Physics Concepts

• What is the difference between s- and p-polarized light?

S&P polarization refers to the plane in which the electric field of a light wave is oscillating. S-Polarization is the plane of polarization perpendicular to the page (coming out of the monitor screen). P-polarization is the plane of polarization parallel to the page (in the plane of the monitor screen). See figure below:

When referring to polarization states, the p-polarization refers to the polarization plane parallel to the polarization axis of the polarizer being used ("p" is for "parallel"). The s-polarization refers to the polarization plane perpendicular to the polarization axis of the polarizer. A linear polarizer, by design, polarizes light in the p-polarization.

https://physics.stackexchange.com/questions/319009/what-is-the-significance-of-different-modes

• Phase Velocity

The phase velocity is the rate at which the phase of the wave propagates in space. It is given by

v_phase=lamda/t=w/k;

where lamda is the wavelength, t is the time period of a wave, w is angular frequency and k is the wavenumber.

• Group Velocity

The group velocity is the velocity with which energy propagates and is defined by 

v_group=dw/dk;

where w is the angular frequency of a wave and k is the wavenumber.

• Relationship:

In a given medium, both the phase velocity and group velocity depend on the frequency and on the medium's material properties (refractive index)

In terms of the real part of the index of refraction,

n_r=c/v_phase=ck/w,

where c is the speed of light and v_phase is the phase velocity. So,

dn_r/dw=-ck/w^2+c/wdk/dw

=c/w(dk/dw-k/w)

=c/w(1/v_group-1/v_phase)

solving for v_group then gives

v_group=(w/c*dn_r/dw+1/v_phase)^-1

Noting, the equation indicates that the group speed is equal to the phase velocity only when the refractive index is a constant (in dependent to the frequency) or the frequency is a constant (independent to the wavenumber). In this case both phase and group velocity are independent of frequency and equal to c/n. 

Otherwise, if they vary with freuqency, the medium is called dispersive, the relation w=w(k) is known as the dispersion relation of the medium.

dw/dk=c/n-ck/n^2*dn_r/dk.

Ref.: http://scienceworld.wolfram.com/physics/topics/WaveProperties.html

Most op­er­a­tors in quan­tum me­chan­ics are of a spe­cial kind called Her­mit­ian. This sec­tion lists their most im­por­tant prop­er­ties.

An op­er­a­tor is called Her­mit­ian when it can al­ways be flipped over to the other side if it ap­pears in a in­ner prod­uct:

That is the de­f­i­n­i­tion, but Her­mit­ian op­er­a­tors have the fol­low­ing ad­di­tional spe­cial prop­er­ties:

• They al­ways have real eigen­val­ues, not in­volv­ing i=sqrt(-1). (But the eigen­func­tions, or eigen­vec­tors if the op­er­a­tor is a ma­trix, might be com­plex.) Phys­i­cal val­ues such as po­si­tion, mo­men­tum, and en­ergy are or­di­nary real num­bers since they are eigen­val­ues of Her­mit­ian op­er­a­tors.
• Their eigen­func­tions can al­ways be cho­sen so that they are nor­mal­ized and mu­tu­ally or­thog­o­nal, in other words, an or­tho­nor­mal set. This tends to sim­plify the var­i­ous math­e­mat­ics a lot.
• Their eigen­func­tions form a com­plete set. This means that any func­tion can be writ­ten as some lin­ear com­bi­na­tion of the eigen­func­tions. (There is a proof in de­riva­tion for an im­por­tant ex­am­ple.  In prac­ti­cal terms, it means that you only need to look at the eigen­func­tions to com­pletely un­der­stand what the op­er­a­tor does.

In the lin­ear al­ge­bra of real ma­tri­ces, Her­mit­ian op­er­a­tors are sim­ply sym­met­ric ma­tri­ces. A ba­sic ex­am­ple is the in­er­tia ma­trix of a solid body in New­ton­ian dy­nam­ics. The or­tho­nor­mal eigen­vec­tors of the in­er­tia ma­trix give the di­rec­tions of the prin­ci­pal axes of in­er­tia of the body.

An or­tho­nor­mal com­plete set of eigen­vec­tors or eigen­func­tions is an ex­am­ple of a so-called “ba­sis.” In gen­eral, a ba­sis is a min­i­mal set of vec­tors or func­tions that you can write all other vec­tors or func­tions in terms of. For ex­am­ple, the unit vec­tors , , and  are a ba­sis for nor­mal three-di­men­sion­al space. Every three-di­men­sion­al vec­tor can be writ­ten as a lin­ear com­bi­na­tion of the three.

The fol­low­ing prop­er­ties of in­ner prod­ucts in­volv­ing Her­mit­ian op­er­a­tors are of­ten needed, so they are listed here:

The first says that you can swap  and  if you take the com­plex con­ju­gate. (It is sim­ply a re­flec­tion of the fact that if you change the sides in an in­ner prod­uct, you turn it into its com­plex con­ju­gate. Nor­mally, that puts the op­er­a­tor at the other side, but for a Her­mit­ian op­er­a­tor, it does not make a dif­fer­ence.) The sec­ond is im­por­tant be­cause or­di­nary real num­bers typ­i­cally oc­cupy a spe­cial place in the grand scheme of things. (The fact that the in­ner prod­uct is real merely re­flects the fact that if a num­ber is equal to its com­plex con­ju­gate, it must be real; if there was an  in it, the num­ber would change by a com­plex con­ju­gate.)

KEY TAKE-AWAYS:

Her­mit­ian op­er­a­tors can be flipped over to the other side in in­ner prod­ucts.

Her­mit­ian op­er­a­tors have only real eigen­val­ues.

Her­mit­ian op­er­a­tors have a com­plete set of or­tho­nor­mal eigen­func­tions (or eigen­vec­tors).