Physics 6 Tutorial 3 - Heisenberg's Uncertainty Principle

 

Contents

Early Quantum Physicists

Uncertainty

Particles and Probability

Quantum Model of the Atom

 

Early Quantum Physicists

Classical physics is sometimes referred to as Newton's Clockwork Universe.  You apply simple and well-defined rules, and you get the result that was predicted.  For most physical phenomena, those involving very large things, it works rather well.  However once we get to the very small, strange things start to happen.  The quantum world is is weird in the way that things happen that cannot be explained by the laws that govern the behaviour of large objects.  A new way of thinking is needed.  It is often said that if we think that we understand the quantum world, we don't.

 

The principles of the quantum world explain things like why atoms don't implode, how the Sun continues to shine, why space is not a complete vacuum, and how particles can be seen to be in two places at once.  The key concept to quantum phenomena is that energy is not continuous, but in discrete amounts call quanta.  (Quantum is a Latin word meaning How Much?.  The plural is quanta.)  Anything that is smaller than an atom behaves in a quantum manner.  If we have a fine enough pair of tweezers, we can just about pick up an atom, although it is rather fuzzy.  Once we start to look at sub-atomic particles, like electrons, neutrons, and protons, we cannot do that.  We have seen neat pictures of protons (usually red) and neutrons (usually black), but these are really a gross simplification.  The situation is a lot more complex.

 

Much of the work that has lead to the explanation of quantum concepts was been due to the work of Werner Heisenberg (1901 - 1956), Wolfgang Pauli (1900 - 1958), and Erwin Schrödinger (1887 - 1961).  They were German and Austrian theoretical physicists who developed the observations of Einstein and Planck.  Much of what they worked on involved complex theoretical arguments.  Although they met from time to time, Pauli and Schrödinger left Germany to escape from the Nazis.  Pauli went to America, and Schrödinger stayed in Ireland.  Heisenberg remained in Germany and was involved with the attempts to produce an fission bomb during the Second World War.  Fortunately the attempts were thwarted due to attacks on the laboratories which were moved several times.  Eventually Heisenberg was captured with several other physicists shortly before the end of the war and debriefed in a farm house near Huntingdon.  Not long after, in January 1946, Heisenberg returned to Germany, having demonstrated that the German physicists had not come close to producing a fission bomb.

 

Quantum Physics has a lot of complex mathematics, which involves probability.

 

 

Uncertainty

The principle of uncertainty says that there are two properties of a subatomic particle.  Firstly there is the position (x) of the particle, and secondly the momentum (p).  The more we know one, the less we know of the other.  In other words, if we are chasing an electron, the closer we get to catching the little brute, the less likely we are to get it.  There is uncertainty in the position, if we know the momentum.  Similarly, if we know the position, we find there is uncertainty in the momentum.  This is called Heisenberg's Uncertainty Principle.  In the Bohr Model of the hydrogen atom (see Physics 6 Tutorial 2), we could predict the position and the momentum of the electron.  However it only gave predictable results in very specific conditions.

 

The minimum uncertainty is the product of the position and the momentum.  This product is greater than the Planck Constant divided by 4 p. We write this as:

 

 

Sometimes you will see the equation written as:

 

The term ħ (with the slash through it ("h-bar")) is sometimes called the shortened Planck Constant, and is related to h by:

 

 

We can, of course, give a value for ħ :

 

ħ = 1.06 × 10-34 J s

 

In the SQA syllabus, we will use the Planck Constant divided by 4 p, which gives us a value of 5.28 × 10-35 J s.  The reason for this is because due to large numbers of particles in the same state, there is uncertainty in their positions and momenta given by the relationship:

 

DxDp = ħ / 2

 

There is also a minimum for the products of the uncertainty for energy and uncertainty for time.  In other words, if you know the energy of the electron, you will have a lot of uncertainty in predicting the time for the little brute:

 

 

Worked example

A neutron has a mass of 1.67 × 10-27 kg and a speed of 1.56 × 106 m s-1.  Calculate the minimum uncertainty for the position.

Answer

Momentum:

p = 1.67 × 10-27 kg × 1.56 × 106 m s-1 = 2.605 × 10-21 kg m s-1

 

Equation:

 

Minimum uncertainty in position = Dx = (6.63 × 10-34 J s) ÷ (4 × p × 2.605 × 10-21 kg m s-1) = 2.03 × 10-14 m

 

 

Although 2 × 10-14 m does not seem that far, it is about ten times the diameter of the neutron. 

