# Wave packet

Last updated

In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. [1] Each component wave function, and hence the wave packet, are solutions of a wave equation. Depending on the wave equation, the wave packet's profile may remain constant (no dispersion, see figure) or it may change (dispersion) while propagating.

In the physical sciences, the wavenumber is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.

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 wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

## Contents

Quantum mechanics ascribes a special significance to the wave packet; it is interpreted as a probability amplitude, its norm squared describing the probability density that a particle or particles in a particular state will be measured to have a given position or momentum. The wave equation is in this case the Schrödinger equation. It is possible to deduce the time evolution of a quantum mechanical system, similar to the process of the Hamiltonian formalism in classical mechanics. The dispersive character of solutions of the Schrödinger equation has played an important role in rejecting Schrödinger's original interpretation, and accepting the Born rule.

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 quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability or probability density.

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 postulated 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.

In the coordinate representation of the wave (such as the Cartesian coordinate system), the position of the physical object's localized probability is specified by the position of the packet solution. Moreover, the narrower the spatial wave packet, and therefore the better localized the position of the wave packet, the larger the spread in the momentum of the wave. This trade-off between spread in position and spread in momentum is a characteristic feature of the Heisenberg uncertainty principle, and will be illustrated below.

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction in three-dimensional space. If m is an object's mass and v is the velocity, then the momentum is

Werner Karl Heisenberg was a German theoretical physicist and one of the key pioneers of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series of papers with Max Born and Pascual Jordan, during the same year, this matrix formulation of quantum mechanics was substantially elaborated. He is known for the Heisenberg uncertainty principle, which he published in 1927. Heisenberg was awarded the 1932 Nobel Prize in Physics "for the creation of quantum mechanics".

## Historical background

In the early 1900s, it became apparent that classical mechanics had some major failings. Isaac Newton originally proposed the idea that light came in discrete packets, which he called corpuscles, but the wave-like behavior of many light phenomena quickly led scientists to favor a wave description of electromagnetism. It wasn't until the 1930s that the particle nature of light really began to be widely accepted in physics. The development of quantum mechanics and its success at explaining confusing experimental results was at the root of this acceptance. Thus, one of the basic concepts in the formulation of quantum mechanics is that of light coming in discrete bundles called photons. The energy of a photon is a function of its frequency,

Sir Isaac Newton was an English mathematician, physicist, astronomer, theologian, and author who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force is carried by electromagnetic fields composed of electric fields and magnetic fields, is responsible for electromagnetic radiation such as light, and is one of the four fundamental interactions in nature. The other three fundamental interactions are the strong interaction, the weak interaction, and gravitation. At high energy the weak force and electromagnetic force are unified as a single electroweak force.

${\displaystyle E=h\nu .}$ [2]

The photon's energy is equal to Planck's constant, h, multiplied by its frequency, ν. This resolved a problem in classical physics, called the ultraviolet catastrophe.

The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation in all frequency ranges, emitting more energy as the frequency increases. By calculating the total amount of radiated energy, it can be shown that a blackbody is likely to release an arbitrarily high amount of energy. This would cause all matter to instantaneously radiate all of its energy until it is near absolute zero - indicating that a new model for the behaviour of blackbodies was needed.

The ideas of quantum mechanics continued to be developed throughout the 20th century. The picture that was developed was of a particulate world, with all phenomena and matter made of and interacting with discrete particles; however, these particles were described by a probability wave. The interactions, locations, and all of physics would be reduced to the calculations of these probability amplitudes.

The particle-like nature of the world has been confirmed by experiment over a century, while the wave-like phenomena could be characterized as consequences of the wave packet aspect of quantum particles (see wave-particle duality.) According to the principle of complementarity, the wave-like and particle-like characteristics never manifest themselves at the same time, i.e. in the same experiment; see, however, the Afshar experiment and the lively discussion around it.

The Afshar experiment is a variation of the double slit experiment in quantum mechanics, devised and carried out by Shahriar Afshar while at the private, Boston-based Institute for Radiation-Induced Mass Studies (IRIMS). The results were presented at a Harvard seminar in March 2004. The experiment gives information about which of two paths a photon takes through the apparatus while simultaneously allowing interference between the two paths to be observed, by showing that a grid of wires, placed at the nodes of the interference pattern, does not alter the beams. Afshar claimed that the experiment violates the principle of complementarity of quantum mechanics, which states roughly that the particle and wave aspects of quantum objects cannot be observed at the same time, and specifically the Englert–Greenberger duality relation. The experiment has been repeated by a number of investigators and its results have been confirmed, but its interpretation is controversial, and some disagree that it violates complementarity, while also disagreeing amongst themselves as to why.

