Glossary
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abstract space
A mathematical “space” is a collection of objects (such as numbers, vectors, or functions) with a structure that defines the relationship between those objects. Just as locations in three-dimensional physical space can be represented by points along three coordinate axes, the objects in an abstract space can be represented by points along an arbitrary number of “generalized coordinate” axes. In physics and engineering, the state or configuration of a system can be represented by points in “state space” or “configuration space”, with generalized coordinate axes representing the parameters of the system.
In quantum mechanics, the wavefunction solutions to the Schrödinger Equation are members of an abstract vector space called a Hilbert space.
Read more about it:
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basis system
A basis system (often called simply a basis) in a vector space is a set of vectors that are linearly independent (that is, the inner produce between them is zero) and that span the space (that is, every other vector within the space may be made up of a weighted linear combination of these basis vectors). If the basis vectors have a norm of one unit, they are called an orthonormal basis, and the orthonormal basis vectors that point along the directions of the coordinate axes are called the standard basis or natural basis.
Read more about it:
http://mathworld.wolfram.com/VectorBasis.html
http://mathworld.wolfram.com/StandardBasis.html
http://fourier.eng.hmc.edu/e102/lectures/orthogonaltransform/node3.html
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bra
A bra is a mathematical object first defined by physicist Paul Dirac, who realized that the “bracket” notation for an inner product (such as \(\langle a|b\rangle\)) could usefully be separated into two distinct objects, a “bra” \(\langle a|\) and “ket” \(|b\rangle\). Since an inner product produces a scalar output, a bra such as \(\langle a|\) serves as a linear functional to map a vector such as \(|b\rangle\) onto the field of scalars. Hence bras and kets inhabit separate “dual†abstract vector spaces, and every bra has a corresponding ket. It’s common to represent a ket by a column vector and the corresponding bra by a row vector with elements that are the complex conjugates of the ket’s elements.
In quantum mechanics, the state of a particle or system may be represented by a ket such as \(|ψ\rangle\) and just as a vector may be expanded into a series of components and basis vectors, a ket may be expanded into a series of components and basis kets. The “amount” of another state represented by ket \(|ψ_1\rangle\) in a state represented by ket \(|ψ\rangle\) can be found by multiplying the bra \(\langle ψ_1|\) by the ket \(|ψ\rangle\) (which is the inner product \(\langle ψ_1|ψ\rangle\)).Read more about it:
http://mathworld.wolfram.com/Bra.html
https://www.mathpages.com/home/kmath638/kmath638.htm
http://www.physics.unlv.edu/~bernard/phy721_99/tex_notes/node6.html
http://karin.fq.uh.cu/qct/MMN/extras/Bra-ket_notation-Wiki.pdf
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classically forbidden
A classically forbidden region is a range of spatial locations at which a particle or system with given characteristics cannot exist (so is “forbidden”) within the laws of physics prior to the development of modern physics (hence “classical”). For example, in classical physics the potential energy of a particle cannot exceed the particle’s total energy (the sum of potential plus kinetic energy), since that would require the kinetic energy to be negative in that region. Regions in which the total energy exceeds the potential energy are called “classically allowed” regions.
The laws of quantum physics do not preclude particles from existing in classically forbidden regions, albeit with exponentially decaying wavefunctions. In spite of the existence of particles in classically forbidden regions, any measurement of kinetic energy will always yield a positive result, and the Heisenberg Uncertainty principle ensures that the particle may have been in a classically allowed region (with positive kinetic energy) when the measurement was made.
Read more about it:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc6.html
http://demonstrations.wolfram.com/QuantumHarmonicOscillatorTunnelingIntoClassicallyForbiddenRe/
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complete set
In a vector space, a set of orthogonal vectors is complete if any vector in that space can be represented as a weighted combination of those vectors. In quantum mechanics, if a quantum state can be uniquely specified by a set of measurements, the operators corresponding to those measurements (that is, the operators whose eigenvalues are the possible outcomes of the measurements) are said to form a complete set of commuting observables (CSCO).
