Is the speed of light different in different medium?

[exchemist]

I think you both have the better explanation. The vids interested me but your explanations are better.

If we think of refraction as a bulk effect acting on the wave velocity then we can almost dispense with trying to trace the path at the level of the atomic interstitial space. Actually in the first vid the prof is saying glass, which is almost always amorphous, but he's drawing crystalline material. Minor point, but it serves to show the advantage of keeping this at the bulk level.

I would have just said that whenever the impedance changes, the phase shifts and hence the velocity changes. I'm pretty sure I can calculate the characteristic impedance if I know the polarization and susceptibility, so I think I'm on the right track. And there is a dual for the velocity change for acoustic waves in air vs water which uses the stiffness (or compliance) and density (I think) which are duals of something like permittivity (or polarization) and permeability (or susceptibility)-- from which we can calculate a characteristic impedance for both of them.

So using your better explanations as the gold standard, I'm betting that if we reduced this to a single parameter it's probably impedance.

Also, velocity (esp c) can be written as the inverse of the geometric mean of permittivity and permeability. And they establish the impedance.

Interesting. The only problem I see with this idea of impedance is that one also has to account for dispersion, i.e. the fact that refractive index is a function of frequency and changes in a rather interesting way as one approaches, passes through, and then retreats from, an absorption resonance frequency. I like the concept of the forced oscillation because the idea of "lead" and "lag" of the driven oscillation, relative to the forcing frequency, gives one a picture that incorporates this and relates it to spectroscopic absorption and emission at the same time.

 

[exchemist]

In my post 15, I commented that the wave model of photons is best most of the time, but when they die, they die as particles in one very small (atomic size or less) spot. CaptBork and others were concerned about speed of an electron in orbit about a nucleus. That "speed" is sort of a particle concept, not even a valid concept for a standing wave

I think it was Bohr who modeled the energy levels of a hydrogen atom with waves going around and their "heads" being required to over lap IN PHASE with their "tails." Sort of a "standing wave" idea or at worst a slowly drifting standing wave.

He had a value for the "wave length of an electron." If one full cycle of it is "wrapped" around a proton, then that circumference is a specified distance from the point proton. - A certain way up the negative potential hill which he could calculate. I think the answer is 13.6 ev. from being free (ionized) for the one wavelength around case.

Like wise if two full cycles of the electron wave length are "wrapped" around the proton, then the circumference is twice as big and twice as far from the proton. He could calculate how deep in the negative potential well that would be and got 13.6 /4 ev.

For three wave lengths wrapped around that depth in the well from being free is 13.6/9 ev and in general the negative energy levels go as 13.6 n-2. ev. with n being the number of full electron wave lengths that wrap around to come back in phase with themselves. A concept that does give the hydrogen energy levels correctly!

So, once again, as I said, the electron is a wave until you kill it - just like photons are until they dies (are fully absorbed, not just slightly red shifted by a scattering , like "Compton scattering" does.) So that energy and momentum conservation are honored by "mother nature."

.

Yes, I have wrestled with this issue of the "speed" of an electron in an atomic orbital, with others, on another forum. For sure, we cannot know what value to put on any notional "speed" of an electron, due to the Uncertainty Principle. But we have, I think, to treat the electron as having motion, because the form of the Hamiltonian is based on there being a continual interchange of kinetic and electrostatic potential energy as the electron circulates about the nucleus. Without at least some notional form of - admittedly indeterminate - motion there would be no kinetic energy, surely? In fact all simple treatments of wave mechanics (the particle in the box, the harmonic oscillator, and so on) assume motion of a particle, in order to derive the wave equation and hence the eigenstates. Don't they?

 

[Billy T]

... we have, I think, to treat the electron as having motion, because the form of the Hamiltonian is based on there being a continual interchange of kinetic and electrostatic potential energy as the electron circulates about the nucleus. ...

Often true, but not necessarily so. The Hamiltonian is just the total energy of the system. Yes, often that conserved constant does at least partially switch between kinetic and potential energy, but it need not. For example a planet (the only one) bound to an isolated star in a circular orbit has KE that is ALWAYS half the magnitude of its negative potential energy.