## Basic behaviors

### Non-dispersive

As an example of propagation without dispersion, consider wave solutions to the following wave equation from classical physics

${\displaystyle {\partial ^{2}u \over \partial t^{2}}=c^{2}{\nabla ^{2}u},}$

where c is the speed of the wave's propagation in a given medium.

Using the physics time convention, exp(−iωt), the wave equation has plane-wave solutions

${\displaystyle u(\mathbf {x} ,t)=e^{i{(\mathbf {k\cdot x} }-\omega t)},}$

where

${\displaystyle \omega ^{2}=|\mathbf {k} |^{2}c^{2}}$, and ${\displaystyle |\mathbf {k} |^{2}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2}.}$

This relation between ω and k should be valid so that the plane wave is a solution to the wave equation. It is called a dispersion relation.

To simplify, consider only waves propagating in one dimension (extension to three dimensions is straightforward). Then the general solution is

${\displaystyle u(x,t)=Ae^{i(kx-\omega t)}+Be^{-i(kx+\omega t)},}$

in which we may take ω = kc. The first term represents a wave propagating in the positive x-direction since it is a function of x − ct only; the second term, being a function of x + ct, represents a wave propagating in the negative x-direction.

A wave packet is a localized disturbance that results from the sum of many different wave forms. If the packet is strongly localized, more frequencies are needed to allow the constructive superposition in the region of localization and destructive superposition outside the region. From the basic solutions in one dimension, a general form of a wave packet can be expressed as

${\displaystyle u(x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }A(k)~e^{i(kx-\omega (k)t)}dk.}$

As in the plane-wave case the wave packet travels to the right for ω(k) = kc, since u(x, t)= F(x − ct), and to the left for ω(k) = −kc, since u(x,t) = F(x + ct).

The factor 12 comes from Fourier transform conventions. The amplitude A(k) contains the coefficients of the linear superposition of the plane-wave solutions. These coefficients can in turn be expressed as a function of u(x, t) evaluated at t = 0 by inverting the Fourier transform relation above:

${\displaystyle A(k)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }u(x,0)~e^{-ikx}dx.}$

For instance, choosing

${\displaystyle u(x,0)=e^{-x^{2}+ik_{0}x},}$

we obtain

${\displaystyle A(k)={\frac {1}{\sqrt {2}}}e^{-{\frac {(k-k_{0})^{2}}{4}}},}$

and finally

{\displaystyle {\begin{aligned}u(x,t)&=e^{-(x-ct)^{2}+ik_{0}(x-ct)}\\&=e^{-(x-ct)^{2}}\left[\cos \left(2\pi {\frac {x-ct}{\lambda }}\right)+i\sin \left(2\pi {\frac {x-ct}{\lambda }}\right)\right].\end{aligned}}}

The nondispersive propagation of the real or imaginary part of this wave packet is presented in the above animation.

### Dispersive

By contrast, as an example of propagation now with dispersion, consider instead solutions to the Schrödinger equation (Pauli 2000, with m and ħ set equal to one),

${\displaystyle i{\partial \psi \over \partial t}=-{\frac {1}{2}}{\nabla ^{2}\psi },}$

yielding the dispersion relation

${\displaystyle \omega ={\frac {1}{2}}|\mathbf {k} |^{2}.}$

Once again, restricting attention to one dimension, the solution to the Schrödinger equation satisfying the initial condition ${\displaystyle ~\psi (x,0)={\sqrt[{4}]{2/\pi }}\exp({-x^{2}+ik_{0}x})}$, representing a wave packet localized in space at the origin, is seen to be

{\displaystyle {\begin{aligned}\psi (x,t)&={\frac {\sqrt[{4}]{2/\pi }}{\sqrt {1+2it}}}e^{-{\frac {1}{4}}k_{0}^{2}}~e^{-{\frac {1}{1+2it}}\left(x-{\frac {ik_{0}}{2}}\right)^{2}}\\&={\frac {\sqrt[{4}]{2/\pi }}{\sqrt {1+2it}}}e^{-{\frac {1}{1+4t^{2}}}(x-k_{0}t)^{2}}~e^{i{\frac {1}{1+4t^{2}}}\left((k_{0}+2tx)x-{\frac {1}{2}}tk_{0}^{2}\right)}~.\end{aligned}}}

