Schrodinger's Equation

• Type of physical science: Atomic physics
• Field of study: Nonrelativistic quantum mechanics

The laws of quantum mechanics are determined by the Schrödinger equation. They replace the classical Newtonian mechanics by a wavelike description of particles.

Overview

Before Erwin Schrödinger wrote his fundamental equation, it was observed that particles have some wavelike properties, at least on atomic scales. Louis de Broglie stated that concept in his work on duality, but Schrödinger was the first to draw the right conclusion from this evidence. He decided to look for an equation that describes wave motion to replace the Newtonian mechanics of particles. His success in finding a wave equation that governs the behavior of particles led to a tremendous leap in the understanding of quantum mechanics and to fundamental discoveries in practically all sciences.

The heuristic derivation of the Schrödinger equation is also a profound step; several such derivations are known, and each constitutes a useful research tool. The first and best-understood derivation is close to the original work of Schrödinger. The second approach is credited to Richard P. Feynman and became very important with the advances in modern quantum (field) theory.

Particles are described by their position and velocity at any given time in classical mechanics. In general, the forces act on the particles; one looks for the resulting positions and velocities of the particles of the system at a later time. The basic relation, Sir Isaac Newton's equations, states that force = ma, where m is the mass of the particle and a is its acceleration (the rate of velocity change per unit time).

Wave phenomena are more difficult to describe. Few equations of waves were known before Schrödinger, notably the equations that describe light waves (Maxwell's equations) and water and sound waves. Unlike particles, waves are described in terms of oscillations in both space and time.

Consider a wave traveling in a water pond. It has an amplitude, namely, its highest level above the calm water. The higher the amplitude, the more energy it carries with it. Another property is its velocity, the speed at which the wave propagates from one end of the pond to the other. Looking at one point in the pond, one can see that it moves up and down as the wave travels through. The number of times each point goes up in a second is called the frequency of the wave; it is another important characterization of the wave. The last quantity needed is the distance between one peak point of the wave and the next one at any given moment of time. This distance measures how twisted the water is by the wave. As this distance gets smaller, the water is more twisted and therefore has more energy. The equation of the wave is the relation between the above quantities: amplitude, frequency, velocity, and the wave number defined as one divided by the distance between the peaks.

Schrödinger's idea was to choose the wave equation such that the relation between its quantities is the same as that in classical Newtonian mechanics. From this point, the derivation is relatively easy, as these relations are well understood in both classical mechanics and wave motion.

The energy of a particle with velocity v moving in free space (free of forces) is proportional to its velocity squared: energy = 1/2 mv². On the other hand, the energy of a wave is proportional to its frequency, which is denoted by ω. Energy equals ħω, where ħ is Planck's constant number to be determined by the medium where the wave phenomenon takes place.

The velocity of a wave is proportional to its wave number (one divided by the distance between consecutive peaks). This results from the fact that a wave is propagating by creating peaks nearby, and the closer they are to one another, the faster they are affecting one another.

Therefore, the equations of quantum waves should satisfy the relation ħω=½m(ħk)², where k is the wave number. The equation of the wave is the relation satisfied by the amplitude (denoted by ψ) of the wave. This amplitude, being a function of both space and time, is called the wave function.

In the connection between the wave and the system it is describing, Schrödinger argued that since (superposition of) waves do not have a definite position or speed, one can talk only about the probability of observing a particular value for the corresponding particle. Therefore, the amplitude of the wave squared (ψ²) is associated with the probability of seeing the particle near the point x at time t. This results because the higher the amplitude squared, the higher the energy the wave has.

Feynman's derivation, on the other hand, can be summarized by relating quantum mechanics to classical description through probability rather than waves. He argues that while the laws of Newtonian mechanics are deterministic, they describe only the most probable behavior of a quantum particle. In classical mechanics, given the position and velocity of a particle at time zero, for example, one can then determine the whole future of the particle; one can tell exactly its path in space at all later times. Feynman then argued that in quantum mechanics, every path is allowed; the difference between them is only in probability. The classical path will be the most probable one, but any other path is also possible.

One then associates to any path the particle may choose, a quantity called action, which is related to the energy needed to carry the particle along that path. One is then given a simple formula to compute the amplitude of each such path. This amplitude squared is the probability for this path to materialize.

In order to determine the probability for the particle to go from point A at time zero to point B at time one, the amplitudes of all paths going from A to B are simply added. This gives the function ψ(B,t).

Unlike the Schrödinger derivation, Feynman's approach is very problematic from a mathematical point of view. Nevertheless, despite its formality, it is very useful, both theoretically and computationally.

