Topic:Advanced Classical Mechanics/The Eikonal Approximation and Classical Particle Motion

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History[edit]

There exists a wave equation such that in the eikonal approximation, wavepackets accelerate down this potential hill as if they were particles.

In 1834, William Rowan Hamilton published a paper[1] that displayed an understanding of the link between the mathematics particle motion and the propagation of linear waves in the eikonal approximation. Centuries later, Herbert Goldstein would remark that Hamilton would have postulated Schrodinger's equation had there been experiment evidence that particles were waves.[2]

For most modern physicists, the connection is understood through the Ehrenfest theorem of quantum mechanics. A much more elementary derivation of how wavepackets can move as classical particles can be found at the end of this discussion.

Statement of theorem in modern form[edit]

Suppose we have a linear partial differential equation involving time, t , and the spatial coordinates x_i. If the coefficients do not depend on t or x_i, then this partial differential equation is a wave equation if it supports infinite plane wave solutions of the form,

\psi (\vec r, t) = A e^{i \vec k \cdot \vec r - i\omega t}

where the dispersion relation is,


\omega = \omega(\vec k)\rightarrow\omega(\vec k, \vec r, t)\, ,

Where \rightarrow represents a generalization of the concept of dispersion relation to include weak homogeniety in space in time. We assume that the medium varies so gently with respect to  \vec r and  t that localized plane waves can exist in the form of wave packets. These wave packets must be many wavelength in size, yet small enough that the medium remains nearly uniform over the extent of the wavepacket. The medium must also vary sufficiently slowly with respect to time. This regime of gently variation in properties of the medium that supports these wave packets is associated with geometrical optics and the eikonal equation. In this approximation, wavelength, angular frequency, and wavenumber of a wave packet all obey the following approximate equations of motion:[3]

\dot{q_i} = \frac{\partial \omega}{\partial k_i}\,,    \dot{k_i} = -\frac{\partial \omega}{\partial q_i }\,,    \dot{\omega} = -\frac{\partial \omega}{\partial t}.

Applications[edit]

  • Plasma physics. These equations also describe the propagation of wave energy and have application in plasma physics applications where microwaves might be used to heat the core of the plasma.[4]
  • Derivation of Schrödinger's Equation. If a wavepacket were to obey Newton's laws of physics, with the force being the gradient of a scalar potential, \ \vec F = -\nabla V , then the dispersion equation must be of the form,
\hbar\,\cdot\,\omega(\vec k, \vec r,t) =  \frac{\hbar^2k^2}{2m} + V(\vec r,t)\, ,
where \hbar is any constant small enough so that the wavepackets act as particles. The Schrödinger equation follows directly.

Elementary example with proof[edit]

This discussion is restricted to one-dimensional motion, in the special case that the Hamiltonian does not depend on time.[5] Suppose we have a wave that obeys a linear (partial differential) wave equation in one dimension (plus time) so that the the two dynamic variables are x and t. If the coefficients of the partial differential depend on x but not on t, the equation is said to be inhomogeneous in x but homogeneous in t. In this case, solutions exist that vary in time as

\psi(x,t)=\psi(x)\exp{(-i\omega t)}.

Consider for example, wave equations of the form,

\frac{\partial^\,\psi(x,t)}{\partial t^2}=f(x)\frac{\partial}{\partial x}\left (g(x)\frac{\partial\psi(x,t)}{\partial x}\right )

If both coefficients are independent of position, f(x)=f_0 and g(x)=g_0, then we take the trial solution, \psi(x,t)= \exp (ikx-i\omega t). This yields a well-known dispersion relation with two branches:

\omega = \pm\left(f_0g_0\right)^{1/2}k

Now suppose that the f(x) and g(x) depend so weakly on position that they are nearly uniform over a scale-length we shall call \Delta L. If \Delta x is the size of the wavepacket, and \lambda is the wavelength (λ=2π/k), then the regime of interest is:

\Delta L >> \Delta x >> \lambda .

This inequality is the eikonal approximation. In modern language, the first allows us to form wavepackets that are many wavelengths long, which permits a quasi-specific value of wavenumber (k) to exist. The second inequality allows us to approximate the wave equation as having constant coefficients (with respect to x). A wavepacket can be constructed from a very large (infinite) collection of waveforms over a small range of angular frequency, ω. As this wavepacket moves over distances larger than L, the average angular frequency will be constrained to within this range. We express this constraint on ω by taking the differential to be zero:

(1)\qquad\Delta\omega (x,k) = 0= \frac{\partial\omega}{\partial x}\Delta x + \frac{\partial\omega}{\partial k}\Delta k

Using the concept of group velocity, we have

(2)\qquad\frac{\Delta x}{\Delta t} = \frac{\partial\omega}{\partial k}

Divide (1) by \Delta t, substitute the expression from (2) into (1) and divide both sides by \partial\omega / \partial k to obtain:

\frac{\Delta k}{\Delta t} = -\frac{\partial\omega}{\partial x}

This simple proof is possible because in the one-dimensional case, conservation of energy is sufficient to determine the path of a particle through phase space. The situation is more complicated in higher dimensions.

References[edit]

  1. Hamilton, ON A GENERAL METHOD IN DYNAMICS (Philosophical Transactions of the Royal Society, part II for 1834, pp. 247–308.) http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/GenMeth.pdf
  2. H. Goldstein, Classical Mechanics, Addison Wesley )Cambridge, Mass.) Chapter 9.
  3. Weinbert S, 1962 Eikonal method in magnetoyhdrodynamics, Phys. Rev. 126 1899-909
  4. M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, 2nd ed., Wiley Interscience, Hoboken, NJ: Wiley, 2005. Page 506
  5. Vandegrift, G. 2002. "The maze of quantum mechanics". European Journal of Physics 23 (5): 513-522.