An impression of the dispersive behavior of this wave packet is obtained by looking at the probability density,

${\displaystyle |\psi (x,t)|^{2}={\frac {\sqrt {2/\pi }}{\sqrt {1+4t^{2}}}}~e^{-{\frac {2(x-k_{0}t)^{2}}{1+4t^{2}}}}~.}$

It is evident that this dispersive wave packet, while moving with constant group velocity ko, is delocalizing rapidly: it has a width increasing with time as 1 + 4t² → 2t, so eventually it diffuses to an unlimited region of space. [nb 1]

The momentum profile A(k) remains invariant. The probability current is

${\displaystyle j=\rho v={\frac {1}{2i}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})=\rho \left(k_{0}+{\frac {4t(x-k_{0}t)}{1+4t^{2}}}\right).}$

## Gaussian wave packets in quantum mechanics

The above dispersive Gaussian wave packet, unnormalized and just centered at the origin, instead, at t=0, can now be written in 3D, now in standard units: [3] [4]

${\displaystyle \psi (\mathbf {r} ,0)=e^{-\mathbf {r} \cdot \mathbf {r} /2a},}$

where a is a positive real number, the square of the width of the wave packet,

${\displaystyle a=2\langle \mathbf {r} \cdot \mathbf {r} \rangle /3\langle 1\rangle =2(\Delta x)^{2}.}$

The Fourier transform is also a Gaussian in terms of the wavenumber, t=0, the k-vector, (with inverse width,

${\displaystyle 1/a=2\langle \mathbf {k} \cdot \mathbf {k} \rangle /3\langle 1\rangle =2(\Delta p_{x}/\hbar )^{2},}$

so that

${\displaystyle \Delta x\Delta p_{x}=\hbar /2,}$

i.e., it saturates the uncertainty relation),

${\displaystyle \psi (\mathbf {k} ,0)=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2}.}$

Each separate wave only phase-rotates in time, so that the time dependent Fourier-transformed solution is

{\displaystyle {\begin{aligned}\Psi (\mathbf {k} ,t)&=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2}e^{-iEt/\hbar }\\&=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2-i(\hbar ^{2}\mathbf {k} \cdot \mathbf {k} /2m)t/\hbar }\\&=(2\pi a)^{3/2}e^{-(a+i\hbar t/m)\mathbf {k} \cdot \mathbf {k} /2}.\end{aligned}}}

The inverse Fourier transform is still a Gaussian, but now the parameter a has become complex, and there is an overall normalization factor. [5]

${\displaystyle \Psi (\mathbf {r} ,t)=\left({a \over a+i\hbar t/m}\right)^{3/2}e^{-{\mathbf {r} \cdot \mathbf {r} \over 2(a+i\hbar t/m)}}.}$

The integral of Ψ over all space is invariant, because it is the inner product of Ψ with the state of zero energy, which is a wave with infinite wavelength, a constant function of space. For any energy eigenstate η(x), the inner product,

${\displaystyle \langle \eta |\psi \rangle =\int \eta (\mathbf {r} )\psi (\mathbf {r} )d^{3}\mathbf {r} ,}$

only changes in time in a simple way: its phase rotates with a frequency determined by the energy of η. When η has zero energy, like the infinite wavelength wave, it doesn't change at all.

The integral ∫|Ψ|2d3r is also invariant, which is a statement of the conservation of probability. Explicitly,

${\displaystyle P(r)=|\Psi |^{2}=\Psi ^{*}\Psi =\left({a \over {\sqrt {a^{2}+(\hbar t/m)^{2}}}}\right)^{3}~e^{-{a\,\mathbf {r} \cdot \mathbf {r} \over a^{2}+(\hbar t/m)^{2}}},}$

in which √a is the width of P(r) at t = 0; r is the distance from the origin; the speed of the particle is zero; and the time origin t = 0 can be chosen arbitrarily.

The width of the Gaussian is the interesting quantity which can be read off from the probability density, |Ψ|2,

${\displaystyle {\sqrt {a^{2}+(\hbar t/m)^{2} \over a}}.}$

This width eventually grows linearly in time, as ħt/(m√a), indicating wave-packet spreading.

For example, if an electron wave packet is initially localized in a region of atomic dimensions (i.e., 10−10 m) then the width of the packet doubles in about 10−16 s. Clearly, particle wave packets spread out very rapidly indeed (in free space): [6] For instance, after 1 ms, the width will have grown to about a kilometer.

