This book presents (and demystifies) for ordinary people the magic world of quantum science, like no other book. She talks about basic scientific concepts, from light particles to matter states and the causes of the negative impact of greenhouse gases, revealing every topic without using specific scientific terminology – examples from everyday life. Of course, a book on quantum physics can not do without a minimal set of formulas and equations, but this is a necessary minimum, understandable to most readers. According to the author, the book popularizing science should be accessible, but not lowered to the level of the reader, but to raise and develop its intellect and general cultural level.

### Quantum racquetball and color Fruit

The key property of electrons associated with atoms and molecules is that their energy states are discrete. We say that the energy of an electron can be quantized, that is, an electron bound to an atom or molecule can have only certain definite energy values. Energy varies stepwise, and these stages have certain discrete dimensions. Energy states are like a ladder. You can stand on one level or climb to the next higher stage. However, it is impossible to stand halfway between the two steps. These discrete, or quantized, energy values are often called energy levels. Unlike ordinary staircases, the intervals between energy levels are usually not the same.

An important area of modern quantum research is the calculation of the electronic states of molecules. This area is called quantum chemistry. Such calculations make it possible to obtain quantized energy levels for electrons in molecules (energy levels), and also to calculate the structure of molecules. The calculation of the structure of the molecule gives the distances between atoms and the positions of all atoms in the molecule with an accuracy limited only by the uncertainty principle. Thus, quantum-mechanical calculations allow us to determine the size and shape of molecules. Such calculations are important for understanding the fundamental principles of binding atoms to molecules and for constructing new molecules. With the development of the quantum theory and the emergence of ever more powerful and complex computers capable of solving time-consuming mathematical problems, more and more large molecules can be studied by the methods of quantum chemistry. One of the most important applications of quantum theory is the development of pharmaceuticals. Molecules can be designed so that they have the right size and "fit" in shape to specific loci of proteins or enzymes.

Quantum chemistry requires very laborious calculations. Even for the simplest hydrogen atom, quantum-mechanical calculations are mathematically very complex. The hydrogen atom consists of one electron bound to one proton. The proton, which is the nucleus of the hydrogen atom, is a positively charged particle, and the electron is negatively charged. The attraction of a negatively charged electron to a positively charged proton keeps them together, securing the hydrogen atom. Details of the calculation of the energy levels of the hydrogen atom will not be presented here, but in the following chapters we will consider some features of the results of these calculations. They give the energy levels of the hydrogen atom and its wave functions. It is the wave functions, that is, the waves of the probability amplitude for the hydrogen atom, that are the starting point for understanding all atoms and molecules. Atoms and molecules are complex because they are absolutely small three-dimensional systems, and it is necessary to consider how protons and electrons interact with each other.

### A particle in a box is a classic case

There is a very simple task related to our topic. It is known as the problem of a particle in a box. To solve it, complicated mathematics is not needed, but this solution allows us to illustrate the important properties of bound electrons, for example, the quantization of energy levels and the wave-like nature of electrons in bound states. Before analyzing the nature of an electron in a one-dimensional box of atomic dimensions, we will discuss the classical problem of an ideal one-dimensional playground for a racquetball in order to reveal the differences between the classical (large) and quantum-mechanical (absolutely small) systems

. 8.1 shows the ideal "box". It is one-dimensional. Its walls are considered infinitely high, infinitely massive and completely impenetrable. There is no air inside the box that would resist movement. In the figure, the interior of the box is denoted by Q = 0, and the outer one by Q = ∞. Earlier it was said that a particle is called free, to which no forces act. Forces arise when a particle interacts with something. For example, a negatively charged particle, such as an electron, can interact with a positively charged proton. Interaction in the form of attraction between oppositely charged particles will generate a force acting on the electron. When controlling electrons in a CRT (see Figure 7.3), the electric field generates a force acting on the electrons and causes them to change direction.

The measure of interaction of a particle with something that affects it, like an electric field, is called the potential and Has the dimension of energy. In the future, the potential will be denoted by the letter Q. Inside the box Q = 0, as in the case of a free particle. This means that the particle does not interact with anything inside the box. There are no electric fields or air resistance. However, outside the box Q = ∞. Infinite potential means that the particle would have to have infinite energy to end up in areas outside the box. The expression Q = ∞ is simply a way of formalizing the statement that the box walls are ideal. A particle can not penetrate walls or jump over them, no matter how great its energy. If you place a particle in such a box, it can not slip away and will always remain inside it. In this sense, the particle is locked in a box. It can be in the region of a space of length L, but nowhere else.

