Schrödinger's equation

To read this article, I recommend that you accompany the equations and sheets listed in the photos in this publication (they are written in Portuguese, but the math can be understood). Based on our knowledge, we can use classical wave theory to describe any wave - from waves on a string up to electromagnetic waves. From the wave equation (equation 1), a second order and linear differential equation, with derivatives in relation to position and time, we can find the solutions that correspond to the description of the wave system (as shown in a generic way in equation 2, in which A is the amplitude of the wave k is the wave number, x is the position, ω is the angular frequency and t is the time). In other words, the solution of the equation describes the behavior of the wave at an x ​​position and at a time t. Equation 1 can be extended to the three-dimensional case (equation 3, written using the Laplacian), with a exponential solution (equation 4). The k-wave number basically tells us how many wavelengths can fit in a 2π range. In equation 3, K is the wave vector, with the same direction and orientation as the wave propagation and with module/magnitude being the k wave number. Waves can overlap, as in the diffraction shown in the double slit experiment, thus generating interference (the amplitudes of the waves are added). If we have two solutions of the wave equation, one with wave vector K1 and the other with wave vector K2, a third solution with wave vector K3 = K1 + K2 is also valid, due to the linearity of the wave equation. But what is linearity? A differential equation (equation in terms of rates of variation/derivatives) is said to be linear when the sums of solutions also constitute valid solutions. In the case of waves, looking at the physical meaning, the linearity condition makes the formation of interference patterns possible. With the classical wave theory, we are able to solve any conventional physics wave problem. It is possible to accurately predict the behavior of wave systems, such as sound waves, electromagnetic waves, on liquid surfaces, etc. However, with the advent of quantum mechanics, it was shown that the classic description was not enough to describe the elementary behavior of matter. The German physicist Max Planck published in 1900 his theory of the quanta of light, in which he explained the radiation of black bodies and that light's energy came in amounts defined by equation 5, where h is the Planck constant and f is the photon's frequency. Planck gave rise to the development of quantum mechanics and, a few years later, Louis de Broglie published the so-called “de Broglie hypothesis” in which he defended the dual wave-particle behavior, relating the linear moment (p = mv) with the wavelength λ, as shown in equation 6. With the evolution of quantum theory, an equation was developed in order to describe quantum systems (be they free particles, simple atoms, etc.). The person responsible for formulating such an equation was the Austrian physicist Erwin Schrödinger, who, in 1926, published it. His equation is very peculiar and cannot be derived from any other basic principle, therefore it is said to have been postulated by the scientist (although it was based on previous knowledge of the analogy between optics and mechanics, by William Rowan Hamilton, as well as in the hypothesis of Louis de Broglie). There are several phrases by famous scientists that illustrate the non-triviality nature of the equation, such as that of Richard Feynman: “Where did we get this equation from? Nowhere. It is not possible to derive it from anything we know. It came from Schrödinger's mind ”. In classical mechanics, a system is conventionally described according to Newton's laws of dynamics, while in quantum mechanics, systems are usually described using the Schrödinger equation. The Schrödinger equation (equation 7) is, like the classic wave equation, a linear and second order differential equation, where ℏ is the Planck constant divided by 2π (h/2π), m is the mass of the particle , V(x) is the potential function and E is the energy. The first term on the left side of the equation represents the kinetic energy of the particle and the second represents the potential energy, so E (right side of the equation) is the total energy of the system. Equation 7 is known as the time independent Schrödinger equation. The fact that the referent equation is linear unfolds in one of the pillars of quantum mechanics, which is the principle of quantum superposition. Solutions of the Schrödinger equation can be added, with their respective coefficients (equation 8), in order to form another possible solution. Physically this means that the system is in an superposition of states (it is not in any specific state) but, if there is a measurement, the system will, randomly “choose” one of the possible states to be the result of the measurement/observation. The probability that the system will end up in a specific state is given by the square of the coefficient of the referring state (equation 9); equation 9 is known as Born's rule. For example: if Ѱ1 and Ѱ2 are possible solutions to the Schrödinger equation, the sum between them can also be a possible solution (given the linearity of the differential equation). Schrödinger developed his equation, but, like all scientists at the time, he did not know what its physical meaning was. It so happens that, to this day, nobody knows the physical interpretation for the solutions of the Schrödinger equation, which are called wave functions. However, in 1926, scientist Max Born postulated