Quantum Laws: When Do They Rule?

Quantum mechanics is a fundamental theory that describes the behaviour of nature at and below the scale of atoms. It is the foundation of all quantum physics and is typically applied to microscopic systems such as molecules, atoms, and subatomic particles. Quantum mechanics can describe many systems that classical physics cannot, and its predictions have been verified experimentally to an extremely high degree of accuracy.

The laws of quantum mechanics, also called wave mechanics, were developed by physicists such as Erwin Schrödinger, Werner Heisenberg, and others in the mid-1920s. These laws explain atomic and subatomic phenomena, including the wave-particle duality of light and matter, and have led to important applications in various fields, including technology, chemistry, optics, and computing.

While quantum mechanics has had enormous success, it has also raised philosophical questions and debates about the nature of reality, determinism, and free will. The probabilistic nature of quantum mechanics and the concept of wavefunction superposition have challenged classical ideas of causality and sparked discussions about the role of observation and measurement in defining reality.

The development of quantum mechanics marked a shift from classical determinism to a more complex understanding of the universe, where probabilities and uncertainties play a significant role. This evolution in our understanding continues to have implications for various fields, including physics, philosophy, and even our understanding of human behaviour and choices.

The probabilistic nature of quantum mechanics

Quantum mechanics is a fundamental theory that describes the behaviour of nature at and below the scale of atoms. It is the foundation of all quantum physics and can describe many systems that classical physics cannot.

A fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only give probabilities. The probabilistic nature of quantum mechanics stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand.

The Born rule, named after physicist Max Born, is used to calculate probabilities. A probability is found by taking the square of the absolute value of a complex number, known as a probability amplitude. For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found.

The Schrödinger equation relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another.

The uncertainty principle is a consequence of the basic quantum formalism. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions for both a measurement of its position and a measurement of its momentum.

Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference, which is often illustrated with the double-slit experiment. In the basic version of this experiment, a coherent light source, such as a laser beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere, producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen.

Another non-classical phenomenon predicted by quantum mechanics is quantum tunnelling: a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential. In classical mechanics, this particle would be trapped.

When quantum systems interact, the result can be the creation of quantum entanglement: their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "..the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought".

While the early conception of atoms from Greek philosophy had been that they were indivisible units, the 19th century saw the formulation of hypotheses about subatomic structure. One important discovery in that regard was Michael Faraday's 1838 observation of a glow caused by an electrical discharge inside a glass tube containing gas at low pressure. Julius Plücker, Johann Wilhelm Hittorf and Eugen Goldstein carried on and improved upon Faraday's work, leading to the identification of cathode rays, which J. J. Thomson found to consist of subatomic particles that would be called electrons.

In 1900, Max Planck proposed the hypothesis that energy is radiated and absorbed in discrete "quanta" (or energy packets). According to Planck, quantities of energy could be thought of as divided into "elements" whose size (E) would be proportional to their frequency (ν):

> E=hν

Where h is the Planck constant.

In 1905, Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect, in which shining light on certain materials can eject electrons from the material. Niels Bohr then developed Planck's ideas about radiation into a model of the hydrogen atom that successfully predicted the spectral lines of hydrogen.

In the mid-1920s, quantum mechanics was developed to become the standard formulation for atomic physics. In 1923, the French physicist Louis de Broglie put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. Building on de Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg, Max Born, and Pascual Jordan developed matrix mechanics and the Austrian physicist Erwin Schrödinger invented wave mechanics. Born introduced the probabilistic interpretation of Schrödinger's wave function in July 1926.

By 1930, quantum mechanics had been further unified and formalized by David Hilbert, Paul Dirac and John von Neumann with greater emphasis on measurement, the statistical nature of our knowledge of reality, and philosophical speculation about the 'observer'.

Wave-particle duality

In the 17th century, Sir Isaac Newton advocated for a corpuscular (particulate) theory of light, while Christiaan Huygens proposed a wave description. Newton, despite favouring the particle theory, was the first to attempt to reconcile both perspectives. In the 19th century, Thomas Young's interference experiments in 1801 and François Arago's detection of the Poisson spot in 1819 validated Huygens' wave models. However, in 1901, Max Planck's law for black-body radiation challenged the wave model by assuming that energy changes in hypothetical electrically charged oscillators were proportional to the frequency of their associated electromagnetic waves. In 1905, Albert Einstein interpreted the photoelectric effect using the concept of discrete energies for photons, indicating particle behaviour. Arthur Compton's experiments from 1922 to 1924 provided further evidence of light's particle-like momentum and energy, contradicting earlier observations of wave-like interference.

