Wave Packets and Uncertainty
The probability amplitude for a free particle with momentum
is the complex wave function
everywhere so this does not represent a localized particle.
In fact we recognize the wave property that, to have exactly one frequency,
a wave must be spread out over space.
We can build up localized
wave packets that represent single particles
by adding up these free particle wave functions (with some coefficients).
(We have moved to one dimension for simplicity.)
Similarly we can compute the coefficient for each momentum
, are actually the state function of the particle in momentum space.
We can describe the state of a particle either in position space with
momentum space with
We can use
to compute the probability distribution function for momentum.
We will show that wave packets like these behave correctly in the classical limit,
vindicating the choice we made for
is a property of waves that we can deduce from our study of localized wave packets.
It shows that due to the wave nature of particles, we cannot localize a particle
into a small volume without increasing its energy.
For example, we can estimate the ground state energy (and the size of) a
Hydrogen atom very well from the uncertainty principle.
The next step in building up Quantum Mechanics is to determine how a wave function
develops with time - particularly useful if a potential is applied.
The differential equation which wave functions must satisfy is called
the Schrödinger Equation.