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Introduction

In the first section of this series of articles, we noted that, for our bowling ball thrown off the top of a building, we can measure the position and speed of the bowling ball to an arbitrary degree of accuracy.  We take it for granted in our every day world that we can measure a number of parameters of a system, and that there are no fundamental limitations against us doing this.  Of course, one limitation is the accuracy of our measuring equipment.  However, in principle, we can gain any desired accuracy (if the appropriate measuring equipment exists) in our measurements, and there is no law of nature that forbids us from doing so.

As you might have guessed, things are not this easy in the quantum world, and to illustrate this, we will consider the problems associated with obtaining the position and momentum of an electron.  First though, we need to talk briefly about the process of measurement.  To measure the speed (and hence momentum) of a bowling ball dropped off a tall building, we need to shine light on it (which may well come from the sun, or from an infra-red or microwave speed gun) and then process the light that is reflected from the ball.  This maybe as simple as letting the reflected rays enter our eyes and using our brain to deduce that the speed is "about 50 mph".  Alternatively, it could be detecting reflected microwaves from a speed gun, which are then processed by computer.  Whatever, the common theme is a source of light or electromagnetic radiation, followed by reflection, followed by processing.

(Before proceeding, a technical note is in order for more advanced readers.  In this article, we will consider how the use of measuring equipment on an electron leads to an inability to determine precisely both the momentum and position of that electron.  It must be emphasised that this does NOT imply that the electron has a definite momentum and position, and that it is only the attempt to measure these quantities that introduces uncertainty in these quantities.  The wave particle duality of the electron means that there is an INTRINSIC uncertainty in these quantities, which is bound by the Heisenberg uncertainty principle.

This intrinsic uncertainty emerges more naturally by considering the wave properties of the de Broglie wave of the electron.  This argument is more complex than the argument presented here, which is based on the particle properties of electrons and what happens when an attempt is made to measure the momentum and position of an electron.  Thus, it must be borne in mind that the discussion here is a simplified account of how the uncertainty principle is manifested).

How the Uncertainty Arises

The following diagram shows a schematic representation of the measurement of the characteristics of a bowling ball, done by shining light on the bowling ball and then observing the characteristics of the reflected light:

Light photons hit bowling ball, the scattered light photons enter the eye of an observer, and because the bowling ball is so massive, it goes about its business without feeling any effects from the photons that were used to observe it.  We therefore can determine the motion of the bowling ball, and more importantly, the results we get are meaningful, because we have done nothing, in the process of measurement, to affect the motion of the bowling ball.

Now let's try the same thing with an electron, as shown in the following figure.

This time, we have a VERY important distinction.  Because electrons are so small and light, when the incident photon interacts with the electron, they change the motion of the electron after the collision.  It's a bit like a collision between pool balls in a game of pool.  The motion of the object ball changes when it is struck by the cue ball.

The Uncertainty Principle

Now, electrons have momentum, and when the photon interacts with it, the electron's momentum is changed.  However, and this is a key point, we can have no knowledge of HOW the electron's momentum is changed.  Why?  Because we are observing a photon that was scattered from an electron whose initial motion was unknown in the first place.

However, we can place bounds on what happens to the electron.  There are two limiting cases that could happen, if we consider out photon to have a momentum of value p.  These are:

1.  The photon misses the electron altogether, and transfers no momentum.

2.  The photon collides "head on", and transfers all of its momentum to the electron.

There is therefore an UNCERTAINTY in the momentum of the electron after the collision that can be taken as equal to the momentum of the photon, p.  That is, the new momentum of the electron is known to an accuracy of about p.  In our example here, we know that the NEW momentum of the electron after the collision (which is equal to its original momentum plus whatever is transferred from the photon) is uncertain by a value equal to p.

Now for the slightly complicated bit, which relates to the uncertainty in the position of the electron after the photon collision.  The only information we have relates to the information contained within the scattered photon.  However, since we are using light (or indeed any form of electromagnetic radiation), it turns out that there is a fundamental limitation on how much information we can gather, and that is determined by the wavelength of the light photon.

It turns out that, if we consider for example visible light, then the amount of resolution we can gain is limited by the wavelength of that light.  For example, we cannot see atoms with the naked eye.  This is because the dimensions of atoms are very much smaller than the wavelength of light.  In our example here, it is therefore quite reasonable to state that the uncertainty in the position of the electron is roughly equal to the wavelength of the incident light.

Let us try to put this on a mathematical footing.  In science, small or uncertain quantities are specified by putting the symbol Δ in front of the quantity.  Thus, we can express the uncertainty in the momentum of the electron with the symbol Δp, and the uncertainty in its position by Δx.  Now, we saw in the previous section that the momentum of a photon is h/λ, where λ is the wavelength of the incident light.  In mathematical terms we have

For the electron momentum:   Δp = p = h/λ

For the electron position:   Δx = λ

If we combine these two equations by substituting for λ from the second equation, we get

Δp × Δx = h

This is one of the fundamental equations of quantum physics, and it expresses what is known as the Heisenberg Uncertainty Principle.  In essence, it states that we CAN NOT know SIMULTANEOUSLY the position and momentum of our electron.  We have to compromise.  We can know a little bit about both, but we cannot know both to an arbitrary degree of accuracy, unlike in the case of our bowling ball!  A staggering result, and again, a result that is completely at odds with the everyday world that we live in.

There are a few things to note about this uncertainty principle:

1.  If we knew the momentum of the electron precisely, then Δp = 0 and that would mean that Δx is infinite.  The price therefore for knowing the momentum of the electron precisely is that the electron could be located anywhere in the universe, and we would not know where!

2.  Note that this result has NOTHING to do with the limitations of the equipment, it is a characteristic of the quantum system that we are considering.

Summary

In the quantum world, the accuracy with which we can know the position and momentum of an object such as an electron is limited by the Heisenberg Uncertainty Principle.  This introduces a reciprocal relationship between position and momentum.  Thus, if the position of an electron is known to a high degree of accuracy, then the momentum of the electron will be very uncertain, and vice versa.

Next:  Wavefunctions and Schrödinger's Equation