====== 1.vi.1 Particles vs. Waves ======

In classical physics, particles and waves are mutually exclusive.

  * A particle is described by its position $\vec{r}(t)$ and momentum $\vec{p}(t)$, and always has a well-defined trajectory in space-time.
  * A wave is an excitation of a field and is specified by its wavefunction, e.g. $\psi(\vec{r},t) = Ae^{i(\vec{k}\cdot\vec{r} - \omega t)}$.  It is spread out over space at all times.

In quantum mechanics, the //same// physical system, e.g. an electron or light, sometimes displays wave-like properties and sometimes particle-like properties, so which is it?
  * Are quantum systems //both// waves and particles at the same time?
  * Are they //sometimes// waves and //sometimes// particles, but never both at the same time?
  * Are they //neither// waves //nor// particles, but something completely different?
  * Is the whole question meaningless?

These are //good// questions and we do not have a completely satisfactory answer to them.  All of the answers have advantages and disadvantages and we cannot conclusively rule out any of them at the moment.  You should not panic about this, as being in a state of uncertainty is normal in the progress of science.  For example, we currently do not have a completely satisfactory answer to how life on earth started or what dark matter is made out of.  This does not make it impossible to do biology or cosmology, nor does it mean we should stop looking for answers to these questions.  The situation with quantum mechanics is the same.  Despite the fact that it does not give us an unambiguous answer to what the world is made of does not prevent us from //using// the theory to make accurate predictions, nor does it mean that we should stop asking questions about the foundations of the theory.  The focus of this course is on how to //use// the theory.  I will point out foundational questions where they occur, but I will not dwell on them.  If you are interested in the foundational aspects of the theory then you can take the course "Philosophy and Foundations of Quantum Mechanics" when it is offered,  [[http://pirsa.org/C19002|watch my masters level lectures on the subject]], or do an undergraduate research project with me or another faculty member on these issues.

====== 1.vi.2 The Double Slit Experiment ======

To illustrate some of the difficulties in interpreting quantum mechanics, we will consider the famous //**double-slit experiment**// from the point of view of both classical and quantum mechanics.  

Richard Feynman described this experiment as:

> a phenomenon which is impossible, //absolutely// impossible to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the //only// mystery. //[[https://www.feynmanlectures.caltech.edu/III_01.html|The Feynman Lectures on Physics, vol. 3, section 1.1]]//

I disagree with this characterization.  There are a few different reasonable explanations of what is going on in the double slit experiment, and we just do not know which, if any, is the correct one.  Nevertheless, the double slit experiment does show why it is difficult to maintain an interpretation in which quantum systems either behave naively like classical particles or classical waves, independently of how we observe them, so it is worth knowing about.

The basic setup is illustrated below.
{{ :doubleslitsetup.png?direct&600 |}}
A source is located to the left of a screen with two slits, labelled 1 and 2, in it.  Depending on the version of the experiment we are considering, the source will be a beam of classical particles, e.g. bullets fired from a gun, a source of classical waves, e.g. a high intensity monochromatic light source, or a source of quantum particles, e.g. a laser beam tuned down in amplitude so that only one photon passes through the apparatus at a time or a beam of electrons as in the Davisson-Germer experiment.  We will consider what happens when only slit 1 is open, when only slit 2 is open, and when both slits 1 and 2 are open.

===== Classical Particles =====

First, imagine firing a stream of classical particles at the slits.  Assume the particles are light enough that we can neglect gravity, or you might imagine the whole apparatus is placed on an air hockey table and you are sliding small pucks towards the slits.  The results of this are illustrated below.

{{ :doubleslitparticles.png?direct&800 |}}

Here $I_1$ is the "intensity" of particles when only slit 1 is open, i.e. the number of particles detected at a given location divided by the total number of particles fired, in the limit of a large number of particles being fired.  Similarly, $I_2$ is the "intensity" when only slit 2 is open and $I$ is the intensity when both slits are open.

