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ECE 4060: Introduction to Quantum and Statistical Mechanics
Handout 6
Simple Harmonic Oscillators (SHO)
Textbook Reading:
Chaps. 9 and 10 (pp. 165 – 202).
6.1
The classical problem of simple Harmonic oscillator
The harmonic oscillator (i.e.,
V(x)
assume a quadratic form or
F(x)
is a linear function
like in the Hook’s law for elasticity) is not only important for the basic formulations of
molecules, circuits, phonons, and radiation fields, but also presents special and powerful methods
for handling quantum effects, so special that the quantum formulation of harmonic oscillator
predates the Schrödinger equation.
We will first review the classical problem, and then use the
Heisenberg’s principle to develop the quantum intuition before we introduce the full solution of
the timeindependent Schrödinger equation (which unfortunately, except for the ground state, is
rather complicated, although there are many mathematical techniques intuitively derived
already).
We will then develop new operators that link the eigenfunctions so that we can
manipulate the expectation and variance values more conveniently.
Finally, we will look back at
the physical meanings for the quantum system we have achieved.
The classical formulation of the harmonic oscillator can be expressed in the Hamiltonian
form as:
)
(
)
(
)
(
)
(
1
)
(
t
kx
x
F
t
p
dt
d
t
p
m
t
x
dt
d
(6.1)
The solution can be easily found as
0
0
0
0
0
0
0
0
)
cos(
)
(
)
sin(
)
(
x
m
p
m
k
t
p
t
p
t
x
t
x
(6.2)
The total energy
E
can be expressed as the sum of the kinetic and potential energies:
)
(
sin
2
)
(
cos
2
2
2
2
2
2
0
2
2
0
2
0
0
2
2
0
2
2
0
2
2
2
2
t
x
m
t
m
p
x
m
m
p
Kx
m
p
V
m
p
E
(6.3)
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We should notice the symmetry not only in
K.E.
and
V
, but also in
p
and
x
.
Except for the
coefficient difference, the Hamiltonian is basically proportional to the square of
p
plus the square
of
x
.
We will expect that working in
(x, t)
will be very similar for working in
(k, t)
through the
Fourier transform in harmonic oscillators.
(
Exercise 6.1
) What function has the same functional form after Fourier transform?
Do you
expect the eigenfunctions of the harmonic oscillator Hamiltonian to be like this function?
Since the maximum
K.E.
must be equal to maximum
V(x)
, we can relate
p
0
and
x
0
by
2
0
2
0
2
0
2
2
x
m
m
p
(6.4)
We can extend the previous results to the Virial theorem in classical mechanics:
If
n
x
x
V
)
(
, then
n
E
K
V
2
.
.
(6.5)
The braket average above means the average in time.
6.2
The link to quantum mechanics through Heisenberg Uncertainty Principle
In classical mechanics, we can simultaneously know the exact value of
p
and
x
, which
takes a symmetrical form in the SHO Hamiltonian.
This however violates the basic postulates in
quantum mechanics as the Heisenberg Uncertainty Principle governing the variances of
x
and
p
:
2
p
x
(6.6)
However, quantum mechanically, due to the symmetry argument, we still expect the
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 '05
 TANG
 ground state, timeindependent Schrodinger equation, Eigenfunctions

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