This linear growth is a reflection of the (time-invariant) momentum uncertainty: the wave packet is confined to a narrow Δx=a/2, and so has a momentum which is uncertain (according to the uncertainty principle) by the amount ħ/2a, a spread in velocity of ħ/m2a, and thus in the future position by ħt /m2a. The uncertainty relation is then a strict inequality, very far from saturation, indeed! The initial uncertainty ΔxΔp = ħ/2 has now increased by a factor of ħt/ma (for large t).

## The Airy wave train

In contrast to the above Gaussian wave packet, it has been observed [7] that a particular wave function based on Airy functions, propagates freely without envelope dispersion, maintaining its shape. It accelerates undistorted in the absence of a force field: ψ=Ai(B(xB³t ²)) exp(iB³t(x−2B³t²/3)). (For simplicity, ħ=1, m=1/2, and B is a constant, cf. nondimensionalization.)

Nevertheless, there is no dissonance with Ehrenfest's theorem in this force-free situation, because the state is both non-normalizable and has an undefined (infinite) x for all times. (To the extent that it could be defined, p⟩ = 0 for all times, despite the apparent acceleration of the front.)

In phase space, this is evident in the pure state Wigner quasiprobability distribution of this wavetrain, whose shape in x and p is invariant as time progresses, but whose features accelerate to the right, in accelerating parabolas B(xB³t ²) + (p/BtB²)² = 0, [8]

{\displaystyle {\begin{aligned}W(x,p;t)&=W(x-B^{3}t^{2},p-B^{3}t;0)\\&={1 \over 2^{1/3}\pi B}~\mathrm {Ai} \left(2^{2/3}\left(Bx+{p^{2} \over B^{2}}-2Bpt\right)\right).\end{aligned}}}

Note the momentum distribution obtained by integrating over all x is constant. Since this is the probability density in momentum space, it is evident that the wave function itself is not normalizable.

## Free propagator

The narrow-width limit of the Gaussian wave packet solution discussed is the free propagator kernel K. For other differential equations, this is usually called the Green's function, [9] but in quantum mechanics it is traditional to reserve the name Green's function for the time Fourier transform of K.

Returning to one dimension for simplicity, with m and ħ set equal to one, when a is the infinitesimal quantity ε, the Gaussian initial condition, rescaled so that its integral is one,

${\displaystyle \psi _{0}(x)={1 \over {\sqrt {2\pi \epsilon }}}e^{-{x^{2} \over 2\epsilon }}\,}$

becomes a delta function, δ(x), so that its time evolution,

${\displaystyle K_{t}(x)={1 \over {\sqrt {2\pi (it+\epsilon )}}}e^{-x^{2} \over 2it+\epsilon }\,}$

yields the propagator.

Note that a very narrow initial wave packet instantly becomes infinitely wide, but with a phase which is more rapidly oscillatory at large values of x. This might seem strange—the solution goes from being localized at one point to being "everywhere" at all later times, but it is a reflection of the enormous momentum uncertainty of a localized particle, as explained above.

Further note that the norm of the wave function is infinite, which is also correct, since the square of a delta function is divergent in the same way.

The factor involving ε is an infinitesimal quantity which is there to make sure that integrals over K are well defined. In the limit that ε→0, K becomes purely oscillatory, and integrals of K are not absolutely convergent. In the remainder of this section, it will be set to zero, but in order for all the integrations over intermediate states to be well defined, the limit ε→0 is to be only taken after the final state is calculated.

The propagator is the amplitude for reaching point x at time t, when starting at the origin, x=0. By translation invariance, the amplitude for reaching a point x when starting at point y is the same function, only now translated,

${\displaystyle K_{t}(x,y)=K_{t}(x-y)={1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2} \over 2t}\,.}$

In the limit when t is small, the propagator, of course, goes to a delta function,

${\displaystyle \lim _{t\rightarrow 0}K_{t}(x-y)=\delta (x-y)~,}$

but only in the sense of distributions: The integral of this quantity multiplied by an arbitrary differentiable test function gives the value of the test function at zero.

To see this, note that the integral over all space of K equals 1 at all times,

${\displaystyle \int K_{t}(x)dx=1\,,}$

since this integral is the inner-product of K with the uniform wave function. But the phase factor in the exponent has a nonzero spatial derivative everywhere except at the origin, and so when the time is small there are fast phase cancellations at all but one point. This is rigorously true when the limit ε→0 is taken at the very end.