Applications

The simplest system to solve is that of one particle moving freely on an infinitely long line (also called a quantum system in one dimension). Since there are no forces acting, the particle should be able to appear at any point of space at any time with equal probability. Therefore, it is described by a wave such that all points oscillate with the same amplitude and frequency. The distance between any two peaks, 1/k, will then be determined by the mass and total energy of the particle through the relation ħω=½m(ħk)², the frequency ω will be determined by the energy and the constant ħ known as Planck's constant, which was found by studying the energy of light waves with different colors.

One can now write the function explicitly as ψ(x,t) in terms of sine and cosine functions: for example, ψ(x,t)=sin(kx+ωt). This example is very important and is used to derive the general form of the Schrödinger equation.

A few observations can be made about the free case. First, the free particle can have any energy chosen simply by taking the frequency ω to satisfy energy equals ħω. In this case, the velocity is also exactly known and is determined by k. Yet, there is no information about the position of the particle; it can be anywhere on the line with equal probability. This is an extreme case of Werner Heisenberg's uncertainty principle: Complete knowledge of the velocity forces total smearing of the particle over the whole line, with no knowledge about its position. To describe a free particle that is localized in space, say in some interval of length of 1 meter, one needs to use another important property of waves not shared by particles: They add up. Two or more waves traveling independently can be composed on top of each other to get new forms; the waves may cancel each other at some places and get stronger and sharper at other points.

Localizing a wave is, therefore, done by adding up many waves with different velocities and amplitudes in such a way that they cancel each other at all places except the interval in which one is interested. This method is another manifestation of the uncertainty principle: The better one tries to localize a wave in a box, the more and more waves with different energies and velocities have to be added up; hence, the probabilities of finding different velocities are getting larger and more widespread. A wave localized in such a way is called a "wave packet."

To explain the effect of measurement, another aspect of quantum mechanics, assume one knows that at time 0, the wave function c describes a particle with some fixed known energy. Clearly, then, it has the same probability of being anywhere on the line. An experiment is conducted to check if the particle is in a given box. Suppose the answer is yes. Then, the wave function of the particle after the experiment is a wave packet in this box. It has undergone a drastic change as a result of the measurement. This is in sharp contrast to classical mechanics, where measurements have no effect on the observed system. Note, however, that the details of the experiment have not been revealed: for example, how long it takes and how much energy was transferred from the measuring apparatus to the system. At present, there is no simple answer to such questions. It goes beyond the issue of the description of the system by Schrödinger's equation. The quick answer to these questions is to add the measuring apparatus and the observer to the observed system and write down a new equation for the combined system of observer and object. This approach is not satisfactory, as it requires solving the equation for a system made of about 10 to the power of 26 particles, which is impossible. Furthermore, it is believed that the results of an experiment should not depend on the observer's fine details.

The second example of a quantum system is known as a "particle in a box." This problem is the same as the free case, but now the particle/wave is confined to moving in a box of length l (in one dimension). One expects, since no forces act on the particle, that it can visit any place in the box with the same probability. The wave function c, however, must vanish on the boundary of the box. If the wave has n equally spaced peaks inside the box, this is possible only if the length l of the box is equal to n times the distance between the peaks. Hence, the distance between peaks is l/n and so k=[(2πn)/(l/n)]=[(2πhn)/l]. From this relation, one finds that the energy of the particle in the box must satisfy E=(k²/2m)=(4π²h²n²)/(2ml²)

This formula exemplifies one of the most striking phenomena of quantum mechanics, namely, that the energy is quantized. Recall that n is an integer (n = +1, 2, 3...). Hence, the only allowed energies for a particle in a box of length l are proportional to n to the power of 2. The lowest possible energy is called the ground state energy. In this case, one finds it by choosing n to be the smallest possible, that is, 1. Therefore, the ground state energy of a particle in a box of length l is given by (4π²h²) / (2ml²)

The next to the lowest possible energy is called the first excited state. In this case, it corresponds to choosing n = 2, and the solution is (4π²h²4)/(2ml²)

The difference between the first excited state and the ground state is the energy that must be added to the ground state to excite it to the next level.

The "particle in a box" model gave a quite useful qualitative description of both atoms and nuclei. An atom can be thought of as an electron trapped in a box formed by the binding force of the protons at its nucleus. Exciting an atom (by heating or hitting) would move the electron from its ground state to its first (or higher) excited state. At a later time, the electron will "fall back" to its ground state, emitting in a form of light radiation the energy difference between its previous state and the ground state. This radiation is the one responsible for the color of materials and the ability to see objects. Different atoms and molecules will have different energy differences between their energy states, and, therefore, the spectra of their radiation will be different. There are very few other cases where one can solve the Schrödinger equation exactly. Most problems of practical interest are solved by approximating them using some of the simple known examples.

