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**General Science / Re: what is Nothing or ?energy landscape.? or waves **

« **on:**11/07/2024 20:17:47 »

Hi.

I wasn't at the same Solid State course, so I don't really know what @varsigma was suggesting.

However, I'm suspicious that you may be wrong @alancavlerd but in a nice way and you have tried to make a reply that is still reasonably sensible.

When considering light travelling into some media from some other media, it is common to model the e-m wave by considering the E field component only. By Maxwells equations, the B field would just be a scalar multiple (with orthogonal direction) anyway. So we need only consider the E field.

So we have, E field at the position z (a 1-dimensional position co-ordinate measured along the direction of travel) and the time t =

E (z,t) = A.e

Where, as usual, k can be identified as the wave number for a component of the e-m radiation that we're interested in. The whole collection of e-m radiation will be expressable as a sum of several different rays with different k numbers.

The wave equations are linear, as @varsigma seemed to be saying, so that multiples of solutions are solutions and sums of solutions are also still solutions. Hence, rather than considering oscillations of the form Cos(ωt) or Sin(ωt) we can consider solutions of the form Cos(ωt) + i.Sin(ωt). With Euler's formula that means we need only consider solutions of the form A.e

Since we can have the A (amplitude) term arbitrary, there's nothing very special that would happen if that was allowed to be a complex number, it always could be.

Alan said:

The amplitude being real or imaginary need not relate to anything physically observable, since as you pointed out it is generally only the square of this Amplitude (the Intensity, I = |A|

However, if we allow the wave number, k, to be imaginary then we do get something useful.

Assume k is complex and write it as the sum of real and imaginary parts as usual, k = K

This is a fair model of how light may penetrate another media. For example, if the new media is perfectly transmissive then k

Permitting a complex valued wave number k, allows various situations to be modelled. Since we can express the wave number k in terms of a refractive index, N, for the media, this is equivalent to using or permitting a complex valued index of refraction.

It is thus, the use of complex numbers

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The rest of what you ( @alancalverd) said looks OK.

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NOTE: The above was fairly well established scientific information and not just something out of my head. It may not match exactly what @varsigma or @alancalverd was talking about but the information is still factually accurate to the best of my knowledge and belief. However, the comments below are speculative. It is just an attempt to engage in discussion on the topic raised by @varsigma and others.

However, the whole thing is a bit of a fudge. There isn't a lot of direct reason to assume a light ray entering a forbidden region (e.g. some media that is opaque to it) would fall off exponentially instead of just abruptly stopping and being completely reflected at the point or place of entry to that forbidden region. As mentioned, the main motivation is from Quantum Mechanics and the observation that the QM wave function does often enter classically forbidden regions and just falls off exponentially. However, since that is our main motivation AND then the results we obtain from the mathematical model of light as oscillations in the E and B fields when we do use complex valued wave numbers really do match experimental observation so well, it is very tempting to think that there is some more fundamental link here.

For example, maybe all of Maxwells equations and what we describe as the E and B fields are just a simplified or limiting case of Quantum Mechanics. So that light is just some QM wave phenomena instead of an oscillation in the classical E and B field. There are many discussions available which suggests that the use of complex numbers is fundamental or

Best Wishes.

I wasn't at the same Solid State course, so I don't really know what @varsigma was suggesting.

However, I'm suspicious that you may be wrong @alancavlerd but in a nice way and you have tried to make a reply that is still reasonably sensible.

When considering light travelling into some media from some other media, it is common to model the e-m wave by considering the E field component only. By Maxwells equations, the B field would just be a scalar multiple (with orthogonal direction) anyway. So we need only consider the E field.

So we have, E field at the position z (a 1-dimensional position co-ordinate measured along the direction of travel) and the time t =

E (z,t) = A.e

^{i(kz - ωt)}[EQUATION 1]

Where, as usual, k can be identified as the wave number for a component of the e-m radiation that we're interested in. The whole collection of e-m radiation will be expressable as a sum of several different rays with different k numbers.

