This section will explore a further property of complex waves, in particular, the spatial and momentum wave functions. These two wave functions are intimately related through the Fourier transform. You may have gotten a hint of the relation in Section 1b on deBroglie waves. These waves have a single momentum and show no localization in space. In later exercises, you found that the addition of a discrete and then a continuous spectrum of deBroglie waves with different momenta yielded a localized wave packet. The wave function represents a particle that has a greater probability of being found in some region of space and not a point in space. It would take a superposition of all momenta from +infinity to -infinity with equal amplitudes in order to confine a particle to a point in space. The representation of a particle either by a single momentum or a single position wave is a physically impossible solution. Thus, a particle is represented by a distribution of position and momenta waves. Along with this distribution comes some uncertainty in the position and momentum of the particle. This property was first realized by Werner Heisenberg in 1927 and played a critical role in the interpretation of quantum mechanics and in showing that there could be no conflict between quantum and classical physics in their respective domains of applicability.

You are familiar with the standard deviation of a distribution
of discrete measurements. We will use the square root of the standard
deviation as a measure of our uncertainty in either the position, delta x,
or the momentum, delta p. Heisenberg derived the
uncertainty relation, **deltax*deltap >hbar/2**.
According to this relation, the more localized in space a particle is, the
greater the uncertainty in momentum. There is also a fixed boundary, hbar/2,**
**to our knowledge of both position and momentum.

The Gaussian wave packet has two interesting properties. The first is that the product delta x * delta p is a minimum. For this packet there is maximum simultaneous localization in position and momentum. The second is that the Fourier transform of a Gaussian distribution is another Gaussian distribution.

Below you will see a plot of a wave function describing a particle at rest at the origin. The second plot is the momentum distribution, the Fourier transform of the position wave function. Notice that both follow a Gaussian shape.

To correctly evaluate the uncertainty in the quantity represented by a wave function, expectation value integrals must be done. However, the uncertainty in a Gaussian can be estimated by measuring the Half Width at Half Maximum (HWHM). This HWHM value is equal to (2 ln(2))^0.5 times the uncertainty,

With this method of estimating the uncertainty, the Heisenberg uncertainty
relation becomes **deltax*deltap
~ 1**.

**Student Exercises:**

- Measure the HWHM for both the position and momentum of the initial condition. Does the product deltax*deltap ~ 1?

In the next two exercises we will be changing the width of the Gaussians to see what happens to the uncertainty relation.

- Measure the uncertainty for both the position and momentum of this particle. What is the product deltax*deltap? Which is more localized compared to the initial case: position or momentum?
- Measure the uncertainty for both the position and momentum of this particle. What is the product deltax*deltap? Which is more localized compared to the initial case: position or momentum?