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One of the questions of some physical interest is ``how can we create
a qunatum-mechanical state that behaves as much as a classical
particle as possible?''
From the uncertainty principle,
this must be a state where
x and
p are both as
small as possible. Such a state is known as a ``wavepacket''. We
shall see below (and by using a computer demo) that its behavior
depends on the Hamiltonian governing the system that we are studying!
Let us start with the uncertainty in x. A state with width
x =
should probably be a Gaussian, of the form
(x, t) = exp
-
A(x). |
(10.14) |
In order for
to be normalised, we need to require
| A(x)|2 = . |
(10.15) |
Actually, I shall show below that with
A(x) = eip0x/ , |
(10.16) |
we have
 
= x0,  
= p0, x = , p = / |
(10.17) |
The algebra behind this is relatively straightforward, but I shall
just assume the first two, and only do the last two in all gory details.
exp
-
A(x) = (p0 + i )exp
-
A(x). |
(10.18) |
Thus
p
=  (p0 + i )exp
-
= p0. |
(10.19) |
Let
act twice,
exp
-
A(x) |
= |
p02 + 2i p0
-  
-
x |
|
| |
|
exp
-
A(x). |
(10.20) |
Doing all the integrals we conclude that
Thus, finally,
This is just the initial state, which clearly has minimal uncertainty.
We shall now investigate how the state evolves in time by usin a
numerical simulation. What we need to do is to decompose our state of
minimal uncertainty in a sum over eigenstates of the Hamiltonian which
describes our system!
Next: 10.6 computer demonstration
Up: 10. Time dependent wave
Previous: 10.4 Simple example