Trying to Beat the Uncertainty Principle
In order to understand the Uncertainty Principle better, lets try to see what goes
wrong when we actually try to measure position and momentum more accurately than
allowed.
For example, suppose we look at an electr
The Gaussian Wavepacket
Fortunately, there is a simple explicit mathematical realization of the addition of plane
waves to form a localized function: the Gaussian wavepacket,
where
. For this wavepacket to represent one electron, with the probability of
f
How the Sum over N Terms is Related to the Complete Function
To get a clearer idea of how a Fourier series converges to the function it represents, it
is useful to stop the series at N terms and examine how that sum, which we
denote
, tends towards
.
So,
Thus the allowed values of p are hn/2L, where n = 1, 2, 3 , and from E = p2/2m the
allowed energy levels of the particle are:
Note that these energy levels become more and more widely spaced out at high
energies, in contrast to the hydrogen atom potential
Plane Wave Solutions
The best way to gain understanding of Schrdingers equation is to solve it for various
potentials. The simplest is a one-dimensional particle in a box problem. The
appropriate potential is V(x) = 0 for x between 0, L andV(x) = infinity
Phase Velocity and Group Velocity
It will immediately become apparent that there are two different velocities in the
dynamics: first, the velocity with which the individual peaksmove to the right, and
second the velocity at which the slowly varying envelo
Exercise: for large N, approximately how far down does it dip on the first
oscillation? (
)
For functions varying slowly compared with the oscillations, the convolution integral
will give
close to
as N increases.
, and for these functions
will tend to
It
States with Moving Probability Distributions
Recall that the Schrodinger equation is a linear equation, and the sum of any two
solutions is also a solution to the equation. That means that we can add two solutions
having different energies, and still have
Definitions of
The standard notation for the expectation value of an operator in a given quantum
state is
In other words,
would be the statistical average outcome of making many
measurements of x on identically prepared systems all in the quantum state
(i
Electron out of the Box: the Fourier Transform
To break down a wave packet into its plane wave components, we need to extend the
range of integration from the
used above to
. We do this by first
rescaling from (, ) to (L/2, L/2) and then taking the limit
We begin with a brief review of Fourier series. Any periodic function of interest in
physics can be expressed as a series in sines and cosineswe have already seen that
the quantum wave function of a particle in a box is precisely of this form. The
importa
How the Uncertainty Principle Determines the Size of Everything
It is interesting to see how the actual physical size of the hydrogen atom is determined
by the wave nature of the electron, in effect, by the Uncertainty Principle. In the
ground state of th