Remember also that the neutron is not a neat black snooker ball with an 'n' written on it.  It is fuzzy.  And it exists as a probability.

 

Question 1

An electron moving in a straight line has an energy of 13.6 eV.  What is the minimum uncertainty in measuring the time it travels a certain distance?

What is the distance covered by the electron in this time, if its speed is 3.50 × 106 m s-1?

Answer

 

The distance travelled by the electron in the time you worked out is about the diameter of an atom.  We cannot express it in terms of the diameter of an electron, as the electron is a point charge with mass and charge only.

 

 

Position and momentum go together, and energy and time.  You cannot put together position and time, or, for that matter momentum and energy.

 

Question 2

Why not?  Explain your answer.

Answer

 

Particles and Probability

This is where things get really strange and we won't go into too much detail.  In the exam, you will only need to give a qualitative description.

 

The key point is that particles like electrons, proton, and neutrons exist as probabilities.  Consider a normal distribution curve, say for the mass of students in a secondary school.

 

If we were looking for students of mass between 72 kg and 78 kg, the probability that they will be found in the shaded box is 1 (i.e. they will all be there).  If we took a single student at random, without knowing its mass, we would know that it would be somewhere under the normal distribution curve.  However, we would need to take an educated guess, looking at the student's height, body build, etc, before we could place it on the normal distribution curve.  There is a probability that we will get the answer right, but a greater probability that we will get the answer wrong.

 

Suppose we consider the energy of electrons.  The energy of electrons is given as 9.0 eV.  We can show this on the normal distribution curve:

 

 

The probability of finding the electron under the graph is 1.  In other words, we know that all the electrons have an energy of some value (really?).  The most likely value for the energy is going to be 9.0 eV.  You may well consider that this is a thesis from the University of the Truly Obvious, but it does have an important implication, in the context of quantum tunnelling.

 

When chemical reactions occur, there needs to be a certain energy to set it going.  This is called the activation energy.  To start that reaction the electrons need to have a certain energy level, to get over the energy hill.  Sometimes the energy hill is quite low, so the electrons can easily vault over it.

 

Now consider this situation, where the electron has a certain amount of energy, but it is much lower than the energy hill.

 

On the left hand side, we have the distribution curve of electron energy (turned on its side).  In classical physics, the electron has achieved it maximum energy level at position 1.  It cannot go over the energy hill.  In quantum physics, there is a small, but definite probability that the electron can be at position 2, and so can roll down the energy hill to position 3.  We imagine that it has burrowed through the hill using a quantum tunnel.

 

This model can be used to explain how electrons in a semiconductor can jump the forbidden zone into the conduction band, when in classical physics, they could not do so.  It also helps to explain how unstable atoms decay. 

 

Erwin Schrödinger quantified the probability functions in an equation:

The strange looking symbol (that looks like a candle holder that you would find on the dining room table of a posh house) is Y, Psi, a Greek capital letter 'Ps'.  (This letter gives us the spelling of words like "psycho".)  It is the physics code for probability.  It combines the quantum aspects of a particle, and the classical aspects.  You will see more of this when you study quantum physics at university.

 

 

Quantum Model of the Atom

We are used to the Bohr shell model, in which the electrons are arranged in shells.  For example, consider neon, which has 10 electrons (as its proton number is 10).  The configuration in the shell model is:

2, 8

 

The idea of quanta has led to a more sophisticated model of the atom than the Bohr Model.  In this case the electrons occupy orbitals, where there is a high probability of finding the electron.  The idea of this is shown in the picture:

 

 

The orbitals are probability clouds.  You know that the electron is there in the orbital, but the closer you are to catching the little brute, the less likely you are to catch it.

 

There are different types of orbitals shown here, the s-orbitals that are spherical, and the p-orbitals that are elongated.  The s-orbitals carry 2 electrons, while the p-orbitals carry up to 6 electrons.  Therefore for neon, the quantum electron configuration is:

 

1s2 2s2 2p6

 

Note that 1s2 is pronounced "one-ess-two", NOT "one-ess-squared".

 

There is more discussion on Quantum concepts at Physics 6 Tutorial 17.