the mentioned “Born's rule”, which added meaning to the square of the wave function module, that being the probabilistic density for a given system. The Schrödinger equation is seen as a type of wave equation, keeping a relationship with the wave-particle duality. However, we must pay attention to certain thinking pitfalls. The “wave” represented by the wave function should not be seen as the spatial charge distribution of the particle, but rather as a probability distribution, so that when measurements are made, the particle gives results consistent with the probability provided by the square of the wave function module. Several interpretations about the meaning of the wave function itself (and not the squared modulus) have been proposed, such as David Bohm's pilot wave theory and Hugh Everett's many worlds interpretation (this approach addresses a context of multiverse), however, it is impossible to say which, if any, is the correct one. Also noteworthy is the fact that the wave function represents everything that can be known about a given quantum system. Quantum mechanics is, for the most part, written in the form of linear algebra, making use of operators. Operators are used in physics to represent entities that act on a certain physical state, in a certain space, in order to return another physical state, within the same state space. Mathematically, operators are functions, very commonly written in the form of matrices. In quantum mechanics there are special types of operators, called hermitian operators, which represent measurable/observable (capable of being measured, such as position and moment) quantities. Hermitian operators, when acting on a state of a certain physical system, return a real number, called eigenvalue. It is important to understand these key concepts in order to be successful in reading literature involving quantum mechanics. It is very common to find the Schrödinger equation written as in equation 9, where ∇ ^ 2 is the so-called Laplacian, operator referring to the rate of change with respect to the variables of the adopted coordinate system (be it Cartesian, cylindrical or spherical). Much of the work on quantum mechanics is written in Dirac's notation (also called “bra-ket” notation), an elegant and advantageous way of developing calculations. Dirac's notation has something like an "own algebra", with relatively easy simplifications due to the conditions of orthonormality (which will be discussed in later articles). Dirac's notation is based on linear algebra, and the wave functions are represented as state vectors of the system, as shown in equation 10. Equation 10 is Schrödinger's equation, written in Dirac notation, where H is the Hamiltonian , operator that represents the total energy of the system. Armed with the Schrödinger equation, we are able to solve several types of problems, based on models that will be mentioned below. The simplest application of the Schrödinger equation is for the case of a free particle, which means that the potential function is 0 (V (x) = 0) and, therefore, the Hamiltonian of the system will have only the portion of the equation equivalent to the kinetic energy of the particle. With that, we just have to solve the differential equation (page 1) to get the solution. The conventional solution of the Schrödinger equation can be written, equivalently, in the exponential form (see page 2). For cases in which the potential function is not null, the solutions have different forms, as will be shown. In addition to the time-independent version of the Schrödinger equation, we also have the time-dependent version, with the addition of the time factor. We can obtain such version of the equation by the multiplication between the ordinary time-independent wave function and the exponential factor exp (-iEt/ℏ), as shown, in detail on pages 3 and 4. When we mention quantum mechanics, one of the most traditional models is that of the particle confined within a "box", also known as "particle in an infinite well". Such model predicts a certain region of length L (“box length”) in which the potential function is zero and the particle can be found there. At the “ends” of the box, the potential function is infinite, therefore, the particle can be found only in the region inside the box, with the wave function being null for x≤0 and x≥L (the particle cannot exist in these regions). The previously imposed conditions are called boundary conditions and are the same as in the case of standing waves and, therefore, it is expected that the “box length” L will be given in the same way as in the case of standing waves (see equation 11), where λ is the wavelength and n can assume positive integers values. Within the box the potential function is null and, therefore, the total energy of the particle is equal to its kinetic energy. When making the necessary substitutions, with the restrictions of possible wavelengths, from the boundary conditions, we can find the energies allowed for the system (the derivation is available on pages 5 and 6). We see that energy cannot assume any arbitrary value, thus it is restricted, quantized. For each value of n there is a corresponding energy level. If a particle gains enough energy to increase its energy level, it will transition to a higher energy level; say that it is initially at level n and moves to level n + 1. Eventually the particle will return to the energy level n, emitting its previously gained energy in the form of a photon, with energy equals to (En + 1) - (En). Regarding the wave function for the particle confined in a box, we expect it to also be equal