## Contents |

If you think back to the last lesson, you'll remember that scientists had a lot of difficulty understanding electron density and the wave function in terms of the wave properties and the particle properties of the electron. Max Born found a way to use the electron wave function to calculate electron density, and that electron density is actually equal to the *probability* of finding an electron at any point in space. This, however, leads to the question, why can't scientists predict where the electron will be with *certainty*. Why can they only predict the *probability* of finding an electron at any given point in space? Is there something wrong with the theory? Is it possible to improve the theory so that scientists *can* predict exactly where an electron is and where it's going?

## Lesson ObjectivesEdit

- Define the Heisenberg Uncertainty Principle.
- Explain what the Heisenberg Uncertainty Principle means in terms of the position and momentum of an electron.
- Explain why the Heisenberg Uncertainty Principle helps to justify the fact that a wave function can only predict the probable location of an electron and not its exact location.

## Heisenberg Proposed the Uncertainty Principle for Behavior of ElectronsEdit

### The Uncertainty PrincipleEdit

When scientists first suggested that the wave function was related to the *probability* of an electron being at a specific point in space, it raised a lot of questions. Most importantly, scientists wondered why the wave function could *only* predict the *probability* of finding an electron at a given location, and not the *exact* location where the electron actually was. Some scientists suggested that the wave function couldn't make exact predictions because it wasn't complete. They believed that the wave function was actually missing information that was necessary to describe electron behavior with "certainty".

Some early scientists thought that perhaps the wave function could only predict the probability of an electron being at a given location because the wave function was missing information. Many scientists spent time looking for "hidden variables" that controlled electron behavior just like the spin controls the movement of a baseball. The assumption was that if the correct "hidden variables" could be found and included in the wave function, then the exact movement, and the exact location of an electron could be predicted, just as easily as we can predict the movement of larger objects like baseballs, cars and planets.

Everything changed, however, in 1926 when a man named Werner Heisenberg (Figure 6.10) proposed what's known as the **Heisenberg Uncertainty Principle**. According to the Heisenberg Uncertainty Principle, it is impossible to measure certain properties, like momentum (speed multiplied by mass) and position at the same time without introducing uncertainty into the measurement. Of course, if you can't make accurate measurements, you can't make accurate predictions either.

What prevents scientists from making accurate predictions about small objects like atoms and electrons? Is it that the machinery used to make the measurements is simply not good enough? Could scientists design better machines and better measurements and *then* be able to predict electron behavior with certainty? According to Werner Heisenberg, the answer is "no – when it comes to small objects, scientists will *never* be able to make accurate measurements and accurate predictions, no matter how good their machinery is". If you find that statement strange, you're not alone. Many scientists, even today, are bothered by the Heisenberg Uncertainty Principle – it seems as if, with improved machines, we should be able to make better measurements and thus better predictions! To some extent, that's true. Better machines can help us to make better measurements and better predictions, but according to the Heisenberg Uncertainty Principle, there is a fundamental limit to how much we can know and how accurately we can know it. It's as if there is "something" in the universe which prevents us from being able to make absolutely, one hundred percent accurate measurements and, as a result, we will always be plagued by some uncertainty. For large objects, like baseballs and planets, the uncertainty is just too small to matter, but for tiny things, like atoms and electrons, the uncertainty becomes important.

### Impossible to Fix Both the Position of an Electron and Its MomentumEdit

The Heisenberg Uncertainty Principle actually applies to a lot of different measurements, but often, scientists are concerned with two in particular – position and momentum. You probably know what position means, but momentum is a term that you don't hear a lot in everyday life. Momentum, p, is the quantity that you get when you multiply an object's mass by its speed (to be truly correct momentum is actually mass times velocity, but we won't worry about the difference between speed and velocity).

In terms of the position and momentum, the Heisenberg Uncertainty Principle is as follows:

“ | There is a fundamental limit to just how precisely we can measure both the position and the momentum of a particle at the same time. | ” |

So how does the Heisenberg Uncertainty Principle relate to the electron and all the problems scientists have interpreting the electron wave function? Well, if you think about it logically, the Heisenberg Uncertainty Principle basically means that it’s impossible to predict *both* exactly what the electron will do or exactly where the electron will be found. Suppose, for instance, that you know the electron's precise position, then according to the Heisenberg Uncertainty Principle, you can't know its precise momentum as well. In other words, *when you know where the electron is, you don’t know where it's going* (since where it's going is determined by the velocity component of its momentum). Suppose, on the other hand, that you know the electron's precise momentum. According to the Heisenberg Uncertainty Principle, you can't know its precise position as well. In other words, *when you know where the electron is going, you don't know where it is*.

Obviously, there is always some uncertainty when it comes to electrons. You either don't know where they are, or else you don't know where they're going. As a result, any theory that claimed to predict exactly *where* the electron was, or exactly which path it would take as it traveled around inside the atom would go against the Heisenberg Uncertainty Principle. Luckily, the wave function description *doesn't* claim to predict the precise behavior of the electron. Instead, it only makes statements about the probability of finding the electron at one place or another.