The classical particles travel in a straight line unless they hit something.  Therefore, when only slit one is open, only those particles with a trajectory that passes through slit 1 will be detected at the screen, apart from a few that bounce off the edges of the slit.  The intensity will have a single peak, behind slit 1.  Similarly, when only slit 2 is open, there will be a single peak behind slit 2.  When both slits are open, all the particles with trajectories that pass through either slit 1 or slit 2 will be detected.  We will get an intensity with two peaks, one behind each slit and $I = I_1 + I_2$

==== Classical Waves =====

Nest, consider classical waves (e.g. water waves or high intensity light).  The results are illustrated below.

{{ :doubleslitwaves.png?direct&800 |}}

For waves, the intensity is proportional to $|\psi(\vec{r},t)|^2$ where $\psi(\vec{r},t)$ is the wavefunction.  Note, in quantum mechanics, we have a habit of calling $\psi(\vec{r},t)$ the //**amplitude**// of a wave, even though we do not do this anywhere else in physics, so I will call it the amplitude here.  Let $\psi_1(\vec{r},t)$ be the amplitude hitting the screen at point $\vec{r}$ when slit 1 is open, and similarly for $\psi_2(\vec{r},t)$ and slit 2.  Let $\psi(\vec{r},t)$ be the amplitude hitting the screen at point $\vec{r}$ when both slits are open.

For waves, it is the amplitudes, and not the intensities, that add.  So $I_1 \propto |\psi_1|^2$ and $I_2 \propto |\psi_2|^2$, but
\[I \propto |\psi_1 + \psi_2|^2 = |\psi_1|^2 + |\psi_2|^2 + 2\text{Re}(\psi_1^*\psi_2),\]
or
\[I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos \delta,\]
where $\delta$ is the phase difference between paths that travel through slit one and those that travel through slit 2.  It can be calculated from the slit separation and leads to the characteristic double-slit interference pattern with maxima where there is constructive interference and minima where there is destructive interference.

In summary, classically waves exhibit interference but particles do not.  For particles, the intensities add.  For waves, the amplitudes add, but the intensities do not.

===== Quantum Particles =====

Now let's consider what happens if we fire a beam of quantum particles at the apparatus.  We could imagine a beam of electrons as in the Davisson-Germer experiment, or a laser emitting photons with the amplitude tuned down so that only one photon traverses the apparatus at a time.  The results are illustrated below.

{{ :doubleslitquantum.png?direct&800 |}}

We find a patter of intensities just like the classical wave experiment
\[I = |\psi_1 + \psi_2|^2 = |\psi_1|^2 + |\psi_2|^2 + 2\text{Re}(\psi_1^*\psi_2) \neq I_1 +I_2,\]
where $\psi_1$ and $\psi_2$ are he wavefunctions corresponding to only having slit 1 or slit 2 open.

The difference is that, as with classical particles, each quantum particle hits the screen at a definite location and $|\psi(\vec{r},t)|^2$ is the //probability// that the particle will arrive at location $\vec{r}$.

The following image shows how the interference pattern builds up over time from individual detection events in a double slit experiment with electrons.

{{ :double-slit_experiment_results_tanamura_four.jpg?direct&600 |}}
**Image Credit**: Dr. Tonomura and Belsazar, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons

So, does each particle spread out like a wave and go through both slits, or does it have a definite trajectory and only go through one slit?  One way of attempting to find out is to try and observe which slit the particle travels through by placing a detector at one of the slits.  The results of this are illustrated below.

{{ :doubleslitquantumdetector.png?direct&800 |}}

Here, I have schematically drawn an eye at slit 2 to represent the detector.  It does not matter what detector we use to observe the particle, so long as it is capable of accurately resolving which slit the particle is travelling through.  When we do this, the interference pattern disappears, and we get $I = I_1 + I_2$, just like for classical particles.

