Physics 6 Tutorial 17 - Interpreting Quantum Theory (*)

The material in this tutorial will only be examined in Paper 3 Section 2.

 Contents

The Double Slit Experiment

Interference of light passing through two close-set parallel slits was first demonstrated by Thomas Young (1773 - 1829) in 1801.  From the section on Waves, we know that when a laser is passed through two slits, an interference pattern is observed:

The pattern observed on the screen was like this:

You may wish to revise this in Waves Tutorial 7, and a more detailed discussion is available in Physics 6 Tutorial 7.

Physicists thought that electrons were particles.  If the electrons passed through double slits, the slits would act as a stencil to give a pattern like this:

Instead, as we saw in Quantum Physics Tutorial 6, accelerated electrons could be made to form diffraction patterns:

We saw that if waves had particle properties (photons), it was very reasonable to assume that particles has wave properties.  This led to the concept of wave-particle duality.  It would also be reasonable to suppose that if electrons could diffract, they could interfere.  And if we set up a similar apparatus to the above to show two slit interference, we see an interference pattern.  The idea is shown below:

The same effect is observed with neutrons.  (Neutrons, being neutral cannot be accelerated by electric fields and do not interact with magnetic fields either.  Instead they are be emitted from fission reactions, and formed into beams.  An alternative way to produce a more intense beam is to strip them from their partnering protons by a deuteron-stripping reaction.  We won't go into this here.)

So far, so good, but know it's going to get weird...

It could be argued that many thousands of electrons jostling against each other could somehow interfere.  Would the pattern be different if one single electron at a time was fired at the screen?  This is what they saw:

If the electron was a particle, it would hit the double slit and be absorbed.  There would be no pattern at all.  Even if the electron passed through one or other of the slits, the pattern would be that of a stencil.  Instead there was an interference pattern, suggesting that the electron was in two places at once.   We could say that the electron was a wave, which would be consistent with wave-particle duality.  However, if an electron was tracked, the pattern observed was that of the stencil, i.e.:

It was hard to explain such a contradiction, but in 1920, an attempt was made by Niels Bohr and Werner Heisenberg.

The Wave Function

The state of quantum systems is represented by a wave function, given the physics code Y, ("Psi", a Greek capital letter 'Ps', as in psycho).  The wave is not mechanical, electromagnetic, or electrical.  Wave function unites the wave and particle properties of quantum particles like electrons using a complex mathematical equation which is beyond what you need to know.  The amplitude represents a probability of finding a particle in a particular place.

Therefore the probability of finding a particle at xC is greater than the probability of finding the particle at xB, which is greater than the probability of finding the particle at xA.  The "wavelength" or "period" has no significance.

The wave function describes the probability of a particle being at a point.  It is given the Physics code Y.  It represents the amplitude of a probability wave, or a de Broglie matter wave.  The units for Y are m-1.5.  The amplitude of such a wave does not have any specific physics significance.  However the term Y2  (Psi-squared) is the probability per unit volume.  The units for Y2 are m-3.

Consider an electron orbiting a hydrogen nucleus (a proton) at the Bohr radius (5.29 × 10-11 m).  The idea is shown below:

At the Bohr radius, the probability of finding the electron is highest.  However there is a small probability that the electron can be inside the Bohr radius, or outside the Bohr radius.  Therefore the cloud is not smooth billiard ball (nor is it light blue).  It is fuzzy.  The probability can be worked out using the equation:

The terms are:

• P(r) - the probability of finding the electron at a radius, r (no units);

• Y2 - the square of the wave function, which is the probability per unit volume (m-3);

• DV - the difference in the volume between where the electron is and the volume at the Bohr radius (m-3).

To get a value for Y, we need to use the Schrödinger equation, which is challenging to say the least.  It is a differential equation that has different solutions in different circumstances.  It also has complex numbers (i2 = -1).   You will meet it at university, but its discussion is beyond the scope of these notes.    For the electron in the ground state the solution is this equation:

The key terms are:

• a - the Bohr radius (5.29 × 10-11 m);

• r - the radius of where the electron is (m).