In other words, the wave function model is consistent with the Heisenberg Uncertainty Principle. Moreover, the Heisenberg Uncertainty Principle suggests that the electron wave function equation is as complete as it can possibly be. It may not entirely predict electron behavior, but that isn’t because the model is wrong, or faulty. It's because, in our universe, there is a limit to how accurately we can actually know what tiny objects like electrons are doing.

### The Problem of Making Very Small MeasurementsEdit

When you use a thermometer to measure the temperature of a volume of water, you place the thermometer into the water and leave it there until the water and the thermometer have adjusted to the same temperature. Almost all solids and liquids expand when they are heated and contract when they are cooled. Each substance, however, expands and contracts by different amounts. In the case of the mercury and glass in a thermometer, the mercury expands and contracts faster than the glass and so as a thermometer is heated, the mercury expands faster that the glass tube and the mercury runs up the tube. The glass has been marked (calibrated) for each temperature so that you can read the correct temperature from the markings on the tube. In the process of measuring the temperature of hot water, the thermometer is placed in the water and the thermometer *absorbs* heat from the water so that its temperature becomes the same as the water and you can read the temperature of the water from the temperature of the thermometer.

You should see, at least theoretically, that when the thermometer absorbs heat from the water, the water is cooled down; that is, the temperature of the water decreases because of the heat lost to the thermometer. Therefore, the temperature you get when you measure the temperature is not the same temperature of the water that was present before you introduced the thermometer. *The act of measuring the temperature of the water changed the temperature*. When the volume of the water is reasonably large, the magnitude of the change caused by introducing the thermometer is not significant so we don't bother to consider it. What about if the volume of the water is very small? If the volume of water is 200 mL and the thermometer absorbs 20 Joules of heat, the introduction of the thermometer might change the temperature of the water by approximately 0.03 °C. Which is certainly not a significant change. But what if the volume of water whose temperature we were measuring was only 2 mL? Introducing the same thermometer into this small volume might change the temperature of the water 3 °C, which would certainly be a significant change. The point is that when we measure very small things, the act of making the measurement may change what we are observing.

Consider the method that humans use to see objects. We arrange for photons (quanta) of light to strike the object and we see the object by the directions, angles, and colors of the photons that bounce off the object and strike us in the eye or other light measuring instrument. If only red photons bounce back, we say the object is red. If no photons bounce back, we say there is no object present. Suppose for a moment that humans were gigantic stone creatures and we used golf balls to "see" with. That is, we would fire off golf balls at our surroundings and the balls would bounce off objects and come back and enter our eyes so we could see the object. If this were true, we could see mountains successfully and large buildings and trees … but could we see butterflies or small flowers? Obviously, the answer is no. The golf balls would simply knock small objects out of the way and continue on … they would not bounce back to our eyes.

In the case of humans trying to look at electrons, the photons we use to see them with are of significant energy compared to electrons and when the photons collide with the electrons, the motion and/or position of the electron would be changed by the collision. Heisenberg's Uncertainty principle tells us we cannot be sure of both the location of the electron and the motion (path) of an electron at the same time. As a consequence, *scientists had to give up the idea of knowing the path the electron follows inside an atom*.

## Lesson SummaryEdit

- The Heisenberg Uncertainty Principle states that it is impossible to measure certain pairs of properties like momentum (mass multiplied by velocity) and position at the same time without introducing uncertainty into one or both of the measurements. In other words, it is impossible to know both the exact momentum and the exact position of a particle at the same time.
- The Heisenberg Uncertainty Principle suggests that the electron wave function is complete and that it does not predict the exact behavior of an electron because it is actually impossible to do so.
- The uncertainty that Heisenberg spoke of is not due to the failure or inadequacies of the measuring equipment, but rather a fundamental limit imposed by our universe.

## Review QuestionsEdit

- What types of things in everyday life are impossible to predict with absolute certainty?
- Why is it impossible to predict the future with absolute certainty?
- Fill in the blank. According to the _________ Uncertainty Principle, it is impossible to know both an electron's _________ and momentum at the same time.
- Decide whether each of the following statements is true or false:
- (a) According to the Heisenberg Uncertainty Principle, we will eventually be able to measure both an electron's exact position and its exact location at the same time.
- (b) The problem that we have when we try to measure an electron's position and its location at the same time is that our measuring equipment is not as good as it could be.
- (c) According to the Heisenberg Uncertainty Principle, we cannot know both the exact position and the exact location of a car at the same time.

- Circle the correct statement. The Heisenberg Uncertainty Principle…
- (a) applies only to very small objects like protons and electrons
- (b) applies only to very big objects like cars and airplanes
- (c) applies to both very small objects like protons and electrons and very big objects like cars and airplanes

## VocabularyEdit

- Heisenberg's Uncertainty Principle
- Specific pairs of properties, such as momentum and position, are impossible to measure simultaneously without introducing some uncertainty.
- momentum (p)
- The quantity you get when you multiply an object's mass by its velocity (which as far as you're concerned is the same as its speed).

This material was adapted from the original CK-12 book that can be found here. This work is licensed under the Creative Commons Attribution-Share Alike 3.0 United States License