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This section allows us to really explore the super cool, but very weird world of quantum physics in some more detail. When we start to look and very small things, they don't behave as you might expect. Many of the scientific greats have themselves had to grapple with things behaving in unexpected and unpredictable ways. Even better, the more you learn about this field the stranger things seem - particles are no longer just particles, waves no longer just waves; they each exhibit characteristics of both.
Unsurprisingly, there's quite a lot here. However, Crash Course have produced a double bill of videos on quantum mechanics which touch on a lot of the content covered here. Take a look.
This section has been divided up as follows:
The Photoelectric Effect - evidence that light is actually a particle, not a wave...
DeBroglie Waves - well, actually things can behave as both waves and particles
Heisenberg Uncertainty Principle - introducing the idea of the probability distribution of matter
Quantum Tunnelling - the weird phenomenon of matter being able to move through a barrier and appear on the other side
We've talked extensively about the behaviour of light waves and specific bits of evidence for light's wave-like nature, including:
However, we have also talked previously about the idea of the photon - a 'packet', or particle, of light energy. Light appears to exhibit behaviour of particles and waves depending on the circumstances. TedEd have a little video which introduces this dual wave-like and particle-like nature.
But what evidence do we have for the particle-like nature of light? Here we start with something called the photoelectric effect. This is one of the key pieces of evidence behind the photon - i.e. light as a particle. This quantisation of light led to the birth of modern physics.
The experimental setup of the photoelectric effect is quite straightforward, and can be demonstrated with a very simple piece of kit - the gold leaf electroscope.
When a charged object (either positive or negative) is touched to the metal cap, the electroscope becomes charged, meaning the gold leaf repels from the metal stem. When the system is discharged, by touching a conductor to the metal cap, the gold leaf returns to it's original position.
If the electroscope becomes negatively charged, the scope can also be discharged by removing electrons from the system by shining a bright light onto the metal cap - providing energy to the electrons to allow them to escape from the metal surface, thereby discharging the electroscope (as shown below). This does not work when the plate is positive charged, why not?
So how does this simple setup, and the discharging of electrons from the plate surface give us evidence for the existence of the photon?
The PHET simulation below demonstrates this. Let's compare Violet light (higher energy) and Red light (lower energy).
Violet light causes the emission of photoelectrons from the surface. The photons give energy to the electrons allowing them to escape the surface. If the intensity of violet light increases, more photoelectrons are released (though with the same average kinetic energy). Each photon provides a one-to-one interaction with an electron, so a higher intensity corresponds to more photons.
Red light does not result in photoelectrons being emitted, as the energy of each photon is insufficient to allow an electron to escape the metal surface. Even when the intensity is increased to 100%, no photoelectrons are emitted.
A really good simulation from PHET here on the Photoelectric effect. Take a look at what happen in the following cases:
Changing the wavelength of the incident light
Changing the intensity of the incident light
Changing the potential difference between the plates
Finding the stopping potential for each material
As shown in the diagram below, when photons of sufficient energy incident on the surface, there a one-to-one interaction of photons with electrons. The energy of the photon (E = hf) is given to the electron, which is excited. A certain amount of this energy is used to 'escape' from the metal and become liberated - the amount of energy required to free the electron is called the work function, Φ, and depends upon the metal surface. Any excess energy gained from the photon goes into kinetic energy of the electron as it zips away from the metal surface.
Mathematically this can be expressed as following:
(Note, here the kinetic energy is the maximum kinetic energy of photoelectrons, there will be a range of photoelectron kinetic energies as some will be liberated from deeper within the bulk metal, therefore more energy is required to liberate them),
A couple of Geogebra simulations to summarise this subsection:
Work Function : How does this graph change if we vary our work function and the wavelength of incident light?
Photoelectric Effect : This simulation allows you to play around with some of these key ideas. Add the battery tab and experiment with the potential difference required to stop the photoelectrons.
Intro to Quantum Weirdness
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The Bohr Model (revisited)
Let's look back at out Bohr model of the atom we looked at in Chapter 7. This is our model of the atom that looks at the energy levels. There are a few key ideas we remember from the Bohr model:
Electrons can only occupy discrete energy levels
Electrons orbit in these energy levels without radiating energy
Electrons can move between energy levels by absorbing or emitting photons (excitation and deexcitation)
We also looked at quantifying these energy levels using the following equation:
So where did this 13.6 eV come from? There is a really neat derivation to this one which brings together multiple parts of the IB curriculum, including the Fields in Chapter 11 and the Coulombs Law and Circular Motion stuff. You don't need to know this for the IB, but I highly recommend taking a look here.
