Probando traducción/en

Fundamentos de la mecánica cuántica

La mecánica cuántica es la teoría física que describe el comportamiento de la materia y de la radiación a escalas atómicas y subatómicas. Se desarrolló durante las primeras décadas del siglo XX para explicar una serie de fenómenos que la mecánica clásica y el electromagnetismo tradicional no podían describir satisfactoriamente.

These phenomena included black-body radiation, the photoelectric effect, atomic spectra, and the stability of atoms. The work of scientists such as Max Planck, Albert Einstein, Niels Bohr, Louis de Broglie, Werner Heisenberg, Erwin Schrödinger, Max Born, and Paul Dirac gradually led to a new conception of nature.

One of the most important features of quantum mechanics is that it abandons the classical idea that all the properties of a physical system have perfectly defined values at all times. Instead, the theory uses a mathematical formalism that makes it possible to calculate the probabilities of obtaining certain results when a measurement is carried out.

Quantization of energy

The historical origin of quantum theory is usually placed in the year 1900, when Max Planck studied the radiation emitted by hot bodies. To correctly reproduce the experimental results, Planck proposed that energy was not exchanged in a completely continuous manner, but rather in discrete amounts later called quanta.

The energy of one of these quanta is given by:

<math>E = h\nu</math>

where:

• <math>E</math> is the energy;
• <math>h</math> is Planck's constant;
• <math>\nu</math> is the frequency of the radiation.

This hypothesis marked a break with the ideas of classical physics. In certain situations, some physical quantities can only take on certain permitted values.

A particularly important example appears in atoms. Electrons bound to an atom cannot possess just any value of energy, but only certain energy levels. When an atom moves from one energy level to another, it may absorb or emit a photon whose energy corresponds to the difference between the two levels.

Wave-particle duality

Another fundamental concept of quantum mechanics is the so-called wave-particle duality. In classical physics, particles and waves are clearly different objects. A particle has a relatively localized position, whereas a wave is distributed over a region of space.

In the quantum world, this distinction is less clear-cut.

Light, which had traditionally been described as an electromagnetic wave, can behave in certain experiments as a set of particles called photons. The photoelectric effect is one of the best-known examples of this corpuscular behavior.

Conversely, Louis de Broglie proposed that material particles could also exhibit wave properties. A wavelength can be associated with a particle of linear momentum <math>p</math>:

<math>\lambda = \frac{h}{p}</math>

This relationship was confirmed experimentally through electron diffraction experiments.

Wave-particle duality does not simply mean that a quantum object is sometimes a particle and sometimes a wave in the classical sense. Rather, it indicates that classical categories are insufficient to fully describe its behavior.

The wave function

In the formulation developed by Erwin Schrödinger, the state of a quantum system is represented by a wave function, usually denoted by the Greek letter <math>\psi</math>.

The time evolution of this function is determined by the Schrödinger equation. In its time-dependent form it can be written schematically as:

<math>i\hbar\frac{\partial}{\partial t}\psi = \hat{H}\psi</math>

where <math>i</math> is the imaginary unit, <math>\hbar</math> is the reduced Planck constant, and <math>\hat{H}</math> represents the operator associated with the total energy of the system, known as the Hamiltonian.

The wave function does not directly represent an observable property. According to the interpretation proposed by Max Born, the square of its modulus,

<math>|\psi|^2</math>,

is related to the probability density of finding a particle in a given region of space.

For example, in the case of an electron located around an atomic nucleus, one does not necessarily speak of a definite trajectory, like the orbit of a planet around the Sun. Instead, a probability distribution is used, indicating the regions where there is a greater or lesser probability of detecting the electron.

Quantum superposition

The superposition principle states that, if a system can be in several possible states, it can also be in a combination of those states.

If <math>\psi_1</math> and <math>\psi_2</math> represent two possible states, a combination such as

<math>\psi = a\psi_1 + b\psi_2</math>

can also represent a valid physical state, provided that the corresponding normalization conditions are met.

This principle gives rise to some of the most characteristic effects of quantum physics. In the famous double-slit experiment, for example, a particle can produce an interference pattern that requires considering simultaneously the different alternatives available during its propagation.