In Fig. 8.2 depicts a ball for playing racquetball, bouncing off the walls of an ideal one-dimensional classic (large) racket area. As already mentioned, these walls are ideal, but there is no air resistance inside. In addition, the ball is also ideal, that is, it has absolute elasticity. When the ball collides with the wall, it compresses like a spring, and straightens again, causing it to rebound. Real balls are not perfectly elastic. When the ball is compressed at impact, not all the energy spent on squeezing it, is pushing away from the wall. Part of the energy spent on squeezing the ball goes to its heating. However, here we will assume the ball is perfectly elastic. When striking against the wall, all the kinetic energy of the ball, which causes its compression, is then spent on pushing the ball away from the wall. Therefore, the speed of the ball before the collision with the wall is equal to the speed of its rebound after the collision.

On this ideal rocket court the ball bounces off the walls without any loss of energy; In addition, there is neither air resistance nor gravity. Therefore, the ball will always move back and forth, bouncing off the walls. It hits the wall at point L, bounces, collides with the wall at point 0, bounces again and continues its movement back and forth. Inside the box, since the potential is zero (see figure 8.1), no forces on the ball act. Therefore, its energy is purely kinetic:

Where m is the mass of the ball, and V is its speed. If the ball experiences weak external influences, its speed will become slightly less and the value of Ek will also decrease slightly. In this ideal rasbol, energy can change continuously. The value of Ek can increase or decrease arbitrarily, depending only on the force of the impact on the ball.

Another important feature of the classic racquetball is the ability to stop the ball so that it lies motionless on the floor. In this situation, its velocity is zero: V = 0. And once V = 0, then Ek = 0. For V = 0, the momentum is also zero, because p = mV, so the pulse is known exactly. If the ball lies on the floor (V = 0), then its position is known. If we mark this position x (see figure 8.2), the value of x will be in the interval from 0 to L. The value of x can not take any other values, since the ball is on the site (in the box) and can not be outside because of the " For ideal walls. The ball can be placed in a specific position x on the floor of the platform, and then its position will be known exactly. This property of the macroscopic playground is even ideal. This is a classical system, and it can accurately and simultaneously know the momentum p and the position x.

The racquetball court has a length of 12 m, the diameter of the ball is 5.6 cm, and its weight is about 0.04 Kg. Obviously, the game of racquetball is described by classical mechanics. With the help of light, you can follow the bounces of the ball back and forth, without affecting them.

### The particle in the box is a quantum case

What will change if we now turn to the quantum racket? The site remains ideal, but now its length is not 12 m, but 1 nm (10-9 m). In addition, the particle has an electron mass equal to 9.1 10-31 kg, rather than 0.04 kg. Thus, it is a problem of a quantum particle in a box.

One can immediately say that the smallest energy of a quantum particle in a box of a nanometer size can not be zero. On the classical racket pad, the speed of the ball V is possible, which is zero, which means that the momentum p = mV can also be zero. In addition, the position of the ball x has a clearly defined value. For example, the ball can lie motionless (V = 0) exactly in the middle of the pad, which corresponds to x = L / 2. In this case, for our classical rasbol ball, Δp = 0 and Δx = 0. The value of the product ΔxΔp = 0 does not correspond to the Heisenberg uncertainty principle, which is normal, since this is a classical system. However, an absolutely small particle in a nanometer-sized box is a quantum object and must obey the uncertainty principle that ΔxΔp ≥ h / 4. If V = 0 and x = L / 2, then we know simultaneously x and p, and hence, ΔxΔp = 0, as in the classical racquetball. For a quantum system, this is impossible. Thus, V can not be zero. The particle can not remain motionless at a given point. And if the value of V is non-zero, then the value of Ek can not be zero. The principle of uncertainty says that the lowest energy of our quantum rasbolnoy ball can not be zero. A quantum ball never stays in motion.

### Quantum particle energy values in a box

What energy can a quantum particle have in a box of nanometer sizes? This question can be answered without complicated calculations, but first we need to return to the waves again. In Chapter 6 we discussed the wave functions of free particles. The wave function of a free particle with a definite momentum p is a wave that extends all over the space. Thus, an electron with an ideally defined momentum is a delocalized wave that covers the entire space. The probability of finding a free electron is everywhere the same. Such an electron has a well-defined kinetic energy Ek = 1 / 2mV2, since it has a well-defined momentum p = mV.