Anyway, Bohr's model of the atom followed a key postulate that he made:
The angular momentum of the electron is quantised (can only take discrete values)
This angular momentum depends upon which energy level the electron occupies. The angular momentum of the electron is given by the following equation.
So we have seen evidence of light exhibiting wave-like behaviour (i.e. Young's Double Slit, interference and superposition). The interference of our light waves produces bright and dark fringes due to the constructive and destructive interference. We can't imagine a series of light particles hitting our viewing screen and 'cancelling out', to create dark regions.
However, our Photoelectric effect shows pretty compelling evidence that light behaves as a particle - the one-to-one interaction between photons and photoelectrons seems to suggest that light is made up of individual 'packets' of energy (which we of course know as photons).
In reality, light exhibits both wave-like and particle-like behaviour - in something known as 'wave particle duality'. Light is not the only thing that follows this duality - things we have previously accepted as being particles (e.g. electrons) can also exhibit wave-like behaviour, such as interference and superposition.
The below video illustrates this point, demonstrating this weird quantum behaviour. It also touches upon a particularly interesting idea about the effect of 'observing' the behaviour and it's effect on the quantum behaviour. This latter idea is completely unintuitive - it is just not possible to translate our macroscopic view of the world onto such weird quantum ideas.
De Broglie stated that matter can exhibit wave-like properties. Particle-y things, like electrons and neutrons have a certain wavelength that depends on their momentum. This is what we call the de Brroglie wavelength, and is given by the following equation.
This de Broglie wavelength is a very useful idea. Our previous discussion of electron shells in the Bohr Model can be also explained using standing waves around the nucleus. If we go back to our ideas of standing waves , we looked at different harmonics with varying numbers of nodes and antinodes. If we apply these ideas to our Bohr model, these electron energy levels correspond to increasing harmonics of standing wave around the nucleus. The PHET Hydrogen atom simulation shown below exhibits this idea - showing the excitation to higher integer harmonics. You can also see the Schrödinger model shown below.
The wave-like behaviour of electrons can be exploited for a number of useful applications. Electron crystallography involves firing a beam of electrons through a crystalline substance (with a regular atomic spacing). As the electron wavelength is of a similar order of magnitude to the atomic spacing a diffraction pattern is observed (similar to the diffraction gratings in Section 9.3). Scientists can use this diffraction pattern to determine different parameters of the crystal structure.
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Wave Function & Heisenberg Uncertainty
Now, with this idea of waves behaving like particles and particles like waves, our classical 'orbital model' of the nucleus isn't looking so useful. We have previously looked at the electron as a 'billiard ball' type point orbiting around a nucleus. Modern physics does not think of the electron as a single point at all, instead it is much more useful to think about an 'electron cloud' around the nucleus. This cloud is a region in which there is a probability of 'finding' the electron, something we call a probability distribution. Around the nucleus there will be certain areas in which the probability of find the electron is higher, and regions where it is less, and we can map these probabilities.
This is exhibited with our diagrams below, our traditional model on the left, our probability cloud on the right..
The shape of this probability distribution can vary with multiple electrons around a nucleus. (In fact for those chemists, is the reason behind different orbital shapes in chemistry).
We can mathematically describe the probability distribution with something called the wave function, Ψ.
We have an equation shown below that is particularly useful. This equation shows us that the wave function squared, Ψ², tells us the probability, P(r) of finding our electron within a certain volume of space, AV.
(The derivation of the wave function comes from something called the Schrödinger Equation, which although beyond the scope of the IB, is a core component of modern physics - learn more here) ​
Let's take a look at this in some sort of context. This graph shows two different probability distributions with distance from the nucleus.
The distribution in red has a high probability of being found close to the nucleus, which decreases with distance.
The distribution in yellow has the highest probability of being found a certain distance from the nucleus instead.
These wave function shapes can take on a number of different shapes, representing the electron probabilities as shown with the hydrogen wave function models below.
