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

First review
Editorial comments:
 * 1) "between the mathematics particle motion and the propagation of linear waves" to "between the mathematics of particle motion and the propagation of linear waves" --Marshallsumter (discuss • contribs) 12:26, 22 May 2017 (UTC)
 * 2) "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." to "Over a century later, Herbert Goldstein remarked that Hamilton would have postulated Schrodinger's equation had there been experimental evidence that particles were waves." --Marshallsumter (discuss • contribs) 15:38, 25 May 2017 (UTC)
 * 3) "partial differential equation that" to "partial differential equation (PDE) that" --Marshallsumter (discuss • contribs) 23:59, 25 May 2017 (UTC)
 * 4) "ignoring the other other ducks' wakes" to "ignoring the other ducks' wakes" --Marshallsumter (discuss • contribs) 00:59, 17 June 2017 (UTC)
 * 5) "the dependence on x indicates that this inhomogeneous in position" to "the dependence on x indicates that this inhomogeneity in position" --Marshallsumter (discuss • contribs) 00:55, 17 June 2017 (UTC)
 * 6) "Let us begin with the completely homogeneous case, using the o-subscript to denote constants." to "Let us begin with the completely homogeneous case, using the 0-subscript to denote constants."
 * 7) "This latter form of the dispersion relation has no meaning for waves in a medium that is truly homogeneous in space in time." to "This latter form of the dispersion relation has no meaning for waves in a medium that is truly homogeneous in space and time."
 * 8) "These wave packets must be many wavelength in size" to "These wave packets must be many wavelengths in length"
 * 9) "This regime of gently variation in properties of the medium" to "This regime of gentle variation in properties of the medium"
 * 10) "and have application in plasma physics applications where microwaves might be used to heat the core of the plasma." to "and have application in plasmas where microwaves might be used to heat its core."
 * "<:ref>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 " to "<:ref name=Hamilton> "
 * 1) "Addison Wesley )Cambridge, Mass." to "Addison Wesley, Cambridge, Mass."
 * 2) "the integral over all space of |&psi;|^2 does not vary with time." to "the integral over all space of $$|\psi|^2$$ does not vary with time."
 * 3) "and this insight leads to the postulate that |&psi;|^2 is probability density in Schrödinger's equation." to "and this insight leads to the postulate that $$|\psi|^2$$ is a probability density in Schrödinger's equation."
 * "<:ref>Note, for example that eigensolutions exist of the form &psi;=u(x)exp(-i&omega;t) " to "<:ref>Note, for example that eigensolutions exist of the form $$\psi=u(x)e^{(-i\omega t)}$$ " --Marshallsumter (discuss • contribs) 03:24, 15 June 2017 (UTC)

Text comments:

"through the Ehrenfest theorem of quantum mechanics." to "through the Ehrenfest theorem of quantum mechanics which relates the position operator x and momentum operator p expected-value time derivatives $$\frac{d}{dt}\langle x\rangle$$ and $$\frac{d}{dt}\langle p\rangle$$ to the force expected value $F = −dV/dx$ on a massive particle moving in a scalar potential,
 * $$m\frac{d}{dt}\langle x\rangle = \langle p\rangle,\;\; \frac{d}{dt}\langle p\rangle = -\left\langle \frac{\partial V(x)}{\partial x}\right\rangle  ~.$$", just a suggestion. --Marshallsumter (discuss • contribs) 22:46, 25 May 2017 (UTC)

"With more than one spacial variable, it is necessary to perform a different calculation along the lines of Eikonal_approximation" to "With more than one spacial variable, it is necessary to perform a different calculation along the following lines (the eikonal approximation):

Wentzel–Kramers–Brillouin (WKB) theory can be a method for approximating the solution of an nth order differential equation whose highest derivative is multiplied by a small parameter $ε$.
 * $$ \epsilon \frac{d^ny}{dx^n} + a(x)\frac{d^{n-1}y}{dx^{n-1}} + \cdots + k(x)\frac{dy}{dx} + m(x)y= 0,$$

using an asymptotic series expansion solution of the form
 * $$ y(x) \sim \exp\left[\frac{1}{\delta}\sum_{n=0}^{\infty}\delta^nS_n(x)\right]$$

in the limit $δ → 0$. The asymptotic scaling of $δ$ in terms of $ε$ is likely $δ$ ∝ $ε$.

Substituting the solution into the differential equation and reducing yields an arbitrary number of terms $S_{n}(x)$.

We can write the wave function $$\Psi$$ of the scattered system in terms of action S:


 * $$\Psi=e^{iS/{\hbar}} $$

For the Schrödinger equation without the presence of a magnetic field we obtain


 * $$ -\frac{{\hbar}^2}{2m} {\nabla}^2 \Psi= (E-V) \Psi$$


 * $$ -\frac{{\hbar}^2}{2m} {\nabla}^2 {e^{iS/{\hbar}}}=(E-V) e^{iS/{\hbar}}$$


 * $$\frac{1}{2m} {(\nabla S)}^2 - \frac{i\hbar}{2m}{\nabla}^2 S= E-V$$

We write S as a power series in ħ


 * $$S= S_0 + \frac {\hbar}{i} S_1 + ...$$

with the zero-th order:


 * $$ \frac{1}{2m} {(\nabla S_0)}^2 = E-V.$$

For the one-dimensional case, $${\nabla}^2 \rightarrow {\partial_z}^2$$, we obtain a differential equation with the boundary condition:


 * $$\frac{S(z=z_0)}{\hbar}= k z_0$$

V → 0, z → -∞.


 * $$\frac{d}{dz}\frac{S_0}{\hbar}= \sqrt{k^2 - 2mV/{\hbar}^2}$$


 * $$\frac{S_0(z)}{\hbar}= kz - \frac{m}{{\hbar}^2 k} \int_{-\infty}^{Z}{V dz'} $$", also just a suggestion. It puts everything before the reader and can be referenced as desired. It tends to complete the History section and ties it to the force expected value $F = −dV/dx$ at the end. --Marshallsumter (discuss • contribs) 01:29, 26 May 2017 (UTC)


 * I do have concerns the reader may not readily see the connection between the description above of the approximation and your example. --Marshallsumter (discuss • contribs) 01:26, 17 June 2017 (UTC)


 * you comments look constructive. The dean has given me some extra work to do this summer, and maybe I will get it done by Christmass. I will keep this on my to do list.--Guy vandegrift (discuss • contribs) 14:28, 19 June 2017 (UTC)