# Quantum Mechanics

In the late 19th century, many physicists thought that their job was almost finished, for well-understood mathematical laws could explain most of the observed physical phenomena. There were only a few things lacking explanation, but physicists thought that they were just details. They believed that soon it would be possible to describe the behavior of any system. (There is the story of a student who asked Lord Kelvin what he would advise him to choose to research in Physics, and Kelvin replied that he should choose another area, for Physics was almost “complete”.)

Fortunately, they were being too optimistic (and wrong, for details are enough to destroy any scientific theory). Investigations on those few details gave us a completely new and revolutionary field in Physics, which show us how far we were from complete understanding of nature. At the beginning of the 20th century appeared two new fields which developed into two amazing and fundamental theories that explore the limits of the spatial scales: the large scale – Relativity, and the small scale – Quantum Mechanics.

One of the problems without solution at the end of 19th century was the black body problem (which I may explain in detail in another article). Planck solved this problem by introducing a new concept in Physics: the energy (in this problem) is quantified (discrete). Then, Albert Einstein proposed the same idea for light in general (not only within this problem), as I explained in the previous article, Einstein’s Invariance.

The photoelectric effect and the Compton effect clarified the notion that light has a dual nature, because it was clear that it was necessary to understand light as a particle and as a wave to explain all the observed phenomena. Then it was found that this fact was not only true for light, but for all matter in general. (See The dual nature of light and matter for more details about this.)

This is one of the first results that lead scientists to the quantum theory: even particles can behave like waves. How can we understand this? It is against our intuition, however we should note that our intuition, or common sense, comes from the world we perceive, that we can observe, and what we can observe is limited to a certain spatial scale. Quantum effects are beyond this scale, so the common sense is no longer applicable. At the quantum scale we have to redefine our common sense. As I showed in The dual nature of light and matter, a simple estimation demonstrates that at our “normal” spatial scale we never observe objects behaving like waves, however, at the quantum scale, at the atomic scale for instance, they do. In this case, it is necessary to take into account a wave function (mathematical function which completely describes the object) to characterize the position of a particle (and consequently any other physical quantity).

From purely mathematical considerations, Heisenberg demonstrated that the fact that the physical quantities of a particle are described by a wave function implies that these quantities form groups of two. Each group has their precision related and limited together. For instance, the position and the momentum (the product between mass and velocity):

where $x_i$ is the position of a particle and $p_i$ its momentum. Thus, the product between the uncertainties of each one is equal or larger than a constant ($\hbar$ is the Planck constant). This implies that the uncertainty related to a physical quantity (like the position) cannot be zero (for in that case the condition wouldn’t be verified), and if we try to increase the precision of one measure, we will decrease the precision of another. In the above example, this means that it is impossible to measure the position and velocity of a particle with infinite precision (independently on the measuring device). This is not a technical limitation, it is nature that imposes this uncertainty on itself!

Einstein didn’t believe this principle, maybe because for him it was inconceivable such limitation in Physics. Then, he proposed an experiment to show that the Uncertainty Principle was wrong:

One could make a system of two symmetric particles, i.e. symmetric positions and velocities, and equal masses. Thus, if we would measure the position of one particle, we would know the position of the other particle, because of symmetry, and the same for velocity. So, we could bypass the Principle, for we could determine both position and velocity with as much precision as we would like by taking advantage of the “auxiliary” symmetric particle. Of course, this would work assuming that the measurements would be independent. However, it did not work out! The Heisenberg Principle survived, for when experimentalists measured the position of one particle, its momentum changed according to the Principle, but the same happened for the other particle! This is basically the basis for quantum teleportation – this “information” transmission is faster than light, it is instantaneous! (Actually, what is transmitted is not considered information due to some “details” that I might explain in another article. If it was information, the Relativity Theory would be wrong.)

In further research, Schrödinger discovered a very important equation – Schrödinger’s equation:

$-\frac{\hbar^2}{2m}\nabla^2\psi(r)+V(r)\psi(r)=E\psi(r)$

where:

• $\hbar$ -Planck constant;
• $m$ – mass of particle;
• $\psi$ – wave function;
• $V$ – potential energy;
• $E$ – total energy;
• $r$ – position;
• $\nabla$– laplacian, the second derivative of position.

Despite the fact that the wave function describes the system, the function itself does not have a direct physical meaning. However, the square of this function does: it defines the probability of finding a particle at a given position.

If we work out the mathematics, we can infer some interesting implications. If a particle have a certain energy E and tries to overcome a barrier of potential energy V, then we would expect that the particle would only be able to overcome it if E>V (let’s say that we cannot jump over a wall if we do not have the energy to reach the top of the wall). This is not true in quantum mechanics: there is a small probability to jump over the wall, even if E<V. In other words, a particle can overcome a barrier even with E<V. This is the so-called quantum tunneling. In the quantum world, you could go through a wall, or give a headbutt (if you were a particle subject to the quantum rules). It’s just a matter of probabilities: you have to try so that you can find whether you are lucky. Needless to say, in our macroscopic world you would never be lucky.

This theory brought serious consequences to our understanding of the universe: not everything is as it seems. A different scale can bring a new paradigm, a new understanding, and a different common sense. There is already technology that depends on our understanding of quantum mechanics, but I would say that the “real” revolution is yet to come.

If you have any doubt, comment, remark, please do it. I also remind you that I’m not a native speaker, so I’ll appreciate any corrections to the text. Thanks!

Marinho Lopes