x .
If any restoring force obeys Hooke’s Law, it must be worth
studying!
Classically, a harmonic oscillator is subject to Hooke's law.
Newton's second law says F = ma. Therefore
d2 x
- k x =m 2 .
dt
d2 x
m 2 +k x=0.
dt
Another differential equation to solve!
The solution to this differential equation is of the form
x = A cos (ωt + φ)
where the frequency of oscillation is f, and
k
ω = 2f =
.
m
Recall from your first semester (mechanics) physics course, that
the harmonic oscillator potential is
1
U(x) = k x 2 .
2
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, let’s 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.
2 2m  1 2 
+ 2  E - kx   = 0 .
2
x
2


Why  instead of ?
If we let y =
all you do is plug in the correct
potential and turn the math crank
2 m f
2E
x and  =
h
hf
then Schrödinger's equation becomes
2ψ
2
+
α
+
y
ψ =0.


2
y
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 =2n+1 where n = 0, 1, 2, 3, ...
For those values of , the wave function has the form
 2mf 
ψn = 

h


1
4
2 n!
n
-1
2
Hn (y) e
 y2 
- 2 


.
A normal human would say this looks nasty, but a mathematician 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 = 2n+1), given by
1

En =  n +  hf ,
2

n = 0,1,2, ...
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 Schrödinger’s
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
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 didn’t 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.
You’re not too late to get into the field.
Howstuffworks is always a good place to go for easy-tounderstand explanations.
Today’s (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 don’t behave like you
want them to.
This is good if you know your physics and can take
advantage of quantum mechanics.
Quantum dots aren’t single-electron devices, but they are
getting there.
You can actually buy quantum dot products
today.