The classical mechanics Hamiltonian is usually the foundation of modern quantum mechanics. The electron bound to the hydrogen atom in the ground state has zero angular momentum. - Thus in a classical POV it is not "going around the proton" but oscillating back and forth from maximum PE with path straight thru the proton and with maximum KE as it passes thru the proton. - Not a very satisfying way to look at it, is it. In a highly excited state with max angular momentum, then you can think of the electron as a "little planet" going around a central proton "star." with no exchange of the total energy between PE & KE forms.

Thus Bohr's picture of the ground state electron a bound standing wave around the proton, but not "going around" the proton, is completely compatible with a Hamiltonian POV if there is zero KE, and as it gives the correct energy levels, has lot to be said for it. In standard terminology that is the 1s electron. If I remember my QM correctly you can not have a 1p electron but can have a 2p electron n the first excited state. It has one unit of angular momentum (neglecting spin which just confuses the point I am making).

You still can say there is no exchange of PE and KE for that 2p electron. The two peaks of circular standing wave in Bohr's model just happen to be rotating around the proton giving the 1p electron (which has mass, of course) one CONSTANT unit of angular momentum and unchanging KE. For the 3p electron there may be sense in saying that there is an exchange of PE and KE, but probably no sense to say that about the 3d electron. All these non-quantum concepts are not applicable - misleading at best or down-right wrong.

Don't try to put 10 pounds of QM in a 5 pound classical bag.

 

[Farsight]

Your "fat finger" hit the G instead of the F...

It certainly did! IMHO the important point with all this is that the many-paths is a non-localized wave, and we make electrons out of photons, and it's the wave nature of matter and quantum field theory. The electron isn't some point-particle thing, its field is part of what it is. So in a detector where a photon is interacting with an electron, you have two extended entities interacting, akin to the wavefront and the lens.

It is fun to have second lens one focal length way from the plane of the transform and then a focal length from that lens another screen which recreates the source (except for components that diverged more than the first lens intercepted. The "fun" (and some education) comes with the burnt black head of a match stuck on a needle that can selectively blocks /removes one component of the FT from the reconstruction of the original image. I found that the more interesting original images to do this with had 2D regularity - a piece of screen wire is very interest to use as the original. Not only can you remove components from the transform plane, but you can warp its regularity too. Also with a set of regular parallel grids you can get quite quantative evaluation of the nature of the lens defects - spatially resolve those defects into components.

All good stuff.

Not a very satisfying way to look at it, is it. In a highly excited state with max angular momentum, then you can think of the electron as a "little planet" going around a central proton "star." with no exchange of the total energy between PE & KE forms.

I think it's better to think of it as more like a hula hoop of electromagnetic field jiggling around at 137th the speed of light.

...In fact all simple treatments of wave mechanics (the particle in the box, the harmonic oscillator, and so on) assume motion of a particle, in order to derive the wave equation and hence the eigenstates. Don't they?

IMHO the hydrogen-atom electron is like a wave in a box, only you're jiggling the box, so the wavelength is longer, hence the negative binding energy.

Also, velocity (esp c) can be written as the inverse of the geometric mean of permittivity and permeability. And they establish the impedance.

Most people know that c = √(1/ε₀μ₀) and Z₀ = √(μ₀/ε₀), and that impedance is resistance to alternating current. But for some strange reason most people don't know that displacement current is "a time-varying electric field", or that an electromagnetic wave is therefore alternating displacement current.

 

[exchemist]

Often true, but not necessarily so. The Hamiltonian is just the total energy of the system. Yes, often that conserved constant does at least partially switch between kinetic and potential energy, but it need not. For example a planet (the only one) bound to an isolated star in a circular orbit has KE that is ALWAYS half the magnitude of its negative potential energy.

The classical mechanics Hamiltonian is usually the foundation of modern quantum mechanics. The electron bound to the hydrogen atom in the ground state has zero angular momentum. - Thus in a classical POV it is not "going around the proton" but oscillating back and forth from maximum PE with path straight thru the proton and with maximum KE as it passes thru the proton. - Not a very satisfying way to look at it, is it. In a highly excited state with max angular momentum, then you can thing of the electron as a "little planet" going around a central proton "star." with no exchange of the total energy between PE & KE forms.

