Schrödinger’s equation

The dual wave-particle 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 so-called “one-dimensional 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 corpuscular-wave 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.

See also:
Electronic theory of conduction
Constant Plank
Quantum tunneling effect
Band theory of conductivity of solids
Molecular orbital theory
Bell’s theorem
Erwin Schrödinger
Erwin Schroedinger, 1887-1961

Austrian theoretical physicist. Born in Vienna, the son of a wealthy industrialist with an interest in science; received a good education at home. While studying at the University of Vienna, Schrödinger did not attend lectures in theoretical physics until his second year, but he defended his doctoral dissertation in this specialty. During the First World War, he served as an officer in the artillery troops, but even then found time to study new articles by Albert Einstein.

After the war, after changing positions at several universities, Schrödinger settled in Zurich. There he developed his theory of wave mechanics, which to this day is the fundamental basis of all modern quantum mechanics. In 1927, he took the position of head of the Department of Theoretical Physics at the University of Berlin, replacing Max Planck in this position. A consistent anti-fascist, Schrödinger emigrated to Great Britain in 1933, became a professor at Oxford University and received the Nobel Prize in physics the same year.

Homesickness, however, forced Schrödinger in 1936 to return to Austria, to the city of Graz, where he began work at a local university. After the Anschluss of Austria in March 1938, Schrödinger was dismissed without warning and hastily returned to Oxford, managing to take only a minimum of his personal belongings with him. This was followed by a chain of literally detective events. Eamon de Valera, Prime Minister of Ireland, was once a professor of mathematics at Oxford. Wanting to get the great scientist to his homeland, de Valera ordered the construction of an Institute for Basic Research in Dublin specially for him. While the institute was being built, Schrödinger accepted an invitation to read a course of lectures in Ghent (Belgium). When the Second World War broke out in 1939 and Belgium was occupied by fascist troops with lightning speed, Schrödinger was unexpectedly taken by surprise in the camp of the enemy. It was then that de Valera came to his rescue, supplying the scientist with a letter of trustworthiness, according to which Schrodinger managed to leave for Ireland. The Austrian remained in Dublin until 1956, after which he returned to his homeland, to Vienna, to head the department specially created for him.

In 1944, Schrödinger published a book “What is life?”, which shaped the worldview of a whole generation of scientists, inspiring them with a vision of the physics of the future as a science, unsullied by the military application of its achievements. In the same book, the scientist predicted the existence of a genetic code hidden in the molecules of life.

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