I should point out that the arguments here are not straightforward, so don't worry if this section is unintelligible to you.  I have summarised the main conclusion at the end of the section.

In the first section of this series of articles, we made some thing of a throwaway remark.  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 simultaneously, and to an arbitrary degree of accuracy.  We take it for granted in our every day world that we can measure more than one quantity at once, and that if we want to be more accurate in our measurements, then we simply apply more sophisticated measuring equipment.  This is a characteristic of our everyday world - there is no fundamental barrier to making simultaneous measurement to arbitrary degrees of accuracy.  Granted, the equipment may not yet exist to enable such measurements to be made, but there is no law of physics that says that such measurement can NOT be made, if the equipment exists.

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 measuring the speed and position of an electron.  First though, we need to talk briefly about the process of measurement.  To measure the speed 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 the reflected microwaves from a speed gun that are then processed by computer.  Whatever, the common theme is a source of light followed, by reflection, followed by processing.

The following diagram shows a schematic representation of such a measurement, for a bowling ball.  Note that in the following, we will be treating light as having a particle character, i.e. as a stream of photons.  We do this because the arguments are slightly simpler this way.

The process is really quite simple.  The photons hit bowling ball, the results are processed, and because the bowling ball is so massive compared with the photons, 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, when the photons hit the electron, they change the motion of the electron after the collision.  It's a bit like having one pool ball hit another in a game of pool.  The motion of the ball that was hit is changed.

Now, electrons have momentum, and when the photon collides 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 this change comes about through our attempts to observe the electron 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 is equal to the momentum of the photon, p.  That is, the new momentum of the electron is known to an accuracy of p.  To give a more everyday example of uncertainty, if I say that there is an uncertainty about my bank balance of £1,000, then it means that I know the value of my bank balance to within £1,000.  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, and you will have to take my word for this!  if the reflected photon is subsequently received by measuring equipment, then we can attempt to determine the position of the electron at the point of collision.  However, since we are using light, it turns out that there is a fundamental limitation on how much information about position that we can gather, and that is determined by the wavelength of the light.  It turns out that, if we consider 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 distances involved are very much smaller than the wavelength of 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 waves whence the photons come.  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 Δx = 0 and that would mean that Δp 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.

If you didn't follow the way in which we derived the uncertainty principle result, don't worry.  The key message to take away is this:

In the quantum world, attempting to make measurements fundamentally changes the system we are trying to observe, and there is NOTHING we can do to prevent that.  This arises because quantum objects are so small and easily disturbed.  The consequence is that we have limitations imposed on us about how much information we can gather about the quantum system.  This is completely different from the everyday world we live in, where measuring things about bowling balls causes us no problems.

On to Wavefunctions and Schrödinger's Equation