Read more about it:
https://en.wikipedia.org/wiki/Complete_set_of_commuting_observables
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complex numbers
A complex number is a number that can be represented as \(a+ib\), in which \(a\) and \(b\) both represent real numbers, and i represents the imaginary unit \(\sqrt{-1}\) Complex numbers may be represented graphically on two perpendicular number lines, a “real” number line (usually horizontal and increasing to the right), and an “imaginary” number line (usually vertical and increasing upward). The plane containing these two number lines is called the complex plane, and multiplying any number by the imaginary unit \(i\) is equivalent to a counter-clockwise \(90^\circ\) rotation in that plane. In quantum mechanics, the wavefunction solutions to the Schrödinger equation are, in general, complex quantities.
Read more about it:
http://hyperphysics.phy-astr.gsu.edu/hbase/cmplx.html#c1
http://mathworld.wolfram.com/ComplexNumber.html
https://en.wikipedia.org/wiki/Complex_number
https://galileospendulum.org/2012/06/09/imaginary-numbers-are-real/
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conjugate variables
Conjugate variables are pairs of variables in which the two members of the pair are related by the Fourier transform; examples include position/momentum and time/energy. An uncertainty relationship always exists between the two members of a pair of conjugate variables, and the operators corresponding to those variables do not commute.
Read more about it:
https://en.wikipedia.org/wiki/Conjugate_variables
https://brilliant.org/wiki/heisenberg-uncertainty-principle/
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covector
A covector (also called a “one-form”) is a mathematical object that inhabits the dual space to that of vectors. A covector acts as a linear functional that, when multiplied by a vector, produces a scalar result. In the Dirac notation of quantum mechanics, the state of a system can be represented by an abstract vector or ket, and the Hermitian conjugate of a vector ket is a covector bra which occupies a dual space and serves as a linear functional when used in an inner product with a ket.
Read more about it:
http://www.euclideanspace.com/maths/algebra/vectors/related/covector/index.htm
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Dirac delta function
The Dirac delta function is a mathematical object that takes in an argument and produces a known output; if the input argument is zero, that output is infinite, but for all other values of the argument, the output is zero. The infinite output value for zero input argument means that the delta function does not meet the strict mathematical definition of a function, so the delta function is technically called a “generalized function” or “distribution”.
The Dirac delta function is useful in quantum mechanics because the Fourier-transform relationship between conjugate variables (such as position and momentum) means that sinusoidal wavefunctions in one domain correspond to delta-function wavefunctions in the conjugate domain.
Read more about it:
http://mathworld.wolfram.com/DeltaFunction.html
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Dirichlet conditions
The Dirichlet conditions specify the requirements that must be satisfied for a function to be represented by a Fourier series (that is, a weighted linear combination of sines and cosines). The Dirichlet conditions include the requirement that such functions cannot have an infinite number of discontinuities or extrema. This is relevant to quantum mechanics because wavefunctions of conjugate variables such as position and momentum are related by the Fourier transform.
Read more about it:
https://en.wikipedia.org/wiki/Dirichlet_conditions
http://mathworld.wolfram.com/DirichletFourierSeriesConditions.html
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distribution
A distribution or “generalized function” is an extension of the concept of ordinary functions to mathematical relationships which are not “well-behaved” (for example, by having infinitely large values or discontinuities). A distribution is defined not by its value at certain points, as is an ordinary function, but rather by its effect on other functions. Examples of distributions in quantum mechanics include the delta-function representation of position eigenfunctions in position space.
Read more about it:
https://en.wikipedia.org/wiki/Distribution_(mathematics)
http://mathworld.wolfram.com/GeneralizedFunction.html
http://www.csci.psu.edu/seminars/springnotes/2006/Farassat2006.pdf
http://physics.unipune.ac.in/~phyed/27.1/1191%20revised(27.1).pdf
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dual space
The dual space to a vector space is the set of all mathematical objects (called “linear functionals” or “convectors”) that maps the vectors to the field of scalars. In quantum mechanics, the space inhabited by bras is dual to the vector space inhabited by kets.