An electron in a nanometer box is similar to our free particle in the inner area of the box, where Q = 0. There is no inside the box Potential, and therefore there are no forces acting on the particle. In this respect, it is very similar to a free particle, on which no forces act either. However, there is an important difference between the particle in the box and the free particle – these are the walls of the box. The electron in the box is located only inside the box. The ideal character of the box does not allow its wave function to spread over the entire space. The particle is inside the box and can never be outside. The wave function defines the probability amplitude for detecting a particle in a certain region of space. This is the Born interpretation of the wave function. If our electron can be detected only inside the box and never outside, then the probability of its detection in the box must be finite, and outside – zero. If the probability of finding a particle outside the box is zero, then the wave function must also be zero at all points outside the box.

So, we came to the conclusion that the wave function of a particle in a box is similar to the wave function of a free particle, but the wave function Function must be zero outside the box. In his interpretation of the nature of the quantum-mechanical wave function, Born imposed some physical limitations on the form that the wave function can take. One of them is that a good wave function must be continuous. This condition means that the wave function must change smoothly from place to place. An infinitesimal change in position can not lead to an unexpected jump in probability. This is a very simple idea. If the probability of detecting a particle in some very small region of space is, for example, 1%, then a shift to an unimaginably small value can not suddenly make the particle detection probability equal to 50%. This is clear from the images of the wave packets in Fig. 6.7. The probability varies smoothly from place to place. This allows us to add something to the description of the wave functions of a particle in a box in addition to the fact that they are waves with finite amplitudes inside the box and zero amplitude outside. Since the wave function must be continuous, it must have a zero amplitude directly at the box wall from the inside to coincide with the zero amplitude of the wave function outside the box.

Fig. 8.3 shows (forbidden) the breaking of the wave function inside the box. The wave function is indicated (the Greek letter "fi"). The amplitude of the wave function is plotted along the vertical axis. The dashed line shows its zero level. Wave functions, representing waves of probability amplitude, can fluctuate between positive and negative values. The wave function shown in Fig. 8.3, has values near the walls that are different from 0. However, the wave function must be zero outside the box, that is, for values of x less than 0 and greater than L it must be zero. In the figure, the wave function suddenly jumps from a non-zero value near the wall inside the box to a zero value just outside the wall outside the box. Thus, the wave function shown in Fig. 8.3, is not admissible, since it is not continuous. This function can not represent a quantum particle in a box.

### The wave function must have a zero value at the walls

In order for the wave functions representing the particle in the box to be physically acceptable, their values at the walls must be zero, and then they will not experience a rupture on the walls. It is not difficult to fulfill this condition. The wave function fluctuates between positive and negative values. Each time, passing from positive to negative or from negative to positive, it passes through zero. In fact, the zero points are separated from each other by half the wavelength. Therefore, to obtain good wave functions of a particle in a box, we must choose waves whose length allows them to fit in the box so that the zero points are just on the walls.

] In Fig. 8.4 shows three examples of waves that are suitable for the role of wave functions for a particle in a box. The lower of them is denoted by n = 1 and consists of one half-wave. It starts at the left on the amplitude 0, passes through a maximum and then again drops to zero on the wall at point L. The next wave, located above and denoted by n = 2, consists of one complete oscillation. It also starts at the left wall at amplitude 0, passes a positive peak, returns to zero, then a negative peak and a return to zero on the wall at point L. The wave, labeled n = 3, contains one and a half periods. Подходит любая волна, содержащая целое число полуволн, то есть 1, 2, 3, 4, 5 и так далее половин длины волны, и расположенная так, чтобы она начиналась на нуле слева и заканчивалась на нуле справа.

Величина n — это число полуволн конкретной волновой функции. При n = 1 длина волны составляет 2L, поскольку длина ящика равна L, а n = 1 соответствует половине длины волны. При n = 2 длина волны составляет L, поскольку ровно одна длина волны помещается между стенками. При n = 3 между стенками помещаются три полуволны, то есть 1,5 = L. В этом случае = L/1,5, то есть = 2L/3. Обратите внимание, что здесь обнаруживается общее правило: = 2L/n, где n — целое число. Для n = 1 получаем = 2L, для n = 2 — = 2L/2, для n = 3 — = 2L/3 и т.д.

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