to the oscillation amplitude of a standing wave, fixed at two points (equation 12). In equation 12, “An” is the so-called normalization constant, which ensures that the total probability of finding the particle is equal to 100%. If we want to be more rigorous, we can obtain the same solution from the Schrödinger equation. Consider the solution for a free particle (inside the well), shown on page 1. We know that at x = 0 the particle cannot exist, so the wave function there will also be 0. When computing Ѱ(0) we get: Ѱ ( 0) = Acos (0) + Bsin (0) = A. It is concluded that the constant “A” is zero, given the condition that Ѱ (0) = 0. Thus: Ѱ (x) = Bsin (kx) (B being the normalization constant). There is a model similar to that of the infinite well, however, with a finite square well, that is, the potential function is not infinite in the regions x≤0 and ex≥L, so there is a chance that the particle will “escape” from the limits of the box/well . The potential function V (x) has a finite value for x <0 and x> L and is 0 in the region bounded by 0 <x <L. We can solve the Schrödinger equation in the three regions to obtain the solutions, as shown on pages 7, 8 and 9. When obtaining the wave functions, we observe that, even if the particle energy is lower than the potential V (V> E) , it is possible for the particle to “penetrate” the barrier, even if this is classically prohibited. Let us think first in the context of classical mechanics. Imagine a pendulum displaced from its equilibrium position by a certain angle. The pendulum is then abandoned and, disregarding air resistance, it is expected that the pendulum will return in exactly the same position as it was abandoned, obeying the principle of energy conservation. If no force is applied to the pendulum, the pendulum will never return to a height other than the one that was abandoned. However, for the case of quantum mechanics, the particle can pass through a barrier/potential even if it does not have enough energy to do so. It would be, in the classic analogy, as if the pendulum were able to return to a height greater than that which was initially abandoned. This strange phenomenon is called quantum tunneling. When penetrating the barrier, there is an exponential decline in the probability of penetration of the barrier according to the thickness “a” (given by equation 13) of the barrier and the square root of the relative height of the barrier (V-E), as shown on page 10 in which T is the probability of transmission through the barrier. In the same way that it is possible for the particle to pass through the barrier, it is possible for it to be reflected, as it would be expected within classical mechanics. Therefore, inside the well, there is the wave that goes through the barrier and the wave that is reflected, so that the wave function inside the box is the sum of the two waves (the one that "hits" the barrier and the one that is reflected), thus, the wave function inside the box is a superposition of a standing wave and a wave traveling towards the barrier. The wave function, after crossing the barrier, resumes its sinusoidal behavior. Note that, once we know how to calculate the transmission probability of a particle, calculating its reflection probability becomes trivial, according to the probability theory. Let R be the reflection probability, then we have that T+R = 1 (in other words, the particle will either be transmitted or reflected, which results in a 100% probability.) There is another way to calculate R , by means of the wave numbers, as shown on page 10. An example for which such modeling can be applied is for alpha particles (composed of 2 neutrons and two protons), which are initially confined to the atomic nucleus and, even without enough energy, there is the possibility that they “escape”, overcoming the potential barrier (which would be the atomic nucleus itself). One of the most widely used models in quantum mechanics, whether by scientists or engineers, is the harmonic oscillator model. In classical mechanics, a harmonic oscillator is commonly associated with a system composed by a body attached to a spring which, when displaced from its equilibrium position, undergoes the action of a force called restoring force, proportional to the x displacement, as shown in equation 14, where k is the spring constant. However, examples of harmonic oscillators are any system that undergoes small oscillations around their equilibrium positions (for small angles, a pendulum is an example of a harmonic oscillator). In the case of the mass-spring system, we classically know that the elastic potential energy is given according to equation 15 and can be rewritten in terms of the angular frequency of the oscillator, ω, according to equation 16. With the potential of the harmonic oscillator defined, we can insert this in the Schrödinger equation in order to obtain the desired solutions, according to equation 17. The Schrödinger equation for the harmonic oscillator will not be solved in detail because it is necessary to develop a somewhat complicated algebra, which will deviate the focus of the article. When we solve the Schrödinger equation with the potential of the harmonic oscillator, we get the energy quantization of that model (equations 18 and 19). Energy is quantized, just as in the case of the infinite well, that is, it cannot assume an arbitrary value. There is, however, a particularity to the harmonic oscillator: the energy levels are equally spaced by the factor h * f (h being Planck's constant and f being the frequency). The wave functions for the levels of the harmonic oscillator