Read more about it:
https://en.wikipedia.org/wiki/Dual_space
https://people.math.osu.edu/gerlach.1/math5101/DualOfAVectorSpace.pdf
https://math.stackexchange.com/questions/3749/why-do-we-care-about-dual-spaces
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Fourier transform
The Fourier transform is a mathematical process that converts a function of a variable in one domain (such as position) to the corresponding function of the conjugate variable in another domain (such as momentum). This process is equivalent to decomposing a given function into a set of sinusoidal basis functions (generally called Fourier analysis), and to producing a desired function by forming a weighted linear superposition of sinusoids (generally called Fourier synthesis).
In quantum mechanics, conjugate variables, which are also called Fourier-transform pairs, include position/momentum and energy/time. Conjugate variables always obey an uncertainty relation, which means that there is a limit to the precision with which both variables can be simultaneously known.
Read more about it:
http://mathworld.wolfram.com/FourierTransform.html
https://quantummechanics.ucsd.edu/ph130a/130_notes/node82.html
http://theoreticalminimum.com/courses/quantum-mechanics/2012/winter/lecture-9
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free particle
A free particle is any particle on which the net force is zero, which means that the potential energy of the particle is constant over the region of interest. In quantum mechanics, the free-particle wavefunctions are plane waves which extend over all space. Such wavefunctions with a single wavenumber are not normalizable, but normalizable spatially limited wavepackets may be synthesized as weighted linear combinations of these wavefunctions.
Read more about it:
https://en.wikipedia.org/wiki/Free_particle
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/Scheq.html#c3
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Hamiltonian operator
The Hamiltonian operator is the total-energy operator, which is the combination of the kinetic-energy operator (which involves the square of the momentum operator) and the potential energy. The time-independent Schrödinger Equation is an eigenvalue equation in which the operator is the Hamiltonian operator, the eigenfunction is the wavefunction \(\psi(x)\), and the eigenvalue is the total energy \(E\).
Read more about it:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hamil.html
https://www.youtube.com/watch?v=Y0Lm3mtXs5o
https://www.southampton.ac.uk/assets/centresresearch/documents/compchem/DFT_L2.pdf
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harmonic oscillator
A harmonic oscillator is a system in which an object undergoes periodic oscillation about an equilibrium position under the influence of a restoring force. The restoring force always points toward the equilibrium position and increases in proportion to the distance of the object from the equilibrium position. Since force is related to the negative gradient of the potential energy, the potential energy of a harmonic oscillator varies as the square of the distance from equilibrium.
In quantum mechanics, the variation of potential with distance makes the Schrödinger Equation for a harmonic oscillator more difficult to solve than constant-potential cases. For the quantum harmonic oscillator, analytic and algebraic approaches may be used to determine the wavefunction solutions, which are proportional to the Hermite polynomials.
Read more about it:
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Heisenberg Uncertainty principle
The Heisenberg Uncertainty principle states that the product of the uncertainties (the widths of the probability distributions) of position and momentum always has a value that is equal to or greater than Planck’s modified constant divided by two. This is a specific case of the general uncertainty relation between any two conjugate variables (that is, variables related by the Fourier transform). Since the product of the uncertainties has a minimum value, the uncertainty principle means that both quantities cannot be simultaneously known with arbitrarily great precision.
The physical meaning of the Heisenberg Uncertainty principle is that a highly localized wavefunction (a narrow wave packet) requires a wide range of wavenumbers (which means a wide range of momenta, since momentum is proportional to wavenumber). Alternatively, narrow spectrum (small range of wavenumbers and momenta) corresponds to a wide position wavefunction (which means the wave extends over a large range of positions).
Read more about it:
http://hyperphysics.phy-astr.gsu.edu/hbase/uncer.html
https://plus.maths.org/content/heisenbergs-uncertainty-principle
https://farside.ph.utexas.edu/teaching/315/Waveshtml/node92.html
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Hermite polynomials
The Hermite polynomials are an orthogonal set of polynomials; the \(n^{th}\) Hermite polynomial (physicist’s version) can be generated by taking the \(n^{th}\) derivative of \(e^(-x^2 )\) with respect to \(x\) and multiplying the result by \((-1)^n\) and \(e^{x^2}\).