Thus Bohr's picture of the ground state electron a bound standing wave around the proton, but not "going around" the proton, is completely compatible with a Hamiltonian POV if there is zero KE, and as it gives the correct energy levels, has lot to be said for it. In standard terminology that is the 1s electron. If I remember my QM correctly you can not have a 1p electron but can have a 2p electron n the first excited state. It has one unit of angular momentum (neglecting spin which just confuses the point I am making).

You still can say there is no exchange of PE and KE for that 2p electron. The two peaks of circular standing wave in Bohr's model just happen to be rotating around the proton giving the 1p electron (which has mass, of course) one CONSTANT unit of angular momentum and unchanging KE. For the 3p electron there may be sense in saying that there is an exchange of PE and KE, but probably no sense to say that about the 3d electron. All these non-quantum concepts are not applicable - misleading at best or down-right wrong.

Don't try to put 10 pounds of QM in a 5 pound classical bag.

I think we are nearly on the same wavelength (as it were).

I actually very much like the picture of the 1s shell as like the harmonic oscillator, with the electron passing through the nucleus, with the uncertainty principle twist that we can't identify a particular line of motion, so it occupies a spherical distribution. That is just how it behaves, i.e. (a) with no angular momentum and (b) with an exposure to the influence of nuclear charge that is consistent with it going right up to the nucleus. The latter has important consequences in chemistry, due to the special behaviour of electrons in s orbitals, so we chemists are very much aware of it. But then it must have KE, on average, just like the harmonic oscillator.

And yes, quite right, there is no 1p, since the azimuthal quantum number (l) has values from zero to n-1.

In practice however, for most purposes, I tend to a view similar to yours, i.e. it seems more natural to think of the electron as a wave first, but which is constrained to interact only in quantized units corresponding to "whole" electrons. I suppose like a lot of people I've tolerated a degree of cognitive dissonance about the model I have in my head. QM tends to do that to you.

 

[exchemist]

IMHO the hydrogen-atom electron is like a wave in a box, only you're jiggling the box, so the wavelength is longer, hence the negative binding energy.

Negative binding energy is merely a result of the convention of setting the energy of charges at infinite separation to zero. The important point is that the Coulomb attraction creates a confining potential for the electron. In the case of the harmonic oscillator, the convention is to set the potential to zero at the mid point of the motion and have the potential rise parabolically to +ve values either side. But whether one chooses to set the top of the potential well or the bottom of it at a nominal zero energy level makes no difference, so long as the potential rises with distance from a central point, since that is what creates the confinement that results in quantization.

The forms of the solutions to the wave equation in the atom are analogous to the spherical harmonics that characterise a rubber ball set in vibration. There are radial waves of compression and rarefaction (i.e with radially disposed nodes), corresponding to the principal shells and then there are superimposed on that, dipolar, quadrupolar etc rotational or surface type waveforms, with nodes in planes intersecting the atom, corresponding to the subshells defined by the azimuthal quantum number. It is natural that these "rotational" waves are associated with orbitals in which the electron has angular momentum (p,d, f etc).

 

[Farsight]

Negative binding energy is merely a result of the convention of setting the energy of charges at infinite separation to zero.

Where did you hear that? It isn't. The mass of the hydrogen atom is less than the mass of the electron plus the mass of the proton. There really is a mass deficit.

The important point is that the Coulomb attraction creates a confining potential for the electron.

Yes, and you have to add energy to get that electron out of there. This energy has a mass equivalence. It isn't a whole lot different to the brick in the Earth's gravitational potential, though the Earth/brick ratio is much bigger than the proton/electron ratio.

In the case of the harmonic oscillator, the convention is to set the potential to zero at the mid point of the motion and have the potential rise parabolically to +ve values either side. But whether one chooses to set the top of the potential well or the bottom of it at a nominal zero energy level makes no difference, so long as the potential rises with distance from a central point, since that is what creates the confinement that results in quantization.

IMHO the convention makes no difference because the mass deficit is real.