So the propagation kernel is the (future) time evolution of a delta function, and it is continuous, in a sense: it goes to the initial delta function at small times. If the initial wave function is an infinitely narrow spike at position y,

${\displaystyle \psi _{0}(x)=\delta (x-y)\,,}$

it becomes the oscillatory wave,

${\displaystyle \psi _{t}(x)={1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2}/2t}\,.}$

Now, since every function can be written as a weighted sum of such narrow spikes,

${\displaystyle \psi _{0}(x)=\int \psi _{0}(y)\delta (x-y)dy\,,}$

the time evolution of every functionψ0 is determined by this propagation kernel K,

${\displaystyle \psi _{t}(x)=\int \psi _{0}(y){1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2}/2t}dy\,.}$

Thus, this is a formal way to express the fundamental solution or general solution. The interpretation of this expression is that the amplitude for a particle to be found at point x at time t is the amplitude that it started at y, times the amplitude that it went from y to x, summed over all the possible starting points. In other words, it is a convolution of the kernel K with the arbitrary initial condition ψ0,

${\displaystyle \psi _{t}=K*\psi _{0}\,.}$

Since the amplitude to travel from x to y after a time t+t' can be considered in two steps, the propagator obeys the composition identity,

${\displaystyle \int K(x-y;t)K(y-z;t')dy=K(x-z;t+t')~,}$

which can be interpreted as follows: the amplitude to travel from x to z in time t+t' is the sum of the amplitude to travel from x to y in time t, multiplied by the amplitude to travel from y to z in time t', summed over all possible intermediate states y. This is a property of an arbitrary quantum system, and by subdividing the time into many segments, it allows the time evolution to be expressed as a path integral. [10]

## Analytic continuation to diffusion

The spreading of wave packets in quantum mechanics is directly related to the spreading of probability densities in diffusion. For a particle which is randomly walking, the probability density function at any point satisfies the diffusion equation (also see the heat equation),

${\displaystyle {\partial \over \partial t}\rho ={1 \over 2}{\partial ^{2} \over \partial x^{2}}\rho ~,}$

where the factor of 2, which can be removed by rescaling either time or space, is only for convenience.

A solution of this equation is the spreading Gaussian,

${\displaystyle \rho _{t}(x)={1 \over {\sqrt {2\pi t}}}e^{-x^{2} \over 2t}~,}$

and, since the integral of ρt is constant while the width is becoming narrow at small times, this function approaches a delta function at t=0,

${\displaystyle \lim _{t\rightarrow 0}\rho _{t}(x)=\delta (x)\,}$

again only in the sense of distributions, so that

${\displaystyle \lim _{t\rightarrow 0}\int _{x}f(x)\rho _{t}(x)=f(0)\,}$

for any smooth test function f.

The spreading Gaussian is the propagation kernel for the diffusion equation and it obeys the convolution identity,

${\displaystyle K_{t+t'}=K_{t}*K_{t'}\,,}$

which allows diffusion to be expressed as a path integral. The propagator is the exponential of an operator H,

${\displaystyle K_{t}(x)=e^{-tH}\,,}$

which is the infinitesimal diffusion operator,

${\displaystyle H=-{\nabla ^{2} \over 2}\,.}$

A matrix has two indices, which in continuous space makes it a function of x and x'. In this case, because of translation invariance, the matrix element K only depend on the difference of the position, and a convenient abuse of notation is to refer to the operator, the matrix elements, and the function of the difference by the same name:

${\displaystyle K_{t}(x,x')=K_{t}(x-x')\,.}$

Translation invariance means that continuous matrix multiplication,

${\displaystyle C(x,x'')=\int _{x'}A(x,x')B(x',x'')\,,}$

is essentially convolution,

${\displaystyle C(\Delta )=C(x-x'')=\int _{x'}A(x-x')B(x'-x'')=\int _{y}A(\Delta -y)B(y)\,.}$

The exponential can be defined over a range of ts which include complex values, so long as integrals over the propagation kernel stay convergent,

${\displaystyle K_{z}(x)=e^{-zH}\,.}$

As long as the real part of z is positive, for large values of x, K is exponentially decreasing, and integrals over K are indeed absolutely convergent.

The limit of this expression for z approaching the pure imaginary axis is the above Schrödinger propagator encountered,

${\displaystyle K_{t}^{\rm {Schr}}=K_{it+\epsilon }=e^{-(it+\epsilon )H}\,,}$

which illustrates the above time evolution of Gaussians.