Multiplying the Hermite polynomials by the Gaussian function \(e^{x^2}\) gives the solutions to the equation \(af”(x)+bx^2f_n(x)=(2n+1)f_n(x)\) in which \(a\) and \(b\) are constants. This is relevant to quantum mechanics because this equation is structurally identical to the Schrödinger Equation for the quantum harmonic oscillator.Read more about it:
https://en.wikipedia.org/wiki/Hermite_polynomials
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Hermitian operator
A Hermitian operator is an operator that produces the same result when applied to either member of an inner product. The eigenvalues of a Hermitian operator are always real, and the eigenfunctions of a Hermitian operator are orthogonal in the non-degenerate case and can be made orthogonal in the degenerate case.
Read more about it:
http://farside.ph.utexas.edu/teaching/qmech/Quantum/node37.html
http://www.eng.fsu.edu/~dommelen/quantum/style_a/herm.html
http://vergil.chemistry.gatech.edu/notes/quantrev/node16.html
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Hilbert space
Hilbert space is an abstract vector space in which the rules of vector addition, scalar multiplication, and the inner product are supported. Additionally, for a space to qualify as a Hilbert space, all members of that space must have finite norm. The concept of a Hilbert space is useful in quantum mechanics because the wavefunction solutions to the Schrödinger Equation are members of an abstract linear vector space (which means that the principle of superposition applies), and those wavefunctions must have finite norm (to ensure finite probability density).
Read more about it:
https://en.wikipedia.org/wiki/Hilbert_space
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Hooke's Law
Hooke’s Law is an equation describing the behavior of a force with magnitude that varies in direct proportion to distance from some equilibrium point; this is a “restoring” force because the direction of the force is always toward the equilibrium point. In a region in which a Hooke’s-law force is operating, the potential energy is proportional to the square of the distance from equilibrium.
Read more about it:
https://phys.org/news/2015-02-law.html
https://phet.colorado.edu/en/simulation/hookes-law
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imaginary unit
The imaginary unit is an operator that carries real numbers to the imaginary number line; this operator is usually designated by \(i\) or \(j\) and has numerical value of \(\sqrt{-1}\).
Read more about it:
https://en.wikipedia.org/wiki/Imaginary_unit
https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/
https://galileospendulum.org/2012/06/09/imaginary-numbers-are-real/
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inner product
The inner product is a mathematical process that is an extension of the “dot” or “scalar” product between two real vectors in three-dimensional space to higher-dimensional abstract vectors and continuous functions. The dot product for vectors expanded in an orthonormal basis system is formed by multiplying corresponding component (such as \(A_xB_x\)) and then summing the results of those multiplications over the three dimensions of physical space \((A_xB_x+A_yB_y+A_zB_z)\). This process may be extended to two N-dimensional abstract vectors by multiplying all corresponding components and summing over N dimensions, and to continuous functions over a specified range by multiplying the functions and integrating the product over the range of interest. In the case of complex vector components or functions, it’s necessary to take the complex conjugate of one of the vectors or functions before multiplication (as explained in the text, this preserves the desirable characteristic that the inner product of a vector or function with itself gives the length (or “norm”) of the vector or function.
Read more about it:
http://mathworld.wolfram.com/InnerProduct.html
https://web.auburn.edu/holmerr/2660/Textbook/innerproduct-print.pdf
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ket
A ket is a mathematical object first defined by physicist Paul Dirac, who realized that the “bracket” notation for an inner product (such as \(\langle a|b\rangle\)) could usefully be separated into two distinct objects, a “bra” \(\langle a|\) and a “ket” \(|b\rangle\). The ket portion of this inner-product bracket represents a vector (that is, a member of a vector space ), and the bra portion of this inner-product bracket represents a linear functional that maps the ket to the field of scalars. Kets and bras inhabit separate “dual” abstract vector spaces, and every ket has a corresponding bra. Kets are inherently basis-independent, but a ket may be expanded as a combination of coefficients and basis kets. Once a basis system has been specified, it’s common to represent a ket by a column vector and the corresponding bra by a row vector with elements that are the complex conjugates of the ket’s elements.
