Advanced Visual Quantum Mechanics – The Uncertainty Principle Part I
In classical physics, you may be familiar with the concept of uncertainty as it relates to measurement. When you measure an object’s position or momentum (or whatever) there
is always some uncertainty in your measurement due to the limitations of your measuring device. In quantum mechanics, the equations you work with have wave-like solutions that represent probabilities of positions of momentums (or whatever). Because of this wave-like and probabilistic nature, there is intrinsic uncertainty in all quantum systems. This uncertainty has nothing to do with the measuring device used – only with the system.
In this interactive engagement, you will experiment with the superposition of waves to learn more about this type of uncertainty.
1.1 Determining a wave’s position and wavelength
(From David J. Griffiths: “Introduction to Quantum Mechanics” Section 1.6)
Imagine that you’re holding one end of a very long rope, and you generate
a wave by shaking it up and down rhythmically. (Scanned image here)
If someone asked you, “Precisely where is that wave?” you’d probably
think he was a little bit nutty: The wave isn’t precisely anywhere – it’s
spread out over 50 feet or so. On the other hand, if he asked you what it’s
wavelength is, you could give him a reasonable answer: It looks like
about 6 feet. By contrast, if you gave the rope a sudden jerk, you’d get a
relatively narrow bump traveling down the line. (Scanned image here)
This time the first question (Where precisely is the wave?) is a sensible
one, and the second (What is the wavelength?) seems nutty – it isn’t even
vaguely periodic, so how can you assign a wavelength to it?
Of course, you can draw intermediate cases, in which the wave is fairly
well localized and the wavelength is fairly well defined, but there is an
inescapable trade-off here: The more precise the wave’s position is, the
less precise is its wavelength, and vice versa. A theorem in Fourier
analysis makes all this rigorous, but for the moment I am only concerned
with the qualitative argument.
This applies, of course, to any wave phenomenon, and hence in particular
to the quantum mechanical wave function.
2. Adding Waves
The ability of waves to interfere both constructively and destructively enables us to combine several different waves to create waveforms with a variety of different shapes.
Even waveforms such as that shown in figure 2a can be constructed by adding simple sinusoidal waves together.
Figure 2a: Typical shape for the probability density of a localized electron.
To see how we might create such a wave look at the addition of waves shown in figure 2b.
Figure 2b: Adding waves a), b), and c) produces the wave shown in e). Adding in another wave, shown in figure d) results in the wave shown in figure f). Figure g) shows the wave packet resulting from the addition of ten waves, each with a different amplitude and wavelength.
In figure 2b waves with different amplitudes and wavelengths are added to produce a series of wave packets. By adding hundreds of waves with carefully selected wavelengths and amplitudes we can come very close to the form in figure 2a. Each of the individual waves in figure 2b has the form,
with different values of n for every wave. As you can see from the figure, as n increases
the wavelength decreases due to the Cos(n x) term. From the de Broglie relation we know
that wavelength is related to momentum. So, when we add wave functions of different wavelengths, we are adding wave functions representing objects with different momenta.
The waveform in figure 2a is produced from a Gaussian or bell shaped distribution of
momenta. The factor ？ in equation (1) is related to the width of the momenta distribution. We will return to this factor later after you have had a chance to create some of your own wave forms.
2.1 Using Wave Packet Explorer to Add Waves
To create wave functions such as this one, you can use the Wave Packet Explorer. (you
can find this program as http://www.phys.ksu.edu/perg/vqm/software/). This program
allows you to add wave functions of different momenta (wavelengths) rather easily.
Activity 2.1a: Start the creation of a wave function by clicking in the upper left window (Amplitude vs. Momentum graph).
A vertical line represents the amplitude and momentum of a sine wave. The wave’s amplitude is proportional to the length of the line, and its momentum is the value on the horizontal axis. The wavelength is calculated from using the de Broglie relation. The wave function appears in the window directly below this line.
Wave Packet Explorer allows the following actions:
• add a wave by clicking in the amplitude-momentum graph.
• look at an individual component by clicking at the top of its amplitude-
• change either the momentum or amplitude by dragging the bar horizontally or
Activity 2.1b: Create a wave function by using 3-6 sine waves of different momenta.
Exercise 2.1a: How does the resulting wave function change as you increase the number of sine waves?
Exercise 2.1b: Sketch both your momentum distribution and resulting wave function below.
Activity 2.1c: Now, find two other groups with wave functions that look somewhat different from yours (or if there are not other groups, use the clear all button and repeat activity 2.1b two or three times trying to do something different each time).
Exercise 2.1c: Sketch these momentum distributions and wave functions below.
Exercise 2.1d: Describe how the momentum distributions contribute to the different forms for the wave functions.
Activity 2.1d: You can also add a large number of waves. With the shift key pressed, click and drag the mouse in some pattern across the top graph.
Exercise 2.1e: Sketch the momentum distribution and resulting wave functions below.
Exercise 2.1f: Compare your resulting wave functions with others in the class (or repeat the experiment yourself two or three times). Again, find a couple that are different from yours. Sketch the momentum distributions and wave functions and describe how these are similar or different.
2.2 Wave Packets and Probabilities
Wave functions which are restricted to a small region of space are called wave packets.
Activity 2.2a: Try an amplitude-momentum distribution similar to the one in figure 1.2a. Fill the whole top graph with this distribution.
Figure 2.2a: An amplitude-momentum distribution that gives a wave function and probability density similar to the one in Figure 2.1a
Exercise 2.2a: Sketch the position wave packet and probability density below.
Exercise 2.2b: Does this wave packet have a large or small range of momenta associated with it? Explain your answer.
Activity 2.2b: Now, try the same shape distribution as figure 2.2a but only in a small part of the top window (so that the distribution is very narrow).
Exercise 2.2c: Sketch the position wave packet and probability density below.
Exercise 2.2d: How is this wave packet different from the previous wave function?
As you can see from these two examples, the range of momenta affects the width of the central peak of the wave packet and, in turn, affects the probability density. A wide range of momenta results in a strong central peak, corresponding to a high probability of the electron being in a very narrow region of space, while a small range of momenta gives us a broad “peak”.
Activity 2.2c: Verify this observation by creating several different wave packets using large and small ranges of momenta.
We see that we can create a wave packet that represents an electron in a small region of space. To do so we must add together many different sine waves. Each sine wave in the summation has a different wavelength. Because wavelength is related to momentum, each sine wave represents a different possible momentum for the electron.
Thus, we need many different momenta to create a wave function for one electron which is localized in space. We interpret this result to mean that these localized electrons have a probability of having many different momenta. The probability of each momentum is related to the amplitude of the sine wave with that momentum (figure 2.2b).
High probability of thismomentum
Figure 2.2b: The probability of each momentum depends on the amplitude in the amplitude-momentum graph.
We must make a similar statement about position. Even when we add waves with many different momenta, we get a wave function similar to figure 2.2c.
High probability of finding an electronhere
Figure 2.2c: The relative probability of finding the electron at various locations for a localized wave function.
Finally, as you have observed, as the graph in momentum gets narrower, the graph in position gets wider and vice versa. This is the essence of the uncertainty principle. The wider either of these graphs are, the more uncertain you are of the object’s position or momentum. When you are more certain of its position, you are less certain of its momentum. When you are more certain of its momentum, you are less certain of its position. You will see the formula representing this principle and mathematical proof of it in your textbook and in later sections. For now, the important thing is to see how it works and where it comes from.