So let's put some numbers in:

 Worked example An electron is orbiting a hydrogen nucleus at the Bohr radius.  What is the value of the wave function at the Bohr radius?  Give the unit.  Hence work out the probability density. Answer We will break up the equation into its bits and work out the value of each bit, and then put it all together. (p0.5 × (5.29 × 10-11)1.5) = 6.82 ×10-16 m-3/2   e-5.29 ×10^-11 ÷ 5.29 × 10^-11 = e-1 = 0.367   Y = (6.82 ×10-16 m-3/2)-1 × 0.367 = 5.38 × 1014 m-3/2.   Probability density = Y2 = 2.90 × 1029 m-3.

This answer seems to be very high, but remember that it's a probability density, probability per unit volume.

Now let's consider the probability of finding an electron that has moved from a radius of 5.29 × 10-11 m to a radius of 5.31 × 10-11 m.

 Worked example An electron is orbiting a hydrogen nucleus at a radius of 5.31 × 10-11 m.  What is the value of the wave function at this radius?  Hence work out the probability density. Answer We will break up the equation into its bits and work out the value of each bit, and then put it all together. (p0.5 × (5.29 × 10-11)1.5) = 6.82 ×10-16 m-3/2   e-5.29 ×10^-11 ÷ 5.31 × 10^-11 = e-0.996 = 0.369   Y = (6.82 ×10-16 m-3/2)-1 × 0.369 = 5.41 × 1014 m-3/2.   Probability density = Y2 = 2.93 × 1029 m-3.

Again this number seems very large, but let's see what happens when put it into the equation:

 Worked Example What is the probability of finding at electron outside the 1s orbital (the Bohr radius) at a radius of 5.31 × 10-11. Answer We need to work out the change in volume: At the Bohr radius: VB = 4/3 × p × (5.29 × 10-11 m)3 = 6.20 × 10-31 m3   At the new radius: Vr = 4/3 × p × (5.31 × 10-11 m)3 = 6.27 × 10-31 m3   DV = 6.272 × 10-31 m3 - 6.200 × 10-31 m3 = 0.072 × 10-31 m3   Now substitute:   P(r) = 2.93 × 1029 m-3 × 0.072 × 10-31 m3 = 0.0021

This gives us a one in five hundred chance of finding the electron outside the 1s orbital.

(Note:  Finding numerical values of Y has been incredibly difficult.  The sources that I have searched have given long and learned expositions in mathematics that have not been that helpful.  I have not yet seen a value for Y anywhere, so my calculation may well be completely wrong.  If you can do better, please tell me (without complicated mathematics) and I will give you credit for it.  Your explanation should be understandable for a pre-university student.)

The Copenhagen Interpretation

This was devised by Niels Bohr and Werner Heisenberg and was given the name as Copenhagen is where they worked.  It is the most widely accepted interpretation of the observations above.

The Copenhagen Interpretation went along these lines.  If a quantum system, for example an electron, is left to its own devices, superposition could occur between the wave functions of the electron as it occupied its quantum space.  This interference patterns could be formed.

If, however, we try to track the electron and measure its behaviour, it immediately reverts to being a particle.  The wave-function collapses and the electron becomes a particle which follows the rules of classical physics.

Needless to say, not all physicists accepted the Copenhagen Interpretation.  Nowadays it would be called a "fudge".

Feynman's Sum-over-Histories

This explanation was proposed by Richard Feynman (whom we know from his diagrams of particle interactions).  Consider a particle going from A to B starting at time tA and getting there at time tB. In classical terms, the particle passes from A to B by the shortest route:

In the Feynman explanation, the particle can go any way it likes:

It can travel at any speed as well.  The diagram does not show the speed.  There are an infinite number of paths that can be travelled.  The idea is that these can be added up, and sometimes the sum gives reinforcement, and other times the sum results in cancellation.  This reminds us of waves.  The sum of the paths gives rise to an interference wave pattern.  This results in the probability wave of detecting the particle. This is called the path integral or the sum over histories.  The mathematical models that underpin this are complex, involving the use of the square root of -1.