In quantum mechanics, the state of a particle or system may be represented by a ket such as \(|\psi\rangle\), and just as a vector may be expanded into a series of components and basis vectors, a ket may be expanded into a series of components and basis kets. The “amount” of another state represented by ket \(|\psi_1⟩\) in a state represented by ket \(|\psi\rangle\) can be found by multiplying the bra \(\langle \psi_1|\) by the ket \(|\psi\rangle\) (which is the inner product \(\langle \psi_1 |\psi⟩\)).Read more about it:
http://www.physics.unlv.edu/~bernard/phy721_99/tex_notes/node6.html
http://karin.fq.uh.cu/qct/MMN/extras/Bra-ket_notation-Wiki.pdf
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Kronecker delta
The Kronecker delta \((\delta_{ij})\) is a mathematical function that takes in two input variables (\(i\) and \(j\) in \(\delta_{ij}\)) and outputs zero if the input variables are different (\(i \neq j\)) and outputs one if the input variables are the same (\(i = j\)).
Read more about it:
https://en.wikipedia.org/wiki/Kronecker_delta
http://mathworld.wolfram.com/KroneckerDelta.html
https://webhome.phy.duke.edu/~rgb/Class/Electrodynamics/Electrodynamics/node19.html
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ladder operator
In quantum mechanics, a ladder operator is a mathematical operator that takes in an input wavefunction – for example, a solution to the Schrödinger Equation for the harmonic oscillator – and produces an output wavefunction that is a solution to the Schrödinger Equation for a higher-energy or lower-energy state. A ladder operator that produces the wavefunction of a higher-energy state is sometimes called a “raising” operator, and a ladder operator that produces the wavefunction of a lower-energy state is sometimes called a “lowering” operator.
Read more about it:
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Laplacian operator
The Laplacian is a spatial second-derivative operator representing the divergence of the gradient of the function upon which it operates. Taking the Laplacian of a scalar field produces a scalar value proportional to the amount by which the field at a point is greater than (negative Laplacian) or less than (positive Laplacian) the average value of the surrounding points.
Read more about it:
https://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node243.html
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L'Hôpital's rule
L’Hôpital’s rule is a mathematical technique for handling some indeterminate forms such the ratio of functions that approaches (0/0) or (\(\frac{\infty}{\infty}\)). According to L’Hôpital’s rule, the ratio \(f(x)/g(x)\) as both functions approach zero for some value of \(x\) can be determined by dividing \(f'(x)\) (the derivative of \(f(x)\) with respect to \(x\)) by \(g'(x)\) (the derivative of \(g(x)\) with respect to \(x\)). In quantum mechanics, L’Hôpital’s rule is useful in finding the value of a wavefunction such as \(sin(kx)/(kx)\) (the Fourier transform of a rectangular function) as \(x\) approaches zero.
Read more about it:
https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/lhopitaldirectory/LHopital.html
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Matrix mechanics
Matrix mechanics is an approach to quantum mechanics developed primarily by Werner Heisenberg as an alternative to the wave mechanics developed by Erwin Schrödinger. In analyzing the radiation spectrum produced by electron transitions between states, Heisenberg began assigning two indices to quantities such as position, momentum, and energy, since the intensity and frequency of the radiation emitted during a transition depends on both the initial and final states. Heisenberg noticed that the commutator involving products of these quantities was not always zero, which led him to develop mathematical rules that Max Born later recognized as matrix algebra. Schrödinger, Dirac, and Von Neumann later showed that the predictions of matrix mechanics are identical to those of wave mechanics.
Read more about it:
https://en.wikipedia.org/wiki/Matrix_mechanics
https://www.mathpages.com/home/kmath698/kmath698.htm
https://www.ias.ac.in/article/fulltext/reso/022/04/0399-0405
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normalization
Normalization is a mathematical process by which the length of a vector (the square root of the dot product of the vector with itself) or norm of a function (the square root of the inner product of the function with itself) is set to unity. This is accomplished by dividing an unnormalized vector or function by its norm.
Normalization is important in quantum mechanics because the norm of a quantum-mechanical position wavefunction over a specified region gives the probability that the particle is located within that region, and that probability must be unity if the norm is taken over all space (since, if the particle exists, it must be located somewhere in space).