The wave equations are linear, as @varsigma seemed to be saying, so that multiples of solutions are solutions and sums of solutions are also still solutions. Hence, rather than considering oscillations of the form Cos(ωt) or Sin(ωt) we can consider solutions of the form Cos(ωt) + i.Sin(ωt). With Euler's formula that means we need only consider solutions of the form A.e

^{i(kz - ωt)}and the complete solution will just be a sum of a few of these rays with some different k wave numbers (that is to say, W.L.O.G. all fundamental solutions can be treated and described in the form of [EQUATION 1] as appears above ).Since we can have the A (amplitude) term arbitrary, there's nothing very special that would happen if that was allowed to be a complex number, it always could be.

Alan said:

giving A an imaginary component allows for, say, reflection (P positive) or transmission (P negative)Which may be how things are done in some text you read but in the texts I've seen, a reflected ray wouldn't be marked out just by having an imaginary amplitude only. Instead a reflected ray will have a wave number, k, that is precisely the negative of the wave number k for a ray travelling in the forward direction. That, I hope you will see, makes sense, the wave number k always does tell you about the direction of travel of the ray.

The amplitude being real or imaginary need not relate to anything physically observable, since as you pointed out it is generally only the square of this Amplitude (the Intensity, I = |A|

^{2}) that we could see anyway.However, if we allow the wave number, k, to be imaginary then we do get something useful.

Assume k is complex and write it as the sum of real and imaginary parts as usual, k = K

_{real}+ i. K_{imag}. Then we see that the net effect in [EQUATION 1] above from a non-zero imaginary part for k, K_{imag}>0, is to convert the amplitude A into something of the form A.e^{-kim.z}~ exponential fall off in Amplitude with distance z along the direction of travel.This is a fair model of how light may penetrate another media. For example, if the new media is perfectly transmissive then k

_{im}= 0 and there is no exponential fall off in Amplitude as the wave travels into it. Otherwise for media that is (on a macroscopic scale) opaque to this wave number k of light, this sort of exponential decay of Amplitude is a better approximation to what actually happens and/or what QM predicts. For example, the solution of the Quantum Mechanical "particle in a square well potential" shows that the wave function does penetrate the region outside the well, a classically forbidden region, but this wave function falls off exponentially.Permitting a complex valued wave number k, allows various situations to be modelled. Since we can express the wave number k in terms of a refractive index, N, for the media, this is equivalent to using or permitting a complex valued index of refraction.

It is thus, the use of complex numbers

**for the wave numbers**{ k } representing the collection of e-m rays that I suspect @varsigma was talking about and not so much the use of complex numbers just for the Amplitude.------

The rest of what you ( @alancalverd) said looks OK.

------

NOTE: The above was fairly well established scientific information and not just something out of my head. It may not match exactly what @varsigma or @alancalverd was talking about but the information is still factually accurate to the best of my knowledge and belief. However, the comments below are speculative. It is just an attempt to engage in discussion on the topic raised by @varsigma and others.

Is a light wave a complex number?For most purposes, we could model the situations using only real valued everythings..... real K wave numbers, real index of refraction, real Amplitudes etc. and just manually or directly add in the exponential damping for the Amplitude where a given k wave number cannot ordinarily continue travelling through some media or through some place. It is just easier to use complex valued wave numbers k so that the exponential damping effect is automatically taken account of.

However, the whole thing is a bit of a fudge. There isn't a lot of direct reason to assume a light ray entering a forbidden region (e.g. some media that is opaque to it) would fall off exponentially instead of just abruptly stopping and being completely reflected at the point or place of entry to that forbidden region. As mentioned, the main motivation is from Quantum Mechanics and the observation that the QM wave function does often enter classically forbidden regions and just falls off exponentially. However, since that is our main motivation AND then the results we obtain from the mathematical model of light as oscillations in the E and B fields when we do use complex valued wave numbers really do match experimental observation so well, it is very tempting to think that there is some more fundamental link here.

For example, maybe all of Maxwells equations and what we describe as the E and B fields are just a simplified or limiting case of Quantum Mechanics. So that light is just some QM wave phenomena instead of an oscillation in the classical E and B field. There are many discussions available which suggests that the use of complex numbers is fundamental or

**essential**in Quantum Mechanics, wave functions must be complex valued. As such maybe light actually REALLY IS best considered as some wave phenomena that must be assigned a complex value.Best Wishes.