The dual waveparticle nature of quantum particles is described by a differential equation.
According to folklore so widespread among physicists, it happened like this: in 1926, a theoretical physicist named Erwin Schrödinger spoke at a scientific seminar at the University of Zurich. He talked about strange new ideas floating in the air, that objects in the microcosm often behave more like waves than like particles. Then an elderly teacher asked for the floor and said: “Schrödinger, don’t you see that all this is nonsense? Or we all do not know that waves – they are waves, in order to be described by wave equations? ” Schrödinger took this as a personal grievance and set out to develop a wave equation to describe particles in the framework of quantum mechanics – and he coped with this task brilliantly.
An explanation needs to be made here. In our everyday world, energy is transferred in two ways: by matter when it moves from place to place (for example, by a moving locomotive or by the wind) – particles are involved in this energy transfer – or by waves (for example, radio waves, which are transmitted by powerful transmitters and caught by the antennas of our televisions). That is, in the macrocosm where you and I live, all energy carriers are strictly divided into two types – corpuscular (consisting of material particles) or wave… Moreover, any wave is described by a special type of equations – wave equations… Without exception, all waves – ocean waves, seismic waves of rocks, radio waves from distant galaxies – are described by the same type of wave equations. This explanation is needed so that it is clear that if we want to represent the phenomena of the subatomic world in terms of probability distribution waves (cm. Quantum mechanics), these waves should also be described by the corresponding wave equation.
Schrödinger applied the classical differential equation of the wave function to the concept of probability waves and obtained the famous equation that bears his name. Just as the usual equation of the wave function describes the propagation of, for example, ripples over the surface of water, the Schrödinger equation describes the propagation of a wave of the probability of finding a particle at a given point in space. The peaks of this wave (points of maximum probability) show where the particle is most likely to be in space. Although the Schrödinger equation belongs to the field of higher mathematics, it is so important for understanding modern physics that I will nevertheless present it here – in its simplest form (the socalled “onedimensional stationary Schrödinger equation”). The aforementioned probability distribution wave function, denoted by the Greek letter ψ (“Psi”), is a solution to the following differential equation (it’s okay if you don’t understand it; the main thing is to take it on faith that this equation indicates that probability behaves like a wave):
Where x – distance, h – Planck’s constant, and m, E and U – respectively the mass, total energy and potential energy of the particle.
The picture of quantum events that the Schrödinger equation gives us is that electrons and other elementary particles behave like waves on the ocean surface. Over time, the peak of the wave (corresponding to the place where the electron is most likely to be located) shifts in space in accordance with the equation describing this wave. That is, what we traditionally considered a particle in the quantum world behaves much like a wave.
When Schrödinger first published his results, a tempest broke out in the world of theoretical physics in a teacup. The fact is that almost at the same time, the work of Schrödinger’s contemporary, Werner Heisenberg (cm. Heisenberg’s uncertainty principle), in which the author put forward the concept of “matrix mechanics”, where the same problems of quantum mechanics were solved in another, more complex from a mathematical point of view, matrix form. The commotion was caused by the fact that scientists were simply afraid that two equally convincing approaches to the description of the microworld contradict each other. The excitement was in vain. Schrödinger himself in the same year proved the complete equivalence of the two theories – that is, the matrix equation follows from the wave equation, and vice versa; the results are identical. Today, most of the Schrödinger’s version is used (sometimes his theory is called “wave mechanics”), since his equation is less cumbersome and easier to teach.
However, it is not so easy to imagine and accept that something like an electron behaves like a wave. In everyday life, we are faced with either a particle or a wave. A ball is a particle, sound is a wave, and that’s it. In the world of quantum mechanics, things are not so simple. In fact – and experiments soon showed this – in the quantum world, entities differ from the objects we are accustomed to and have different properties. Light, which we used to think of as a wave, sometimes behaves like a particle (which is called photon), and particles like an electron and a proton can behave like waves (cm. The principle of complementarity).
This problem is commonly referred to as dual or dual corpuscularwave nature quantum particles, and it is inherent, apparently, to all objects of the subatomic world (cm. Bell’s theorem). We must understand that in the microcosm our ordinary intuitive ideas about what forms matter can take and how it can behave is simply inapplicable. The very fact that we use the wave equation to describe the motion of what we used to think of as particles is striking proof of this. As noted in the Introduction, there is no particular contradiction in this. After all, we have no compelling reason to believe that what we observe in the macrocosm should be accurately reproduced at the level of the microcosm. Nevertheless, the dual nature of elementary particles remains one of the most incomprehensible and disturbing aspects of quantum mechanics for many people, and it would not be an exaggeration to say that all the troubles began with Erwin Schrödinger.
1900

Electronic theory of conduction

1900

Constant Plank

1924

Quantum tunneling effect

1926

Band theory of conductivity of solids

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Molecular orbital theory

1964

Bell’s theorem
