![]() Schrödinger subsequently showed that the two approaches were equivalent. Those who applied the methods of linear algebra included Werner Heisenberg, Max Born, and others, developing " matrix mechanics". Those who used the techniques of calculus included Louis de Broglie, Erwin Schrödinger, and others, developing " wave mechanics". In the 1920s and 1930s, quantum mechanics was developed using calculus and linear algebra. The equations represent wave–particle duality for both massless and massive particles. I ℏ ∂ ∂ t | ψ ( t ) ⟩ = H ^ | ψ ( t ) ⟩, now called the De Broglie relation, holds for massive particles, the chief clue being Lorentz invariance, and this can be viewed as the starting point for the modern development of quantum mechanics. Historical background Part of a series of articles about However, the wave function in quantum mechanics describes a kind of physical phenomenon, as of 2023 still open to different interpretations, which fundamentally differs from that of classic mechanical waves. This explains the name "wave function", and gives rise to wave–particle duality. The Schrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. The inner product between two wave functions is a measure of the overlap between the corresponding physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. These values are often displayed in a column matrix (e.g., a 2 × 1 column vector for a non-relativistic electron with spin 1⁄ 2).Īccording to the superposition principle of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a Hilbert space. ![]() When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for each possible value of the discrete degrees of freedom (e.g., z-component of spin). ![]() Some particles, like electrons and photons, have nonzero spin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom other discrete variables can also be included, such as isospin. The information represented by a wave function that is dependent upon position can be converted into a wave function dependent upon momentum and vice versa, by means of a Fourier transform. Wave functions can be functions of variables other than position, such as momentum. One has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called normalization. In one common form, it says that the squared modulus of a wave function that depends upon position is the probability density of measuring a particle as being at a given place. The Born rule provides the means to turn these complex probability amplitudes into actual probabilities. For example, a wave function might assign a complex number to each point in a region of space. Wave functions are composed of complex numbers. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively). In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The wave function of an initially very localized free particle. Panels (G–H) further show two different wave functions that are solutions of the Schrödinger equation but not standing waves. Panels (C–F) show four different standing-wave solutions of the Schrödinger equation. Rather, it is represented as a wave here, the vertical axis shows the real part (blue) and imaginary part (red) of the wave function. The quantum process (C–H) has no such trajectory. The classical process (A–B) is represented as the motion of a particle along a trajectory. Comparison of classical and quantum harmonic oscillator conceptions for a single spinless particle.
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