are shown by equation 20, with Hn being a set of functions. Transitions between the given energy levels are possible, in a similar way to the infinite well, therefore, when it happens, there is the emission of photons. The quantum model of the harmonic oscillator can be used, for example, in atoms contained in molecules such as H2 and HCl, as the atoms in these molecules oscillate around their equilibrium positions. So far we have explored the Schrödinger equation for the one-dimensional case, however, we can extend it, without much difficulty, to the three-dimensional case. In three dimensions, the Laplacian takes the form given by equation 21, with variations in the three dimensions. The Schrödinger equation, with the Laplacian written explicitly, is given by equation 22. For a free particle, the solution of the three-dimensional Schrödinger equation can be given by equation 23. For the case of the infinite square well, the boundary conditions will extend up for all three variables (x, y and z), such that V (x, y, z) = 0 for 0<x <L, 0 <y <L and 0<z <L and V (x, y, z) = ∞ for x, y, z <0 and for x, y, z> L. In the one-dimensional analysis, the square of the wave function (indicating the probability of finding the particle in a defined region from x to x + dx) is indicated by equation 24. The same applies for the other two dimensions “y” and “Z”, indicated by equations 25 and 26. Again we can resort to the theory of probability, which states that the probability that a number “m” of independent events will occur, is given by the product of the individual probabilities for each event to occur . Thus, the probability of finding the particle in a defined region of x + dx, y + dy and z + dz is given by equation 27. Equation 27 is the probabilistic density (square of the wave function module), therefore, the wave function itself will be the square root of the probabilistic density, as shown in equation 28. We can also calculate the energy for the three-dimensional case, according to pages 11, 12 and 13. There are special solutions to the Schrödinger equation, which are very useful for illustrating electron movement, called wave packets. Such solutions are obtained by a linear combination of plane wave solutions, with their respective coefficients. Wave packages will be explored in further detail in the future. As we saw for several models, the Schrödinger equation was solved according to the appropriate Hamiltonian (operator that represents the total energy of the system). However, how do we deal with systems composed of more than one particle? The Hamiltonian for a system composed of two particles of mass m would be given as shown in equation 29. For a system with N particles, the generic Hamiltonian would have the form of equation 30. However, special care must be taken for multi-particle systems, as we have to take into account the matter of symmetry. In quantum mechanics, wave functions are said to be symmetric if they satisfy the condition shown in equation 31. Fermions (semi-integer spin particles, such as electrons and quarks) do not have symmetric wave functions, but antisymmetric ones, as shown in equation 32. Bosons (whole spin particles, such as photons and alpha particles) have symmetric wave functions, shown in equation 31. Details on the matter of symmetry as well as the Pauli exclusion principle will be explored in future articles. In quantum mechanics, especially in quantum field theory, to describe systems of particles that interact with each other, we make use of special operators called annihilation and creation operators. In short, the creation operator, when acting on a null ket state, which is a representation of a state with no particle, “creates” a particle in that state, as shown in equation 33. The annihilation operator has the opposite effect, as shown in equation 34. Although the use of such operators may seem, at first glance, redundant, some of the relations they have, such as anti-commutation, greatly simplifies calculations. The Schrödinger equation for the hydrogen atom has a more complicated form (equation 35), because instead of using Cartesian coordinates, we use polar coordinates in order to obtain the atom's orbitals. The hydrogen atom will also be explored in the future. It is noteworthy that the Schrödinger equation is not 100% efficient, with flaws in some situations. As mentioned, even for the simplest element, hydrogen, the equation already takes on a complicated aspect, which gets even more difficult for more complex elements. The equation also fails for particles with relativistic speeds (significant fractions of the speed of light). However, the brilliant Paul Dirac was able to adapt the Schrödinger equation so that it agreed with special relativity, which gave rise to the so-called Dirac equation (equation 36). Dirac's equation was also able to predict the existence of antimatter. Before Schrödinger, what there was for the description of quantum systems was the peculiar Heisenberg's matrix mechanics, which was considered "abstract" by some physicists. The Schrödinger equation defined a milestone in the history of science, being widely accepted among physicists, who were already familiar with classical wave mechanics - whose similarity was highlighted here.

Reference material: Quantum Mechanics for Scientists and Engineers (David A. B. Miller) Modern quantum mechanics (J.J. Sakurai and Jim Napolitano) 50 Quantum Physics Ideas You Really Need to Know (Joanne Baker) The quantum universe (Brian Cox and Flavio Demberg) Physics for scientists and engineers volume 3 (Paul A. Tipler and Gene Mosca)