Read more about it:
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orthogonality relations
An orthogonality relation is a mathematic statement that the inner product of two functions (often taken to be a sine and cosine or two sinusoids with different frequencies) is zero over some range. This is analogous to two perpendicular vectors having zero dot product.
Read more about it:
https://www.macalester.edu/aratra/edition2/chapter1/chapt1d.pdf
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Planck constant
The Planck constant, usually written as \(h\), is a fundamental constant of nature, established by Max Planck as the constant of proportionality between the energy (\(E\)) of a quantum of radiation (now called a photon) and its frequency (\(f\)). Planck’s hypothesis that energy could be absorbed or emitted only in discrete amounts (quanta of energy \(E=hf\)) led to the development of quantum mechanics in the early part of the 20th century. In his work on matter waves in the 1920’s, Louis de Broglie hypothesized that the momentum (\(p\)) carried by a matter wave is proportional to the wavenumber (\(h=\frac{2\pi}{\lambda}\)) of the wave, with the modified (also called “reduced”) Planck constant (\(\hbar=\frac{h}{2\pi}\)) as the constant of proportionality (so \(p=\hbar {k}\)).
The Planck constant has dimensions of energy x time (which is mass x length\(^{2}/\)time and SI units of Joules-sec (which is kg m\(^2/\)sec). The numerical value of Planck’s constant in SI units is \(h=6.626\times10^{-34}{J}{s}\) and the modified Planck’s constant in SI units is \(\hbar=1.055\times 10^{-34}{J}{s}\).Read more about it:
https://www.britannica.com/science/Plancks-constant
http://scienceworld.wolfram.com/physics/PlancksConstant.html
https://physicsworld.com/a/nist-4-watt-balance-weighs-in-on-plancks-constant/
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Planck-Einstein relation
The Planck-Einstein relation is a proportionality relation between energy and frequency, with proportionality constant \(\hbar\) (known as the modified Planck’s constant). In attempting to explain the behavior of radiation from a blackbody, Planck concluded that the radiation was produced by a collection of oscillators that could absorb or emit energy only in discrete quanta, and the energy in each quantum of radiation was proportional to the frequency of the radiation. Although not derived from first principles, this relation is consistent with Albert Einstein’s conclusion that the photoelectric effect could be explained using quanta of light energy (called “photons”), with the energy of a photon proportional to its frequency. The combination of discrete amounts of energy (a feature usually attributed to particles) and frequency (a feature usually attributed to waves) is the essence of wave-particle duality in quantum mechanics.
Read more about it:
https://en.wikipedia.org/wiki/Planck%E2%80%93Einstein_relation
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plane wave
A plane wave is a propagating periodic wave in which the surfaces of constant phase are flat planes. A wave is a disturbance from an equilibrium condition, and a propagating periodic wave is a wave in which the amount and direction of the disturbance oscillates over space and time. The phase of the wave determines which portion of the cycle the wave has at a given location and time, and a surface of constant phase is the locus of points in space at which the wave’s phase has the same value at a given time. The wavefunction for a plane wave propagating along the x-axis can be written as \(Ae^{i\phi}\), in which \(A\) represents the wave’s amplitude and \(\phi\) represents the wave’s phase. The change in phase over space and time is given by \(kx- \omega t\), in which the \(kx\) term gives the phase change over space (\(k\) represents the wave’s wavenumber \(2\pi/\lambda\)) and \(\omega t\) term gives the phase change over time (\(\omega\) represents the wave’s angular frequency \(2\pi/T\)).
Although a single-frequency plane wave extends over all space with uniform amplitude and are therefore not normalizable, plane waves with different amplitudes and wavelengths may be combined to produce spatially limited waves that may be normalized.Read more about it:
https://en.wikipedia.org/wiki/Plane_wave
http://farside.ph.utexas.edu/teaching/qmech/Quantum/node16.html
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Pythagorean theorem
The Pythagorean theorem is a mathematical formula relating the square of the length of the hypotenuse and the sum of the squares of the lengths of the sides of a right triangle. In a two-dimensional vector space, this relationship can be used to find the length of a vector in terms of its Cartesian coordinates. This is relevant to quantum mechanics because an extended version of the Pythagorean theorem can be used to find the “length” of a vector in a higher-dimension abstract vector space. In that extended version, the square of the “length” of a vector is related to the sum of the squares of the vector’s N coefficients in an N-dimensional orthonormal basis system.