One way of understanding why this happens is to realize that in order to detect the location of the quantum particle, it has to interact with something.  For example, we might shine light at the slit.  In order to resolve which slit the particle is travelling through, the wavelength of the light must be smaller than the separation between the two slits.  But then, when the light hits the particle, there will be Compton scattering, which will cause the momentum of the quantum particle, and hence its wavelength, to change.  These random changes of wavelength will cause the phase to change randomly, so there is no longer a fixed phase difference between the paths travelling through the two slits.  Thus, the interference pattern will disappear.

I want to emphasize that this semi-classical way of understanding what is going on when we detect which slit the particle is travelling through is not the real story according to full-blown quantum mechanics.  When the particle is detected at one of the two slits, it becomes localized and, in quantum mechanics, particles that are localized no longer have a definite momentum (and hence wavelength).  It does not matter if we use light or anything else to detect the location of the photon.  The mere fact of having detected its location means that there will not be an interference pattern.

We observe a wave-like interference between the two possible trajectories //only// if we cannot tell which trajectory was actually taken.  Otherwise, we observe particle like properties, with no interference patter.

Niels Bohr introduced the principle of //**complementarity**// to describe this:  We need both particle and wave concepts to describe quantum systems, but we can only ever observe one of them at a time, depending on the experimental arrangement.  It is not a case of either/or or both/and.  They are complementary aspects of the same physical system.

Bohr, along with many physicists, would say that it is meaningless to ask which trajectory a quantum particle takes when we are observing an interference pattern.  Since you cannot observe the trajectory at the same time as observing the interference, it simply does not have a trajectory.  Note that in popular science accounts, it is often said that the particle travels along //both// trajectories.  But the conventional view is not //both//, not //either/or//, and not //neither//, but that the particle is in a new kind of physical state where the question itself is meaningless.

Whether you buy this conventional view is, to some extent, a matter of taste.  There are interpretations of quantum mechanics in which each electron does in fact go through one or the other slit, and interpretations where it does, in some sense, go through both slits.  All we have shown is that you cannot //detect// which slit it goes through without disturbing the interference pattern, and it may be the case that there are things that exist that we cannot directly detect.  Bohr's view of this experiment is optional, although it is the conventional view in the physics community.

====== 1.vi.3 The Superposition Principle ======

In quantum mechanics, the //**superposition principle**// states two things:

  * //Existence of superpositions//: If $\psi_1(\vec{r},t_0)$ and $\psi_2(\vec{r},t_0)$ are solutions to the equations of motion of quantum mechanics at time $t_0$ then so is
  \[\alpha \psi_1(\vec{r},t_0) + \beta \psi_2(\vec{r},t_0),\]
  for arbitrary complex coefficients $\alpha$ and $\beta$.
  * //Preservation of superpositions//: When a system is isolated (not interacting with its environment or being measured) then if, according to the equations of motion, the solution $\psi_1(\vec{r},t_0)$ evolves to $\psi_1(\vec{r},t)$ and the solution $\psi_2(\vec{r},t_0)$ evolves to $\psi_2(\vec{r},t)$ for $t>t_0$, then the solution $\alpha \psi_1(\vec{r},t_0) + \beta \psi_2(\vec{r},t_0)$ evolves to $\alpha \psi_1(\vec{r},t) + \beta \psi_2(\vec{r},t)$.

The superposition principle is responsible for the wave-like interference effects we have been discussing.  It holds because the equation of motion of quantum mechanics (the Schrödinger equation) is a linear, homogeneous differential equation, just like the wave equation in classical physics.

{{:question-mark.png?direct&50|}}
====== In Class Activity ======
  - In light of the double slit experiment, many physicists are inclined to say either:
    * Only the wavefunction exists.  Other properties only come into existence when observed.
    * Nothing exists until observed.

Consider the experiment depicted below.
{{ :einsteinsscreen.png?direct&600 |}}

In this experiment, which principle of physics would be violated by the "only the wavefunction" or "nothing exists" points of view?