The forms of the solutions to the wave equation in the atom are analogous to the spherical harmonics that characterise a rubber ball set in vibration. There are radial waves of compression and rarefaction (i.e with radially disposed nodes), corresponding to the principal shells and then there are superimposed on that, dipolar, quadrupolar etc rotational or surface type waveforms, with nodes in planes intersecting the atom, corresponding to the subshells defined by the azimuthal quantum number. It is natural that these "rotational" waves are associated with orbitals in which the electron has angular momentum (p,d, f etc).

No problem. Interestingly the phrase spherical harmonics was coined by Thomson and Tait, see the history section of the Wikipedia article. Their "vortex atoms" are arguably the first example of a spinor, see Smoke rings and nineteenth-century atomism. Sadly they predated the discovery of the electron.

 

[Billy T]

Negative binding energy is merely a result of the convention of setting the energy of charges at infinite separation to zero....

No setting the POTENTIAL to zero (or any value) there is an arbitrary choice, but binding or "bound" means energy must be given to "unbind" or "free" the bound particle or planet etc. We set the potential to zero there for hydrogen atom as in makes the equation for the energy levels very simple I.e. they are for hydrogen just 13.6ev /n^2 where n is 1 for the ground state, 2 for first (lowest energy) excited state, etc.

Thus the energy for a transition between two states m & n is 13.6ev( m^-2 - n^-2) and for the visible lines the lower state has n=2 and m equal or greater than 3. that gets "messy" for potential not zero when just free from being bound.

 

[exchemist]

No setting the POTENTIAL to zero (or any value) there is an arbitrary choice, but binding or "bound" means energy must be given to "unbind" or "free" the bound particle or planet etc. We set the potential to zero there for hydrogen atom as in makes the equation for the energy levels very simple I.e. they are for hydrogen just 13.6ev /n^2 where n is 1 for the ground state, 2 for first (lowest energy) excited state, etc.

Thus the energy for a transition between two states m & n is 13.6ev( m^-2 - n^-2) and for the visible lines the lower state has n=2 and m equal or greater than 3. that gets "messy" for potential not zero when just free from being bound.

OK I may have expressed myself clumsily. What I meant, as I hope the context of my remarks bears out, is simply that the depth and shape of the potential well is what determines the quantization, in the atom just as in the harmonic oscillator. So whether one sets the bottom of the well or the top to "zero", and hence whether the potential energy of the particle at some level within the well is said to be positive or negative with respect to this zero level is not important. You are quite obviously right that in any potential well, the energy rises as the particle moves outward from the centre - that's what we mean by a "well", after all.

As I'm sure you know, but just for any other readers, with the harmonic oscillator the well has a bottom but no top, because the well is parabolic, so we choose to set the bottom as "zero" potential. With the Coulomb potential there is a top (at least the potential flattens out asymptotically to a maximum height), but has no bottom, as it descends to infinity. So we set the upper maximum level to "zero", as of course we also conventionally do for gravitational potential.

 

[Forceman]

According to Newtonian mechanics anything with particles or made out of particles will have a speed that varies. Having a speed that varies limits radiation infinity ranges and makes near infinite speed radiation of high energy move down to the photonic level as both a wave and a photon and travel at both the speed of light and a lower speed (curved space offers different realities). Two different linear speeds can provoke final acceleration and a photon will proceed through a vector and may become sound or vibrations because of wormholes in time and finally emerge at any speed it wants to emerge at. Light is only one of the lowest-energy EMR's and will proceed through space at the speed of a photon which is not the same as the speed of light and will travel at sound wave speed because of the multiple dimensions of the atom and will annihilate with an antiphoton that has no sound because of no sound wave paring.

 

[exchemist]

According to Newtonian mechanics anything with particles or made out of particles will have a speed that varies. Having a speed that varies limits radiation infinity ranges and makes near infinite speed radiation of high energy move down to the photonic level as both a wave and a photon and travel at both the speed of light and a lower speed (curved space offers different realities). Two different linear speeds can provoke final acceleration and a photon will proceed through a vector and may become sound or vibrations because of wormholes in time and finally emerge at any speed it wants to emerge at. Light is only one of the lowest-energy EMR's and will proceed through space at the speed of a photon which is not the same as the speed of light and will travel at sound wave speed because of the multiple dimensions of the atom and will annihilate with an antiphoton that has no sound because of no sound wave paring.

Ah, a troll. Reported.