Read more about it:
https://en.wikipedia.org/wiki/Pythagorean_theorem
https://www.cut-the-knot.org/Generalization/pythagoras.shtml
http://www.math.brown.edu/~banchoff/Beyond3d/chapter8/section02.html
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quantum state
A quantum state is a mathematical description of the condition of a particle or system subject to the rules of quantum mechanics; the quantum state contains all of the information that can be known about a quantum system. Although the terminology is not standardized and “quantum state” is sometimes used interchangeably with “quantum wavefunction”, in many texts the quantum state is represented by a Dirac ket, such as \(|\psi>\), and a quantum wavefunction is defined as the expansion of that quantum state in a specific basis system, such as \(\psi(x)\).
Read more about it:
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radiation pressure
In the early 1870’s, well before the discovery of the quantum nature of electromagnetic radiation, James Clerk Maxwell predicted that because electromagnetic waves carry momentum, light striking a surface should produce a force on that surface. That force (per unit area) is called radiation pressure, which was experimentally observed approximately 30 years after Maxwell’s prediction.
Read more about it:
http://scienceworld.wolfram.com/physics/RadiationPressure.html
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Schrödinger equation
The Schrödinger equation is the fundamental non-relativistic equation of quantum mechanics which describes the behavior of a quantum particle or system over space and time. Specifically, the time-dependent Schrödinger equation relates the time rate of change of the system wavefunction (first-order temporal derivative) to the spatial curvature of the wavefunction (second-order spatial derivative) and the potential energy in the region of interest. The Schrödinger equation may be written as an eigenvalue equation; solving this equation is equivalent to finding the eigenfunctions of the Hamiltonian (total-energy) operator.
Read more about it:
https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/schr.html
http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/OneDimSchr.htm
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sinc function
The sinc function is a mathematical function of the form \(\sin(ax)/(ax)\) in which \(a\) is a constant. Sinc functions appear in quantum mechanics because they are the Fourier transform of rectangular functions, so if the wavenumber or momentum spectrum is rectangular (uniform amplitude over some range and zero elsewhere), the position waveform will be a sinc function.
Read more about it:
https://en.wikipedia.org/wiki/Sinc_function
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standard deviation
The standard deviation of a distribution is a measure of the average difference between the value of each member of the distribution and the average value of all members of the distribution. In quantum mechanics, the uncertainties that appear in the Heisenberg Uncertainty principle represent the standard deviation of a set of measurements of a quantum observable over an ensemble of identically prepared systems.
Read more about it:
http://mathworld.wolfram.com/StandardDeviation.html
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transcendental equation
A transcendental equation is an equation involving transcendental functions such as logarithmic, exponential, or trigonometric functions. Unlike equations involving algebraic functions, transcendental equations cannot be solved algebraically; instead, numerical or graphical methods must be employed. In quantum mechanics, the boundary conditions for a finite rectangular potential well lead to an equation involving the tangent function (for even solutions) and the cotangent function (for odd solutions), and these transcendental equations are solved graphically in most quantum texts.
Read more about it:
https://en.wikipedia.org/wiki/Transcendental_function
https://web.ma.utexas.edu/users/m408n/m408c/CurrentWeb/LM4-8-6.php
http://nitkkr.ac.in/docs/15-%20Solutions%20of%20Algebric%20and%20Transcendental%20Equations.pdf
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vector
A vector is any member of a vector space, which means that a vector obeys the rules (such as addition and multiplication) of that space. Vectors may be real vectors in three-dimensional physical space or abstract vectors in a higher-dimensional abstract space, and vectors may be expanded as a series of coefficients (components) and basis vectors (directional indicators). Most physics students learn that vectors are “objects with magnitude and direction,” but more general definitions are based on the way vectors transform between coordinate systems, in which case vectors are defined as tensors of rank 1.
