Basics of quantum physics

Classically, light is well described as waves: A light beam can reflect from a surface, refract from one medium into another, diffract around obstacles and interfere with another light beam. On the other hand, certain phenomena, such as black-body radiation or the photoelectric effect, cannot be explained within the wave model. In 1900, German physicist Max Planck proposed a revolutionary idea that the black-body radiation problem can be solved if light is thought of as a flux of particles (photons) carrying fixed tiny portions (quanta) of energy, and thus established a wholly new field of quantum physics. The idea that, in different experiments, light can behave either as a wave or as (a flux of) particles is known as particle-wave duality. This duality, as it turned out later, holds for all matter, but is best observed for subatomic-size objects such as elementary particles. If the fact that the same object manifests itself differently depending on the situation still seems paradoxical, think of it in a more abstract way: An object is an entity that can be described mathematically by a quantum state (or a wave function), such that certain mathematical operations on the state determine the behaviour of the object.

The particle-wave duality was initially thought of as a unique property of light. But come to think about it this way: If light, which classically behaves like a wave, can, in certain experiments, behave like a particle, then why can’t we have it the other way round? Why can’t a particle likewise behave as a wave in some experiments? This idea was theoretically investigated in the 1924 doctoral thesis of French physicist Louis de Broglie, where he introduced the notion of matter waves, suggesting that all matter has wave properties. After three years, clear experimental evidence supporting de Broglie’s hypothesis was found: It was shown that electrons can diffract on a crystal, behaving like waves.

A system, however big or small it is, can be found in one of several (possibly infinitely many) states. Take the previous example of a single-photon double-slit experiment. After the photon (as a particle) has got to the other side of the plate with the two slits, it is in either of the two states –${\displaystyle |0⟩}$ or ${\displaystyle |1⟩}$^[1]– depending on the slit (0 or 1) it has passed through on its way. Classically, the photon must always be exclusively in either state ${\displaystyle |0⟩}$ or ${\displaystyle |1⟩}$(known as basis states), but the quantum mechanical description allows us to superimpose the two states as well. The actual quantum mechanical state of the photon after it has passed through the slits is thus the superposition of states ${\displaystyle |0⟩}$ and ${\displaystyle |1⟩}$, and can be formally written as ${\displaystyle |\psi ⟩=c_0|0⟩+c_1|1⟩}$, where ${\displaystyle c_0}$ and ${\displaystyle c_1}$ are some numbers related to the probability of finding the photon in the corresponding state. More specifically, the probability that the photon has passed through slit 0 is ${\displaystyle {\left|c_0\right|}^2}$ and the probability that it has passed through slit 1 is ${\displaystyle {\left|c_1\right|}^2}$. As the photon must have passed through either of the slits, ${\displaystyle {\left|c_0\right|}^2+{\left|c_1\right|}^2=1}$.

Photons are not the only particles that can be found in a superposition of states. Pretty much any object – as small as an electron or as huge as a whole Universe – shares this property. For macroscopic systems though, it is close to impossible to distinguish between different states, so quantum effects are best observed for microscopic objects.

In a classical world, we can measure some quantity of a system (say, the length of a table) as many times as we like and the measurement process will not affect the state of the system. In a quantum world, it turns out not to be the case. The result (or outcome) of the measurement is a number (with units, if applicable) that corresponds to one of the basis states. In the previous example, ${\displaystyle |0⟩}$ and ${\displaystyle |1⟩}$ were two basis states, and the corresponding outcomes might have been, say, 0 (meaning that the photon has passed through slit 0) and 1 (meaning that the photon has passed through slit 1). In the process of measurement, all uncertainty is lost. Suppose the outcome of the measurement is 1. That means that right after the measurement, there is a 100% probability that the system is in state ${\displaystyle |1⟩}$ and 0% probability that it is in state ${\displaystyle |0⟩}$. But that would mean that the state of the system has changed from ${\displaystyle |\psi ⟩=c_0|0⟩+c_1|1⟩}$ to ${\displaystyle |{\psi }^{\prime }⟩=|1⟩}$. We can say that the original state was destroyed, that is, any measurement of a quantum system is destructive.

Hence, there is a major issue. Suppose the system is in the state ${\displaystyle |\psi ⟩=c_0|0⟩+c_1|1⟩}$. In order to estimate the values of coefficients ${\displaystyle c_0}$ and ${\displaystyle c_1}$, the system should be measured several times to gather statistics. The problem though is that you cannot non-destructively measure the system several times, as each measurement will alter ${\displaystyle c_0}$ and ${\displaystyle c_1}$. This issue can at least partly be overcome in a number of ways. The most straightforward one is to prepare several (the more the better) identical copies of the system and then measure them independently. The relative frequency of each outcome (0 or 1) can then be taken as an estimate of squares of magnitudes of corresponding weights ( ${\displaystyle {\left|c_0\right|}^2}$ or ${\displaystyle {\left|c_1\right|}^2}$). A physical realisation of this task, though, is far from trivial.

1. The probability to measure the system to be in state ${\displaystyle |0⟩}$ is ${\displaystyle {\left(|0.6|\right)}^2=36\mathrm{\%<!--\; \%\; -->}}$, while the probability to measure the system in state ${\displaystyle |1⟩}$ is ${\displaystyle {\left|0.8\right|}^2=64\mathrm{\%<!--\; \%\; -->}}$. The result of the measurement is thus non-deterministic.
2. The measurement ‘selects’ one of the basis states, in this case ${\displaystyle |0⟩}$, and the original state collapses to ${\displaystyle |0⟩=1|0⟩+0|1⟩}$, so the probability to measure the system to be in state ${\displaystyle |0⟩}$ is ${\displaystyle {\left|1\right|}^2=100\mathrm{\%<!--\; \%\; -->}}$, and the probability to measure the system to be in state ${\displaystyle |1⟩}$ is ${\displaystyle {\left|0\right|}^2=0\mathrm{\%<!--\; \%\; -->}}$. That means that in this particular case, the measurement outcome will deterministically be ${\displaystyle 0}$.
3. The probability to measure the system to be in state ${\displaystyle |0⟩}$ is ${\displaystyle 36\mathrm{\%<!--\; \%\; -->}}$ and the probability to measure the system in state ${\displaystyle |1⟩}$ is ${\displaystyle 64\mathrm{\%<!--\; \%\; -->}}$[see question 1]. That means that approximately ${\displaystyle 1000\times <!--\; \times \; -->0.36=360}$ outcomes will be ${\displaystyle 0}$ and approximately ${\displaystyle 1000\times <!--\; \times \; -->0.64=640}$ outcomes will be ${\displaystyle 1}$.
4. The mean value by definition is the sum of the products of the outcome and the probability of that outcome. Thus, the mean value of the outcome is ${\displaystyle 0\times <!--\; \times \; -->0.36+1\times <!--\; \times \; -->0.64=0.64}$.