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**Wave mechanics** may refer to:

- the mechanics of waves
- the
*wave equation*in quantum physics, see Schrödinger equation

In physics, mathematics, and related fields, a **wave** is a disturbance of a field in which a physical attribute oscillates repeatedly at each point or propagates from each point to neighboring points, or seems to move through space.

The **Schrödinger equation** is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who derived the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

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**Quantum mechanics**, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

In particle physics, the **Dirac equation** is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.

The **de Broglie–Bohm theory**, also known as the **pilot wave theory**, **Bohmian mechanics**, **Bohm's interpretation**, and the **causal interpretation**, is an interpretation of quantum mechanics. In addition to a wavefunction on the space of all possible configurations, it also postulates an actual configuration that exists even when unobserved. The evolution over time of the configuration is defined by the wave function by a guiding equation. The evolution of the wave function over time is given by the Schrödinger equation. The theory is named after Louis de Broglie (1892–1987) and David Bohm (1917–1992).

In quantum mechanics, **wave function collapse** is said to occur when a wave function—initially in a superposition of several eigenstates—appears to reduce to a single eigenstate due to interaction with the external world; this is called an "observation". It is the essence of measurement in quantum mechanics and connects the wave function with classical observables like position and momentum. Collapse is one of two processes by which quantum systems evolve in time; the other is continuous evolution via the Schrödinger equation. However, in this role, collapse is merely a black box for thermodynamically irreversible interaction with a classical environment. Calculations of quantum decoherence predict *apparent* wave function collapse when a superposition forms between the quantum system's states and the environment's states. Significantly, the combined wave function of the system and environment continue to obey the Schrödinger equation.

A **wave function** in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters *ψ* or Ψ.

The **Klein–Gordon equation** is a relativistic wave equation, related to the Schrödinger equation. It is second order in space and time and manifestly Lorentz covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pi mesons are unstable and also experience the strong interaction, the practical utility is limited.

**Matter waves** are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter can exhibit wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave. The concept that matter behaves like a wave was proposed by Louis de Broglie in 1924. It is also referred to as the *de Broglie hypothesis*. Matter waves are referred to as *de Broglie waves*.

In quantum mechanics, the **principal quantum number** is one of four quantum numbers which are assigned to all electrons in an atom to describe that electron's state. As a discrete variable, the principal quantum number is always an integer. As *n* increases, the number of electronic shells increases and the electron spends more time farther from the nucleus. As *n* increases, the electron is also at a higher energy and is, therefore, less tightly bound to the nucleus. The total energy of an electron, as described below, is a negative inverse quadratic function of the principal quantum number *n*.

The **measurement problem** in quantum mechanics is the problem of how wave function collapse occurs. The inability to observe such a collapse directly has given rise to different interpretations of quantum mechanics and poses a key set of questions that each interpretation must answer.

In theoretical physics, the **pilot wave theory**, also known as **Bohmian mechanics**, was the first known example of a hidden-variable theory, presented by Louis de Broglie in 1927.
Its more modern version, the **de Broglie–Bohm theory**, interprets quantum mechanics as a deterministic theory, avoiding troublesome notions such as wave–particle duality, instantaneous wave function collapse, and the paradox of Schrödinger's cat. To solve these problems, the theory is inherently nonlocal and non-relativistic.

The **superposition principle**, also known as **superposition property**, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input *A* produces response *X* and input *B* produces response *Y* then input produces response.

In theoretical physics and applied mathematics, a **field equation** is a partial differential equation which determines the dynamics of a physical field, specifically the time evolution and spatial distribution of the field. The solutions to the equation are mathematical functions which correspond directly to the field, as functions of time and space. Since the field equation is a partial differential equation, there are families of solutions which represent a variety of physical possibilities. Usually, there is not just a single equation, but a set of coupled equations which must be solved simultaneously. Field equations are not ordinary differential equations since a field depends on space and time, which requires at least two variables.

A **differential equation** is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

In physics, there are equations in every field to relate physical quantities to each other and perform calculations. Entire handbooks of equations can only summarize most of the full subject, else are highly specialized within a certain field. Physics is derived of formulae only.

Physics deals with the combination of matter and energy. It also deals with a wide variety of systems, about which theories have been developed that are used by physicists. In general, theories are experimentally tested numerous times before they are accepted as correct as a description of Nature. For instance, the theory of classical mechanics accurately describes the motion of objects, provided they are much larger than atoms and moving at much less than the speed of light. These "central theories" are important tools for research in more specialized topics, and any physicist, regardless of his or her specialization, is expected to be literate in them.

In quantum mechanics, energy is defined in terms of the **energy operator**, acting on the wave function of the system as a consequence of time translation symmetry.

In gravity and pressure driven fluid dynamical and geophysical mass flows such as ocean waves, avalanches, debris flows, mud flows, flash floods, etc., **kinematic waves** are important mathematical tools to understand the basic features of the associated wave phenomena.
These waves are also applied to model the motion of highway traffic flows.