Wave function

Extracted Attributes

• Field

  Quantum physics

• Space

  Hilbert space

• Nature

  Complex-valued probability amplitude

• Symbol

  ψ, Ψ (lower-case and capital psi)

• Function

  Mathematical description of the quantum state of an isolated quantum system

• Condition

  Normalization (integral of squared modulus equals 1)

• Variables

  Position, momentum, time, spin, isospin

• Governing Equation

  Schrödinger equation

• Measurable Aspects

  Relative phase and relative magnitude

• Mathematical Property

  Can be added and multiplied by complex numbers (superposition principle)

• Probabilistic Interpretation

  Born rule (squared modulus gives probability density)

• Transformation between representations

  Fourier transform

Timeline

• The concept of the wave function was introduced with the help of the Schrödinger equation. (Source: web_search_results)

  1925

• John Martinis, who clarifies complex concepts like the Wave function, is the winner of the Nobel Prize in Physics. (Source: related_documents)

  2025

Wave function

In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively). According 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. The inner product of 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. 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. This explains the name "wave function", and gives rise to wave–particle duality. However, whether the wave function in quantum mechanics describes a kind of physical phenomenon is still open to different interpretations, fundamentally differentiating it from classic mechanical waves. Wave functions are complex-valued. For example, a wave function might assign a complex number to each point in a region of space. The Born rule provides the means to turn these complex probability amplitudes into actual probabilities. 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 integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called normalization. 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. 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. Wave functions can be functions of variables other than position, such as momentum. 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. 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. 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). 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).

Web Search Results

• quantum mechanics - What is a wave function in simple language?

  A "wave function" is a mathematical model (or representation) of a given wave. A "function" is represented by the symbol $f$. It can be a function of distance (x), time (t), space (r), etc. and is usually represented by an equation. If the equation represents a wave, then the function is a wave function. [...] A wave function is a complex-valued function $f$ defined on ${\mathbb R}^1$ (if your electron is confined to a line) or on ${\mathbb R}^2$ (if your electron is confined to a plane) or ${\mathbb R}^3$ (if your electron ranges over three-space), and satisfying $$\int |f|^2=1$$ (where the integral is defined over the entire line or plane or 3-space). Every electron has an associated wave function, and any function satisfying the above can be the wave function associated to some electron. [...] The wave function is the solution to the Schrödinger equation, given your experimental situation. With a classical system and Newton's equation, you would obtain a trajectory, showing the path something would follow: the equations of motion. For a quantum mechanical system you get a wave function, and the rules it obeys over time. With this you can determine the odds for your particle to be someplace, which is as close as you can get to a trajectory.

• 1.1: Quantum Mechanics and Wavefunctions

  A wavefunction (Ψ) is a mathematical function that relates the location of an electron and the energy of the electron. A wavefunction uses three variables to describe the position of an electron in space (as with the Cartesian coordinates x, y, and z). A fourth variable specifies the time at which the electron is at the specified location. [...] The wavefunction is a mathematical expression that describes the electron. It can be plotted as a three-dimensionional graph. The three-dimensional plot of the wavefunction is sometimes called an orbital. Often, chemists find it useful to look at pictures of orbitals in order to gain some sense of where electrons may be and how they may behave. In another sense, orbitals are probability maps that reveal the probability of an electron being located at a particular position in space.

• Wave Function

  ## What is Wave Function? In quantum physics, a wave function is a mathematical description of a quantum state of a particle as a function of momentum, time, position, and spin. The symbol used for a wave function is a Greek letter called psi, 𝚿. [...] By using a wave function, the probability of finding an electron within the matter-wave can be explained. This can be obtained by including an imaginary number that is squared to get a real number solution resulting in the position of an electron. The concept of wave function was introduced in the year 1925 with the help of the Schrodinger equation. | | [...] Wave function is a mathematical description of a quantum state of a particle as a function of time, position, momentum, and spin. Q3 ### Schrodinger Equation is named after which physicist? It is named after Erwin Schrodinger. Q4 ### How is time-dependent Schrodinger equation represented? The magnitude of the vector product is represented as: \(\begin{array}{l}ih\frac{\partial }{\partial t}\Psi (r,t)=[\frac{h^{2}}{2m}\bigtriangledown ^{2}+V(r,t)]\Psi (r,t)\end{array} \) Q5

• Wave function - Wikipedia

  ### Vector space structure [edit] A wave function is an element of a function space partly characterized by the following concrete and abstract descriptions. [...] 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. This explains the name "wave function", and gives rise to wave–particle duality. However, whether the wave function in quantum mechanics describes a kind of physical phenomenon is still open to different interpretations, fundamentally differentiating it from classic mechanical waves. [...] Wave functions can be functions "Function (mathematics)") of variables other than position, such as momentum. 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. Some particles, like electrons and photons, have nonzero spin "Spin (physics)"), and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete

• The Quantum Wavefunction Explained

  Overall the wavefunction is a mathematical tool which keeps track of all of the properties of a quantum particle, and explains our observations of the probabilistic nature of where particles appear when we do experiments on them. From the wavefunction we can calculate all of the observables like position, momentum, energy or spin, by applying mathematical operators on it. So in quantum physics, the wavefunction encodes all of the information about the quantum system, and valid wavefunctions are [...] of finding the particle at any point in this one-dimensional space so we can extract the position of a particle from this wave function actually the wave function doesn't just tell you about the probability of position but all other measurable physical quantities like momentum or energy or spin we just need to do different mathematical operations on the wave function for each one but overall the wave function is a mathematical tool which keeps track of all of the properties of a quantum [...] Introduction in quantum mechanics particles are things we only see when we measure them but how they move around is described by a wave function a wave function that satisfies the schrodinger equation which i talked about in my last video it's important to note that wave functions are not unique to quantum mechanics we use wavefunctions in many other systems like the motion of ripples of water sound waves vibrations on a string or electromagnetic waves and each of these systems has their own

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 and Ψ (lower-case and capital psi, respectively). The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state. For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier transform. Some particles, like electrons and photons, have nonzero spin, and the wave function for such particles include spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as isospin. 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) – 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). According 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. 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. 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. This explains the name "wave function", and gives rise to wave–particle duality. However, the wave function in quantum mechanics describes a kind of physical phenomenon, still open to different interpretations, which fundamentally differs from that of classic mechanical waves. In Born's statistical interpretation in non-relativistic quantum mechanics,the squared modulus of the wave function, |ψ|2, is a real number interpreted as the probability density of measuring a particle as being at a given place – or having a given momentum – at a given time, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation. This general requirement that a wave function must satisfy is called the normalization condition. 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; 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.