From the fundamental identity of exponentiation, or path integration,

${\displaystyle K_{z}*K_{z'}=K_{z+z'}\,}$

holds for all complex z values, where the integrals are absolutely convergent so that the operators are well defined.

Thus, quantum evolution of a Gaussian, which is the complex diffusion kernel K,

${\displaystyle \psi _{0}(x)=K_{a}(x)=K_{a}*\delta (x)\,}$

amounts to the time-evolved state,

${\displaystyle \psi _{t}=K_{it}*K_{a}=K_{a+it}\,.}$

This illustrates the above diffusive form of the complex Gaussian solutions,

${\displaystyle \psi _{t}(x)={1 \over {\sqrt {2\pi (a+it)}}}e^{-{x^{2} \over 2(a+it)}}\,.}$

## Remarks

1. By contrast, the introduction of interaction terms in dispersive equations, such as for the quantum harmonic oscillator, may result in the emergence of envelope-non-dispersive, classical-looking solutions—see coherent states: Such "minimum uncertainty states" do saturate the uncertainty principle permanently.

## Notes

1. From a general pedagogy web-site by Curtright.

## Related Research Articles

In quantum mechanics, a Hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system. It is usually denoted by , but also or to highlight its function as an operator. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

In quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example, a particle trapped inside a large box can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow, quantum effects become important. The particle may only occupy certain positive energy levels. Likewise, it can never have zero energy, meaning that the particle can never "sit still". Additionally, it is more likely to be found at certain positions than at others, depending on its energy level. The particle may never be detected at certain positions, known as spatial nodes.

In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

In physics and mathematics, the heat equation is a partial differential equation that describes how the distribution of some quantity evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower. It is a special case of the diffusion equation.

In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. In quantum mechanics, it means a region of uniform potential, usually set to zero in the region of interest since potential can be arbitrarily set to zero at any point in space.

The Schrödinger–Newton equation, sometimes referred to as the Newton–Schrödinger or Schrödinger–Poisson equation, is a nonlinear modification of the Schrödinger equation with a Newtonian gravitational potential, where the gravitational potential emerges from the treatment of the wave function as a mass density, including a term that represents interaction of a particle with its own gravitational field. The inclusion of a self-interaction term represents a fundamental alteration of quantum mechanics. It can be written either as a single integro-differential equation or as a coupled system of a Schrödinger and a Poisson equation. In the latter case it is also referred to in the plural form.

Surface states are electronic states found at the surface of materials. They are formed due to the sharp transition from solid material that ends with a surface and are found only at the atom layers closest to the surface. The termination of a material with a surface leads to a change of the electronic band structure from the bulk material to the vacuum. In the weakened potential at the surface, new electronic states can be formed, so called surface states.

In quantum mechanics, the probability current is a mathematical quantity describing the flow of probability in terms of probability per unit time per unit area. Specifically, if one describes the probability density as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. This is analogous to mass currents in hydrodynamics and electric currents in electromagnetism. It is a real vector, like electric current density. The concept of a probability current is a useful formalism in quantum mechanics.

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

The Gross–Pitaevskii equation describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model.

In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The particle follows the path of a semicircle from to where it cannot escape, because the potential from to is infinite. Instead there is total reflection, meaning the particle bounces back and forth between to . The Schrödinger equation for a free particle which is restricted to a semicircle is

The Gamow Factor or Gamow-Sommerfeld Factor, named after its discoverer George Gamow, is a probability factor for two nuclear particles' chance of overcoming the Coulomb barrier in order to undergo nuclear reactions, for example in nuclear fusion. By classical physics, there is almost no possibility for protons to fuse by crossing each other's Coulomb barrier, but when George Gamow instead applied quantum mechanics to the problem, he found that there was a significant chance for the fusion due to tunneling.

The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin (1999) as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. The term fractional Schrödinger equation was coined by Nick Laskin.

In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator.

In fluid dynamics, Beltrami flows are flows in which the vorticity vector and the velocity vector are parallel to each other. In other words, Beltrami flow is a flow where Lamb vector is zero. It is named after the Italian mathematician Eugenio Beltrami due to his derivation of the Beltrami vector field, while initial developments in fluid dynamics were done by the Russian scientist Ippolit S. Gromeka in 1881.

Different subfields of physics have different programs for determining the state of a physical system.