 It is tempting to think about the paths in terms of the random paths taken by molecules in a gas.  However the random paths are caused by collisions with many other molecules.  Between collisions, the molecules travel in straight lines.  The behaviour of the molecules is governed entirely by classical physics. There is only one particle in this explanation.  Its path is random as a result of the probability.

The Many Worlds Interpretation

This was worked out in 1957 by the American physicist, Hugh Everett (1930 - 1982).  He promoted the idea of many parallel universes.  Each universe would have one out of the possible outcomes for a particular event.   At each event, the universe split into two different universes, and each resulting universe carries each particular outcome.  The idea is shown in the diagram below:

Therefore a physicist in one universe measures the classical behaviour of a particle.  In the parallel universe, a second physicist can analyse the wave probability properties.  However we cannot be aware of ourselves in parallel universes.  Suppose I decided not to go out in the car one weekend.   My car's absence on the road that afternoon resulted in there not being an accident in which I was seriously injured.  Instead I ended up doing some wood-turning in my workshop.  In the parallel universe, I took the car out and had a bad accident.  I could be spending that afternoon in hospital with serious injuries, wishing that I was in my workshop.  The consequences could be that I could never use my workshop again.

Another event could be that I went for an interview for a job I really wanted.  In one parallel universe, I might have got the job, but found they didn't like me that much, and I would find myself being fired.  In the second parallel universe, I wouldn't have had a successful interview.  I would be very disappointed.  My conception would be that it was going to be great and I would thrive.  I wouldn't know that I would not have lived up to their expectations and end up being "let go".

The Austrian physicist, Erwin Schrödinger (1887 - 1961) found the Copenhagen Interpretation problematic.  At what point does a quantum system cease to be a superposition of many states, and the wave function collapses to become one state or another in order to be observed?  He illustrated his concern in 1935 using a famous thought experiment, called Schrödinger's Cat.  The idea is shown below:

Image by Dhatfield, Wikimedia Commons

A cat is placed in a sealed box with a radioactive source.  When a nucleus decays, the radioactive emission is detected by a Geiger counter that trips a mechanism that drops a hammer on a flask of hydrogen cyanide (HCN) solution.  The flask is smashed to release hydrogen cyanide which will kill the cat.  The box is kept shut until an observer opens it.  The cat can have two states, dead or alive.  It cannot be both dead and alive.  When the box is opened, either the nucleus has not decayed, and the cat is alive, or the nucleus has decayed and the cat is dead.  The cat is alive or dead long before the box is opened.  (No cats were harmed in this experiment.)

This thought experiment can be interpreted in the Everett Many Worlds Interpretation.  In one parallel universe, it is alive.  In a second, it's dead.

More thought experiments have been carried out to try to explain the difficulty of making observations in quantum systems.  On such is Quantum Suicide devised by Max Tegmark (1967 - ) in 1997.  Thought experiments are useful to discuss and debate fundamental concepts.  Factors can be changed easily and "what if?" scenarios can be used to probe different aspects of the debate.

Heisenberg's Uncertainty Principle

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 2p. We write this as:

• Dx - the uncertainty in the position (m);

• Dp - the uncertainty in the momentum (kg m s-1);

• h - Planck's constant (= 6.63 × 10-34 J s).

In the Pre-U syllabus, we will use the Planck Constant divided by 2p, which gives us a value of 1.06 × 10-34 J s.   In some texts you will see the equation written as:

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:

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

Which is why we can write:

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) ÷ (2 × p × 2.605 × 10-21 kg m s-1) = 4.05 × 10-14 m

Although 4 × 10-14 m does not seem that far, it is about twenty 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.

