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,
(10.13) |
this must be a state where and 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 . A state with width should probably be a Gaussian, of the form
(10.14) |
In order for to be normalised, we need to require
(10.15) |
Actually, I shall show below that with
(10.16) |
we have
(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.
(10.18) |
Thus
(10.19) |
Let act twice,
Doing all the integrals we conclude that
(10.21) |
Thus, finally,
(10.22) |
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!