Superposition is also one of the foundations of quantum computing. Whereas a classical bit takes the value 0 or 1, a qubit can be in a superposition of the states associated with 0 and 1.

Measurement and probabilities

Measurement plays a particularly important role in quantum theory. Before a measurement is carried out, a system may be in a superposition of different possible outcomes. Quantum mechanics makes it possible to calculate the probability of obtaining each one of them.

For example, if a particular property of a particle is measured, such as its energy, the result obtained will be one of the values allowed by the system. If the experiment is repeated many times with identically prepared systems, a statistical distribution of results will emerge that is consistent with the probabilities predicted by the theory.

This probabilistic character gave rise to numerous philosophical debates among the founders of quantum mechanics. Albert Einstein, for example, repeatedly expressed his discomfort with the idea that probabilities were a fundamental element of nature. Nevertheless, the theory's probabilistic predictions have been verified experimentally with extraordinary precision.

Uncertainty principle

Werner Heisenberg discovered that there are certain pairs of physical quantities that cannot simultaneously possess arbitrarily precise values. The best-known example is the position and the linear momentum of a particle.

The uncertainty relation can be expressed as:

<math>\Delta x,\Delta p \geq \frac{\hbar}{2}</math>

where <math>\Delta x</math> represents the uncertainty in position and <math>\Delta p</math> the uncertainty in momentum.

Heisenberg's uncertainty principle should not be interpreted simply as a limitation caused by imperfect measuring instruments. It is a feature inherent to the quantum formalism. A state in which the position is very well defined necessarily presents a broader distribution of possible momentum values, and vice versa.

Quantum entanglement

Two or more quantum systems can be in a joint state known as entangled. In that case, certain properties of the individual systems cannot be described independently, even when the particles are spatially separated.

The correlations associated with entanglement were the subject of a famous debate initiated by Einstein, Podolsky, and Rosen in 1935. Later, the physicist John Stewart Bell developed mathematical inequalities that made it possible to experimentally compare the predictions of certain local hidden-variable theories with those of quantum mechanics.

Numerous experiments have confirmed the correlations predicted by quantum theory.

Entanglement currently has potential and actual applications in areas such as:

• quantum cryptography;
• quantum computing;
• teleportation of quantum states;
• quantum communication networks;
• certain high-precision metrology methods.

Interpretations of quantum mechanics

Although the fundamental equations of quantum mechanics are extraordinarily well verified, there is debate about their conceptual interpretation.

The so-called Copenhagen interpretation, associated mainly with Niels Bohr and Werner Heisenberg, assigns a fundamental role to the measurement process and holds that the theory describes the probabilities of the different observable outcomes.

There are, however, other interpretations. Among them are the following:

1. the many-worlds interpretation;

1. hidden variable theories;

1. objective collapse interpretations;

1. various informational and relational interpretations.

These proposals may offer very different conceptual pictures of reality, although in many cases they make the same predictions for ordinary experiments.

The importance of quantum mechanics

Quantum mechanics is not merely a theory applicable to highly specialized experiments. A large part of modern technology depends, directly or indirectly, on quantum phenomena.

Its applications include transistors, integrated circuits, lasers, light-emitting diodes, photovoltaic cells, numerous sensors, and various techniques for the diagnosis and analysis of materials.

In addition, the combination of quantum mechanics with special relativity gave rise to quantum field theory, which constitutes the mathematical basis of the Standard Model of particle physics.

Conclusion

Quantum mechanics profoundly changed our understanding of nature. Concepts such as quantization, wave-particle duality, the wave function, superposition, uncertainty, and entanglement show that microscopic reality does not always follow the intuitions developed from our everyday experience.

Despite its counterintuitive nature, quantum mechanics is one of the most precisely tested scientific theories. Its principles are fundamental to understanding the structure of atoms, the properties of matter, the interaction between light and elementary particles, and a large part of contemporary technology.

See also

• Quantum physics
• Classical mechanics
• Schrödinger equation
• Heisenberg uncertainty principle
• Quantum entanglement
• Quantum computing
• Quantum field theory