To define the notation, let us briefly recap the dynamics of the classical oscillator: the
constant energy is
or
.
The classical motion is most simply described in phase space, a two-dimensional plot
in the variables
. In this space, the point
correspondi
Time Development
Turning now to the time development of the state, it is convenient to use the ket
notation
with
denoting a minimum uncertainly wave packet (with the same spatial width
as the ground state) having those expectation values of position and m
The set of normalized eigenstates
discussed above are of course solutions to
the time-independent Schrdinger equation, or in ket notation eigenstates of the
Hamiltonian
Putting in the time-dependence
explicitly,
. It is necessary to include the time depen
Notation
We have chosen to work with the original position and momentum variables, and the
complex parameter expressed as a function of those variables, throughout. We could
have used the dimensionless variables introduced in the lecture on the simple
har
Hermite Polynomials
so
It follows immediately from the definition that the coefficient of the leading power is
2n.
It is a straightforward exercise to check that Hn is a solution of the differential
equation
so these are indeed the same polynomials we fou
Heisenberg Representations
Assuming a Hamiltonian with no explicit time dependence, the time-dependent
Schrdinger equation has the form
and as discussed above, the formal solution can be expressed as:
Now, any measurement on a system amounts to measuring
Heisenberg Annihilation
For the simple harmonic oscillator, the equations are easily integrated to give:
We have put in the H subscript to emphasize that these are operators. It is usually clear
from the context that the Heisenberg representation is being
Free Particle Propagator
To gain some insight into what the propagator U looks like, well first analyze the
case of a particle in one dimension with no potential at all.
Well also take
to make the equations less cumbersome.
For a free particle in one dime
Free Particle Eigenstate Problem
Notice first that
is constant throughout space. This means that the
normalization,
! And, as weve seen above, the normalization stays
constant in timethe propagator is unitary. Therefore, our initial wave function must
hav
Consider a macroscopic simple harmonic oscillator, and to keep things simple assume
there are no interactions with the rest of the universe.
In fact, the ground state of a simple harmonic oscillator is a minimum uncertainty
state. This is not too surprisi
Normalizing the Eigenstates in x-space
The normalized ground state wave function is
where we have gone back to the x variable, and normalized using
.
To find the normalized wave functions for the higher states, they are first constructed
formally by apply