The Schrödinger description of quantum mechanics made many new contributions to the actual solution of systems. The "particle in a box" model and combinations of such models could be understood only by the Schrödinger description.

Consider, for example, a particle moving on an infinite line, where at some place, there is a bump of length l sub b. It is possible to solve this case exactly in the Schrödinger picture and demonstrate the ability of a particle, originally in one side of the bump, to pass to the other side even if it does not have enough energy to climb to the top of the bump. This is called "tunneling." It is easily explained by Feynman's description of quantum mechanics. While the classical paths, in this case, are all of particles reflected backward by the bump, classically forbidden paths have finite probability to materialize; hence, they are responsible for the ability of the particle to penetrate through the barrier. The tunneling effect is the basis of operation of many electronic devices: The emission of electrons from the heated filament in electronic valves is caused by tunneling.

Most important, the basic device of electronics-the transistor-can be described as a source of electrons coming from the part called the emitter. The electrons are trying to pass to the other part, called the collector. In order to pass, they have to tunnel through a barrier. The thickness of the barrier is controlled by another part called the base. By controlling the voltage of the base, one can, therefore, control the flow of current from the emitter to the collector in a very efficient way; this is one of the most important basics of electronics.

Context

At the beginning of the twentieth century, there were clear indications of the failure of classical mechanics to explain atomic behavior. As a result, quantum mechanics was developed, beginning with de Broglie's idea of duality between waves and particles in 1924. Schrödinger then developed the modern quantum theory by describing the mechanics of particles in terms of a wave equation. Earlier formulations, while formally equivalent, did not allow the explosive new applications and understanding that the Schrödinger equation had provided. The Schrödinger equation gave rise to a new interpretation of quantum mechanics: It allowed the development of new approximation techniques to solve practical problems in atomic and nuclear physics, chemistry and solid-state physics, and radiation theory. After resolving many of the puzzles of atomic physics, such as the stability of atoms and their spectral radiation, scientists were able to answer more complex questions, gaining some understanding of the chemistry and physics of molecules. Later, developments centered on the nucleus of the atom and formed the theoretical basis for nuclear physics. Then came the applications and extensions of both the ideas and methods of quantum mechanics to the theory of radiation. Quantum mechanics explained the nature and properties of the interaction between radiation and matter, culminating in laser physics. Further, it is evident that quantum mechanics is prominent in the behavior of large systems such as solids and very cool or very hot materials. In particular, superconductors are analyzed by trying to reconstruct the wave function, which describes the joint collective movement of all the molecules of the superconductor.

In the twenty-first century, quantum mechanics has been responsible for advancements in fields such as quantum computing and quantum cryptography. Schrödinger's wave equation remains foundational to quantum information science. It has helped researchers understand quantum states and develop qubits. Qubits harness the principles of quantum superposition and entanglement to perform specific types of calculations exponentially faster than traditional computers. Additionally, quantum communication has enabled ultra-secure data transmission through quantum key distribution. This theoretically ensures encryption that is immune to data breaches. 

Like Newtonian mechanics, which had a significant impact on mathematics, quantum mechanics in general and the Schrödinger theory in particular have generated and will continue to generate a new mathematics. Advances in quantum mechanics later appeared in many new mathematical fields, such as partial differential equations, operator theory, probability theory, and geometry. These advances are expected to continue to be a source of new ideas and applications in all fields of mathematics.

The effects on the biological sciences are no less essential, though slower to penetrate. Important early applications were more concerned with the development of new scientific and medical tools for biological experiments and analysis. Nuclear magnetic resonance machines, for example, are based on purely quantum effects, as well as electronic microscopes and genetic deciphering machines. In the twenty-first century, quantum biology emerged as an interdisciplinary field exploring the role of quantum mechanics in biological processes such as photosynthesis, enzyme function, and human cognition.

Principal terms:

AMPLITUDE: the maximum displacement of a medium from its equilibrium position

ENERGY: the ability to perform work; for example, a moving particle has energy proportional to its mass and to its velocity squared

FREQUENCY: the number of times a repetitive phenomenon occurs in one second; for example, the number of times a drop of water reaches its highest point in one second in a stormy sea

PARTICLE: a point mass moving in space; its position and velocity completely describe its state

WAVE: the oscillatory perturbation of a medium

WAVE FUNCTION: the amplitude of a wave as it depends on space and time

WAVE NUMBER: one divided by the distance between consecutive peaks of a medium where a wave propagates

Bibliography

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