Wave packet

Ideas related to wave packets – modulation, carrier waves, phase velocity, and group velocity – date from the mid 1800s. The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877.[3]

Erwin Schrödinger introduced the idea of wave packets just after publishing his famous wave equation.[4] He solved his wave equation for a quantum harmonic oscillator, introduced the superposition principle, and used it to show that a compact state could persist. While this work did result in the important concept of coherent states, the wave packet concept did not endure. The year after Schrödinger's paper, Werner Heisenberg published his paper on the uncertainty principle, showing in the process, that Schrödinger's results only applied to quantum harmonic oscillators, not for example to Coulomb potential characteristic of atoms.[4]^: 829

The following year, 1927, Charles Galton Darwin explored Schrödinger's equation for an unbound electron in free space, assuming an initial Gaussian wave packet.[5] Darwin showed that at time ${\displaystyle t}$ later the position ${\displaystyle x}$ of the packet traveling at velocity ${\displaystyle v}$ would be

$${\displaystyle x_{0}+vt\pm {\sqrt {\sigma ^{2}+(ht/2\pi \sigma m)^{2}}}}$$

where ${\displaystyle \sigma }$ is the uncertainty in the initial position.

Later in 1927 Paul Ehrenfest showed that the time, ${\displaystyle T}$ for a matter wave packet of width ${\displaystyle \Delta x}$ and mass ${\displaystyle m}$ to spread by a factor of 2 was ${\textstyle T\approx \Delta x{\sqrt {m/\hbar }}}$. Since ${\displaystyle \hbar }$ is so small, wave packets on the scale of macroscopic objects, with large width and mass, double only at cosmic time scales.[4]^: 830

Quantum mechanics describes the nature of atomic and subatomic systems using Schrödinger's wave equation. The classical limit of quantum mechanics and many formulations of quantum scattering use wave packets formed from various solutions to this equation. Quantum wave packet profiles change while propagating; they show dispersion. Physicists have concluded that "wave packets would not do as representations of subatomic particles".[4]^: 829

Superposition of 1D plane waves (blue) that sum to form a Gaussian wave packet (red) that propagates to the right while spreading. Blue dots follow each plane wave's phase velocity while the red line follows the central group velocity.

1D Gaussian wave packet, shown in the complex plane, for a=2 and k=4

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:[9][10]

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

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, 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 satisfies 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

The inverse Fourier transform is still a Gaussian, but now the parameter a has become complex, and there is an overall normalization factor.[6]

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,

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.[11]

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):[12] 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/m√2a, and thus in the future position by ħt /m√2a. 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).

A 2D gaussian quantum wave packet. The color (yellow green blue) indicates the phase of the wave function ${\displaystyle \psi }$, its brightness indicates ${\displaystyle |\psi |^{2}/|\psi |_{max}^{2}}$. ${\displaystyle k_{0x}=k_{0}}$, ${\displaystyle k_{0y}=0}$

A gaussian 2D quantum wave function:

${\displaystyle \psi (x,y,t)=\psi (x,t)\psi (y,t)}$

${\displaystyle \psi (x,t)=({\frac {2a^{2}}{\pi }})^{1/4}{\frac {e^{i\phi }}{(a^{4}+{\frac {4\hbar ^{2}t^{2}}{m^{2}}})^{1/4}}}e^{ik_{0}x}exp[-{\frac {(x-{\frac {\hbar k_{0}}{m}}t)^{2}}{a^{2}+{\frac {2i\hbar t}{m}}}}]}$

where

${\displaystyle \phi =-\theta -{\frac {\hbar k_{0}^{2}}{2m}}t}$

${\displaystyle tg(2\theta )={\frac {2\hbar t}{ma^{2}}}}$ [13]

In contrast to the above Gaussian wave packet, it has been observed[14] 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:

$${\displaystyle \psi =\operatorname {Ai} (B(x-B^{3}t^{2}))\,e^{iB^{3}t\left(x-{\tfrac {2}{3}}B^{3}t^{2}\right)}\,.}$$

(For simplicity, ħ = 1, m = 1/2, and B is a constant, cf. nondimensionalization.)

Truncated view of time development for the Airy front in phase space. (Click to animate.)

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(x − B³t²) + (p/B − tB²)² = 0,

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

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.

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,[15] 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,

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

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

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

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 goes to a delta function

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

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 function ψ₀ is determined by this propagation kernel K,

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.[16]

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 ρ_t is constant while the width is becoming narrow at small times, this function approaches a delta function at t=0,

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

again only in the sense of distributions, so that

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

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,

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

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,

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

• Learning materials related to wave packet motion at Wikiversity
• The dictionary definition of wave packet at Wiktionary
• 1d Wave packet plot in Google
• 1d Wave train and probability density plot in Google
• 2d Wave packet plot in Google
• 2d Wave train plot in Google
• 2d probability density plot in Google
• Quantum physics online : Interactive simulation of a free wavepacket
• Web-Schrödinger: Interactive 2D wave packet dynamics simulation
• A simulation of a wave package in 2D (According to FOURIER-Synthesis in 2D)