The contradictory evidence for electrons was observed in the opposite order. Experiments by J.J. Thomson, Robert Millikan, Charles Wilson, and others demonstrated that free electrons exhibited particle properties, such as the measurement of their mass. In 1924, Louis de Broglie introduced his theory of electron waves, suggesting that electrons could be thought of as standing waves or waves associated with particles. He proposed that particles are bundles of waves (wave packets) with an effective mass and group velocity dependent on energy. Erwin Schrödinger further developed the wave equation of motion for electrons, which became known as the Schrödinger equation. In 1926, Max Born supported the wave-particle duality of electrons by citing experimental data from Clinton Davisson, who had conducted electron diffraction experiments. In 1927, the wave nature of electrons was empirically confirmed by the Davisson-Germer experiment at Bell Labs and the work of George Paget Thomson and Alexander Reid at Cambridge University. These experiments demonstrated that wave behaviour is not unique to electrons but is a general property of matter at a microscopic scale.

The uncertainty principle

In mathematical terms, the uncertainty principle can be expressed as a relationship between the standard deviation of position (σx) and the standard deviation of momentum (σp):

Σxσp ≥ ħ/2

Where ħ is the reduced Planck constant. This inequality states that the product of the standard deviations of position and momentum is greater than or equal to half of the reduced Planck constant.

Quantum entanglement

When two particles, such as a pair of photons or electrons, become entangled, they remain connected even when separated by vast distances. This means that a change in one particle instantly influences the other, regardless of the distance between them. For example, if one particle is measured as spinning upwards, its distant entangled partner will always be found to be spinning downwards (or vice versa, depending on the experiment). This correlation is a fundamental aspect of quantum entanglement, and it is what makes the phenomenon so intriguing and mind-boggling.

The concept of quantum entanglement was first introduced in 1935 by physicist Erwin Schrödinger, who emphasised its role as a defining trait of quantum mechanics, setting it apart from classical lines of thought. Schrödinger's work built upon earlier theories proposed by Albert Einstein, who had demonstrated that no information could travel faster than the speed of light. This contradiction led Einstein to famously refer to entanglement as "spooky action at a distance".

Despite its seemingly mysterious nature, quantum entanglement is not a means of communicating faster than the speed of light. Experiments have shown that the phenomenon does not violate Einstein's special theory of relativity, and it cannot be used for faster-than-light communication. Instead, quantum entanglement is better understood as a form of information, where the measurements of one particle provide information about its entangled partner without any direct influence or communication between them.

Today, quantum entanglement is considered a crucial aspect of quantum physics and is expected to play a significant role in the development of future quantum technologies. It is also used in practical applications such as cryptography and deep-space communications.

Quantum superposition

In other words, the state of a system is given by a linear combination of all the eigenfunctions of the Schrödinger equation governing that system. This means that a quantum system can act as if it is in multiple states at the same time until it is measured. This is known as a superposition of waves.

For example, an electron might be in a superposition of two different velocities or in two places at once. This is a difficult concept to visualise, but it can be understood by thinking of an analogy with waves on the surface of a pond. If you touch the surface of a pond at two different points at the same time, waves will spread out from each point and overlap to form a more complex pattern. Similarly, in quantum science, objects such as electrons and photons have wavelike properties that can combine and become superposed.

The concept of superposition is important in the development of quantum technology, such as quantum computers. In quantum computers, a qubit is the analog of the classical information bit, and qubits can be superposed. Unlike classical bits, a superposition of qubits represents information about two states in parallel.

The double-slit experiment is often used to demonstrate the concept of superposition. In this experiment, a beam of coherent light, such as a laser, is aimed at a barrier with two vertical slits. The light passes through the slits, and the resulting pattern is recorded on a photographic plate. When one slit is covered, the pattern shows a single line of light, aligned with the open slit. However, when both slits are open, the pattern on the photographic plate separates into multiple lines of lightness and darkness in varying degrees. This shows that the light is acting as waves, and the two waves are interfering with each other, adding together in some places and cancelling out in others. The same pattern is shown if photons or electrons are used.

If the experiment is repeated with only single photons or electrons sent through the two slits, one at a time, the same interference pattern is shown, even though only one particle is present. It is as if the particle goes through both slits at the same time and interferes with itself. In order to do this, it must be in a superposition state.

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