Werner Heisenberg built upon this idea of the wave function by developing something called the Heisenberg Uncertainty principle. Despite this he still is probably most famous for being the alias of Walter White in Breaking Bad. TedEd have a little video to introduce this.
The Heisenberg uncertainty principle is a set of equations, as follows:
The right hand side of the equation is a constant (and a very small number at that!). The left hand side is the product of the uncertainty in two quantities. So let's understand a little bit about what these mean.
The first equation states that the uncertainty in position multiplied by the uncertainty in momentum will be above a certain value (given by our constant). This means that if we manage to work out to a high degree of precision the momentum of an electron, we will inherently have a certain uncertainty in our position. This is nothing to do with some limit in the uncertainty of the equipment, even with perfect equipment this uncertainty is an inherent and unavoidable property. Conversely, we have talked about the probability distribution about the area where we might find our electron - if we pinpoint the location with high precision, we must have a certain uncertainty in our momentum. There is always an inherent uncertainty in these two quantities that is unavoidable. The more we know one, the less we know the other.
The other equation is a funny one that we will explore below when looking at pair production. The proof for these ideas are beyond the scope of the IB. However, this online lesson goes into some greater depth about the uncertainty principle.
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Pair Production & Annihilation
We looked in Chapter 7 at the idea of mass energy equivalence. This makes use of Einstein's famous equation:
We also looked at the standard model, which includes particles of matter and antimatter. If a matter particle and its antimatter equivalent meet, they will annihilate. The combined mass of the particle/ anti-particle willpair will be converted into photons of an equivalent total energy. There must be at least two photons produced, such that conservation of momentum is obeyed (each photon has a momentum in a different direction).
The Feynman diagram for annihilation of an electron and positron is shown here. The combined mass of the positron and electron is released as photons of an equivalent energy 1.022 MeV.
Conservation of momentum must also continue to be observed. In order for this to be possible, two photons must be produced (with momenta in perpendicular directions).
The Feynman diagram for pair production of an electron and positron is shown here. Pair production of other particle/ antiparticle pairs is possible (e.g. tau, muons) if higher energy photons are used. This always occurs next to a nucleus/ orbiting electron in order for conservation of momentum to be observed (in which case this particle will gain momentum).
Going back to our Heisenberg uncertainty equations above, our second equation gave the product of an uncertainty in energy and an uncertainty in energy that will always be above our critical value. This law leads to some interesting possibilities occurring. With pair production, a matter-antimatter pair can be produced from a photon with sufficient energy. It is seemingly possible that conservation of energy is violated, and that the matter-antimatter pair have a greater energy than the photon from which they were produced, provided their lifetime is short enough that the still obey Heisenberg's uncertainty.
Pair Production and Annihilation
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There are some baffling and counter intuitive implications of the Heisenberg Uncertainty Principle and the Wave Function. which lead to some very weird effects of the quantum world. One of these is quantum tunnelling.
The wave function allows us to work out the probability of finding an electron in a certain place. If there is a thin barrier (which you can imagine as a thin potential 'wall'), such that the wave function gives a probability of finding the electron on the other side of that barrier, the electron is able to pass through that barrier, and appear on the other side. This is known as quantum tunnelling.
The probability of quantum tunnelling can be increased by:
Decreasing the width of the barrier
Reducing the energy of the potential barrier
The phenomenon of quantum tunnelling is exploited by the scanning tunnelling microscope to image materials on the nano-scale.
We can use our ideas of probability distribution to visualise an electrons ability to tunnel. If the probability distribution extends beyond the width of the physical barrier, there is a probability that the electron can tunnel through the barrier.
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Definitions and Key Words : Chapter 12
A set of Quizlet flashcards of the key words and definitions for this chapter is provided here.
A question by question breakdown of the IB papers by year is shown below to allow you to filter questions by topic. Hopefully you have access to many of these papers through your school system. If available, there may be some links to online sources of questions, though please be patient if the links are broken! (DrR: If you do find some broken links, please contact me through the site)
Questions on this topic (Section 12) are shown in light purple.
Use this grid to practice past IB questions topic by topic. You can see from the colours how similar the question topic breakdown is year by year. The more you can familiarise yourself with the IB question style the better - eventually you will come to spot those tricks and types of questions that reappear each year.