Read more about it:
http://www.math.com/tables/oddsends/vectordefs.htm
https://en.wikipedia.org/wiki/Vector_(mathematics_and_physics)
https://www.doitpoms.ac.uk/tlplib/tensors/what_is_tensor.php
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vector space
A vector space is a collection of objects that obey mathematical rules for operations such as addition and multiplication. Vectors representing physical quantities such as force, velocity, and electric field are members of a Euclidean vector space with three dimensions, but concepts such as distance and direction in Euclidean space may be extended to abstract vector spaces with any number of dimensions. This means that in addition to the vectors mentioned above, objects such as matrices and functions may also be members of a vector space.
Read more about it:
https://en.wikipedia.org/wiki/Vector_space
http://mathworld.wolfram.com/VectorSpace.html
http://www.math.toronto.edu/gscott/WhatVS.pdf
http://www.physics.miami.edu/~nearing/mathmethods/vector_spaces.pdf
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Wave mechanics
Wave mechanics is an approach to quantum mechanics developed primarily by Erwin Schrödinger, who felt that the behavior of the “matter waves” postulated by Louis de Broglie should be governed by some type of wave equation. Modifying the classical real second-order differential wave equation, Schrödinger developed a complex differential equation with first-order temporal and second-order spatial derivatives; he called the solutions to this equation “psi functions.” Today the equation is called the Schrödinger Equation, its solutions are called wavefunctions, and Schrödinger’s approach is called wave mechanics. An alternative approach called matrix mechanics was developed by Werner Heisenberg, although Schrödinger, Dirac, and Von Neumann later showed that the predictions of matrix mechanics are identical to those of wave mechanics.
Read more about it:
http://physics.mq.edu.au/~jcresser/Phys201/WaveMechanicsLectureSlides.pdf
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wave-particle duality
Wave-particle duality is the notion that objects at quantum scales exhibit complementary wave and particle behavior. For example, light waves behave like particles when interacting with a conducting surface in the photoelectric effect, and particles such as electrons display wavelike interference and diffraction effects when passing through small slits.
Read more about it:
https://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality
http://vergil.chemistry.gatech.edu/notes/quantrev/node6.html
http://farside.ph.utexas.edu/teaching/qmech/Quantum/node13.html
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wavefunction collapse
One of the tenets of the Copenhagen Interpretation of quantum mechanics is that the wavefunction of a quantum particle or system that is the superposition of eigenfunctions will “collapse” to one of those eigenfunctions when an observation of the particle or system is made. The result of a measurement of a quantum observable will be the eigenvalue of the eigenfunction to which the wavefunction collapses, and the probability of each measurement outcome is determined by the amount of each eigenfunction in the wavefunction.
Read more about it:
http://farside.ph.utexas.edu/teaching/qmech/Quantum/node29.html
https://plus.maths.org/content/schrodingers-equation-what-does-it-mean
https://www.southampton.ac.uk/~doug/quantum_physics/collapse.pdf
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wave packet
A wave packet is a disturbance from an equilibrium condition that is localized in time and/or space. Whereas an individual sinusoidal wavefunction extends over all space, a wave packet is made up of a group of waves with different wavelengths that combine limit the extent of the packet by combining constructively at some locations and destructively at others.
In the case of propagating waves, the speed of propagation of individual component waves (the phase speed) is given by the ratio of the angular frequency (\(\omega\)) to the wavenumber \((k)\), while the speed of propagation of the packet (the group velocity) is given by \(\frac{\partial \omega}{\partial k}\).Read more about it:
http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/wpack.html
http://farside.ph.utexas.edu/teaching/qmech/Quantum/node25.html
https://quantummechanics.ucsd.edu/ph130a/130_notes/node80.html
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Weber equations
A Weber equation is a differential equation of the form \(af”(x)+bf(x)+cx^2f (x)=0\), and solutions to this equation have the form of the product of a Gaussian function and Hermite polynomials. This is relevant to quantum mechanics because the Schrödinger Equation for the quantum harmonic oscillator is a type of Weber equation.
Read more about it:
https://www.encyclopediaofmath.org/index.php/Weber_equation#
http://mathworld.wolfram.com/WeberDifferentialEquations.html
https://onlinelibrary.wiley.com/doi/pdf/10.1002/9783527634927.app8