Advanced Classical Mechanics/The Eikonal Approximation and Classical Particle Motion

History
In 1834, William Rowan Hamilton published a paper in which he displayed an understanding between the mathematics particle motion and the propagation of linear waves in the  eikonal approximation. Over a century later, Herbert Goldstein would remark that Hamilton would have postulated Schrodinger's equation had there been experimental evidence that particles were waves. For most modern physicists, the connection is understood through the Ehrenfest theorem of quantum mechanics.

Hamilton's discovery was that wavepackets of all linear wave equations obey what we now call Hamiltonian equations of motion, provided the wavelength is sufficiently short compared with the distance over which the medium varies.

It should be noted that not only did Hamilton's original calculation differ significantly than as presented below, the methods used in the simple one-dimensional wave do not allow one to recover Hamilton's theory. With more than one spacial variable, it is necessary to perform a different calculation along the lines of Eikonal_approximation

The terms in a simple linear wave equation
Let us write a linear partial differential equation that is designed to illustrate the eikonal approximation to wavepacket motion:


 * $$\frac{\partial^2\,\psi}{\partial t^2}=a(x)\cdot\frac{\partial^2 \psi}{\partial x^2} +

b(x)\cdot\frac{\partial^2\psi}{\partial x\partial t} + c(x)\cdot\psi$$



This PDE is linear in wave amplitude, $$\psi = \psi(x,t)$$, and the three coefficients (a, b, c) depend on position.

The wave is linear because the terms only vary as the first power of amplitude, $$\psi = \psi(x,t)$$. In virtually all mechanical systems, there is some non-linearity in the wave equation, for example with term like $$d(x)\psi^2$$. But at sufficiently low amplitude, $$d(x)\psi^2<<c(x)\psi$$ and the nonlinearity may be ignored. Not only are nonlinear equations more difficult to solve, their solutions lack certain properties that yield deep insights into the system.

In the image to the right, linearity permits us to simplify the analysis, making the linear approximation of low amplitude waves. To the extent that this approximation holds, we may apply the superposition principle and calculate the wake of a single duck, ignoring the other other ducks' wakes as well as the gentle wave through which they are swimming.

Homogeneous case
In general, the coefficients could depend on both position and time (x and t), but for simplicity we assume only a spatial dependence. These coefficients represent the medium that supports the wave, and the dependence on x indicates that this inhomogeneous in position (meaning that the medium changes as one moves along x). Let us begin with the completely homogeneous case, using the o-subscript to denote constants. This "trial solution",


 * $$\psi(x,t) = A_0\exp\left(ik_0x-i\omega_0t\right)$$

substituted into,


 * $$\frac{\partial^2\,\psi}{\partial t^2}=a_0\cdot\frac{\partial^2 \psi}{\partial x^2} +

b_0\cdot\frac{\partial^2\psi}{\partial x\partial t} + c_0\cdot\psi$$

yields this polynomial for angular frequency, $$\omega_0$$, and wavenumber, $$k_0$$:


 * $$\omega_0^2 + k_0\omega_0b_0 + \left(c_0-k_0^2a_0\right)=0$$

Our wave equation was chosen so that the polynomial is an easily solved quadratic equation that supports two waves or "branches":


 * $$\omega_0 = -\frac 1 2 k_0b_0 \pm\sqrt{\frac 1 4 k_0^2b_0^2 + k_0^2 a_0 - c_0} $$

A dispersion relation for the weakly inhomogeneous case
If the system is weakly inhomogeneous, the coefficients (a, b, c) are nearly uniform over many wavelengths of a wavepacket. Hence, over the duration of the wavepacket this dispersion relation should be approximately true:


 * $$\omega(k,x) = -\frac 1 2 k(x)b(x) \pm $$$$\sqrt{\frac 1 4 k(x)^2b(x)^2 + k(x)^2a(x) - c(x)} $$

Let $$L$$ represent a length over which these coefficients remain nearly unchanged, $$\lambda=2\pi/k$$ be the wavelength, and $$\Delta x$$ be the size of the wavepacket. The eikonal approximation requires that the wavepacket is large compared to one wavelength, yet small compared with the distance one must travel before the coefficients (a, b, c) change significantly:


 * $$\Delta L >> \Delta x >> \lambda .$$

This inequality is the eikonal approximation. Also, we have assumed that the medium does not change with respect to time, which implies that the frequency, &omega;, does not change as it moves through the medium. Thus, k must evolve in such a way that &omega; remains constant. From a well known relationship for differentials of functions of two or more variables, we have:


 * $$\Delta\omega (x,k) = 0= \frac{\partial\omega}{\partial x}\Delta x + \frac{\partial\omega}{\partial k}\Delta k$$

Using the concept of group velocity, we have


 * $$\frac{\Delta x}{\Delta t} =\frac{dx}{dt}=\frac{\partial\omega}{\partial k}$$, and  $$\frac{\Delta k}{\Delta t} =  \frac{dk}{dt}=-\frac{\partial\omega}{\partial x}$$

Here we have replaced &Delta;k, &Delta;x, and &Delta;t, by dk, dx, and dt, respectively.

Application to the Schrödinger equation
According to De Broglie's matter wave hypothesis, energy is, E = ħ&omega; = V + p2/2m, where V=V(x) is potential energy, and momentum is, p = mv = ħk. Expressing angular frequency in terms of this expression for energy, we have:


 * $$\hbar \omega (k,x) = \frac{\hbar^2 k^2}{2m} + V(x)$$

It is straightforward to recover Newtonian mechanics from this. For example, the Newtonian force is, F = –&part;V/&part;x, and


 * $$\frac{\partial\omega}{\partial k} = \frac{\hbar k}m = \frac p m = v = \frac{dx}{dt}$$  and


 * $$-\frac{\partial\omega}{\partial x} = -\frac{1}{\hbar}\frac{\partial V}{\partial x} = \frac{F}\hbar = \frac{mdv/dt}\hbar=\frac 1\hbar \frac{dp}{dt}=\frac{dk}{dt}$$ (as expected)

Conclusion
Had Hamilton been asked in 1834 to find a wave that obeyed Newton's second law of motion for a conservative force, he would have written the Schrödinger equation and said that ħ needs to be a small number. If you had asked "how small", he might have said "very small" and wondered why you wanted such a wave.

Statement of the theorem in modern form
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,

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

This latter form of the dispersion relation has no meaning for waves in a medium that is truly homogeneous in space in time. But now suppose 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:

$$\dot{q_i} = \frac{\partial \omega}{\partial k_i}$$

$$\dot{k_i} = -\frac{\partial \omega}{\partial q_i}$$

$$\dot{\omega} = -\frac{\partial \omega}{\partial t}$$

Applications

 * 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.


 * 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.