# Quantum Interference

## How to visualize the quantum wave nature of matter?

Reminder: What is a wave?

Many phenomena in our environment are described as waves: sound, radio waves, surface waves on water and more. These are oscillatory phenomena that propagate in space: Oscillations of air pressure, of the electromagnetic field or the number of molecules at a given surface location.
The basic vibration mode is mathematically captured by a sine curve with the following key properties:

• The wavelength measures the spatial period of the sine wave, i.e. the distance between repeating wave forms.
• The frequency measures the how often a sine wave repeats itself per second.
• The amplitude measures the height of the wave crest over the zero line.
• The Phase determines the position of a reference point on the wave to a second point in space, in units of the wavelength.

Macroscopic wave phenomena can be easily visualized.

Reminder: How to measure wavelength?

You can characterize surface waves on the sea by your bare eye. If you take a snapshot with your webcam you can even determine the wavelength on that photograph. Radio waves may be meters long and their field can be traced out in space using an antenna. It gets more difficult for very small wavelengths: Green light is an electromagnetic wave with a spatial period of about 0.0005 millimeters and a frequency of $$6\times 10^{14} \mathrm{Hz}$$. Eine Photodiode, eine CCD Kamera oder das Auge sehen nur die Intensität des Feldes, also das Quadrat der Amplitude.

 Wave 1 Wave 2 Amplitude $$A$$ Wavelength $$\lambda$$ Phaseshift $$\Delta \varphi$$

## Interference

In order to measure very small distances it often useful to employ wave interference. When two wave fields are superposed their wave crests may add up (constructive interference) while the encounter of a crest and a trough tends to cancel the wave (destructive interference). The pattern of constructive and destructive interference in space allows to determine the wavelength.

It is a particular trait of quantum physics that wave functions may be associated not only with dense ensembles of particles but even with an ensemble that has been diluted to a single particle in the machine at any instant. The wave function seemingly still describes the individual quantum object. This is why one often says that ‘each particle interferes with itself’.

Quantum theory can only predict probabilities for a certain outcome. Which of the many possibilities is finally assumed in a measurement on an initial superposition of options and states is entirely random. Only many measurements under identical conditions reveal the strictly deterministic distribution of probabilities which is also obtained from solving Schrödinger’s equation

$$i\hbar\frac{\partial}{\partial t}\psi(\mathbf{r},t) \;=\; \left(- \frac{\hbar^2}{2m}\Delta + V(\mathbf{r},t)\right)\psi(\mathbf{r},t)$$

All experiments so far have confirmed Born’s rule: the squared modulus $$|\psi|^2$$ of the state function $$\psi$$ represents the probability to find a quantum object at time $$t$$ at position $$r$$ with all other parameters contained in $$\psi$$.