The Heisenberg Uncertainty Principle limits what we know about the way a quantum system works.  Heisenberg himself carried out a thought experiment whereby he had a gamma ray microscope.  In order to observe with gamma photons, the photons would have to interact with the electrons.  Photons have high energy and when they interact with an electron, they will pass on that energy.  Since energy in an electron is only kinetic, the increased kinetic energy leads to a higher speed.  Therefore the momentum will increase.   If Heisenberg could track an electron down to observe it with the gamma microscope, he could get a lot of information about the position of the electron, but there would be a lot of uncertainty about the momentum of the electron.

No gamma-ray microscope has ever been invented, but that's the beauty of a thought experiment.  Werner Heisenberg was also hopeless at practical physics.

There is a whole branch of philosophy that discusses determinism.  Philosophers think that things will happen as a series of pre-determined and consequential events, just like a train going down a track will pass a series of stations, level crossings, and tunnels.  I am not very philosophical at all, so I am not going to go down that track.  There are two things that are certain, death and taxes.

Instead, I will confine things to how determinism is relevant to Physics.

The kind of physics that we experience from day to day is classical or Newtonian physics.  Determinism in this context means cause and effect, for example:

• A force applied to an object causes it to accelerate in the direction of that force;

• Work is done when a force is moved through a certain distance in the direction of that force;

• Two electrical currents come together at a junction and sum to make a third current.

The results are predictable and can be easily observed.

Quantum Physics is indeterministic.  There is no direct cause and effect.  It works on probability.  Even if the starting point is known, there are many possibilities as to where the process may lead.  Some paths are more probable, while other are less so.  But they are not impossible.

Consider this argument:

"Fingers" is a criminal.  He is in prison, but he doesn't think he should be inside.  He wants to be outside to commit more crimes.  This is our starting point.

In the real world, Fingers can jump, but not that high.  He could not jump from the Inside to the Outside.  So he remains in the nick.  The possible ways for Fingers to get to the Outside include:

• He does his time;

• Some other villains on the outside lift him over the wall with a helicopter (yes, it has been done before);

• He tunnels out;

• He escapes in a laundry basket (that has been done before as well).

All of these are the result of cause and effect.

But in the quantum world, Fingers is in a probability cloud.

Fingers' probability cloud extends to just over the prison wall.  So there is a tiny, but real, probability that he could happen to be just on top of the prison wall, so he could make good his escape.

Fingers does not need to do anything in the probability world, except to wait for the moment that probability takes him over the wall.  We would observe that he has tunnelled through the wall, except that there is no evidence of a tunnel.

Quantum tunnelling is a real world application of probability.  For further discussion of this, go to Physics 6 Tutorial 3.

Einstein and Quantum Theory

Albert Einstein (1879 - 1955) was one of the founding fathers of quantum physics.  He proved the photon nature of light as we have seen in the section on Quantum Physics.  However, as quantum physics evolved in the early part of the Twentieth Century, he found it harder and harder to accept the indeterministic nature of the quantum world.  In 1916 Einstein had predicted the stimulated emission of light, which is now used in the laser.  In this a photon bumps into an electron and knocks it to a lower level, releasing a photon.  But then he discovered that an atom could emit a photon randomly without the interaction of a passing photon, using a very similar mechanism to stimulated emission.  Being a random process, the photon could go in any direction and this could not be predicted.  This troubled him.

As Einstein produced more papers along with Satyendra Nath Bose (1894–1974) and Erwin Schrödinger, he became more troubled by the probability nature of the quantum world.  He particularly disliked the idea that a quantum particle did not move along a predictable physical path.  And he disliked the idea that one could not measure physical parameters like momentum and energy at any particular moment in time at any point (Heisenberg's Uncertainty Principle).

To sum up Einstein's objections:

1. Quantum Theory was indeterministic, so all outcomes were based on probability.  There was no cause and effect.  ("God does not play dice.")

2. Quantum Theory was not local.  The behaviour of an electron in one atom could affect another atom a long way away (quantum entanglement).

3. Quantum Theory could not predict accurately the motion, energy, or other properties of a particle.

The nature of the quantum world still continues to baffle physicists to this day.