I think there may be a bit of a language issue here. I'm not quite sure what you are asking. For one thing, quantum particles don't exist. In fact, in quantum mechanics neither particles nor waves exist; everything is wave-like but rapidly becomes localized (in other words, like a particle or system of particles). Also, it seems as if you are saying that something we describe using classical physics "behaves like" something from quantum physics, but I'm not even sure that's what you are saying (and it isn't true; we can derive classical physics from quantum physics, but not the reverse). Also I'm not sure whether or not by "quantum physics" you mean QED here. Finally, last time I checked there was no empirical evidence that Ohm's law failed to hold at the sub-atomic level, but rather that it so far it has continued to hold even for extremely small systems of a few atoms (also, while I am quite familiar with scattering, I don't recall coming across the term "scatterer"; like I sad, I think there might be some language issues here).

Thanks for your kind reply.Dear sir What my question is that if we have two scatterer,like two boxes we can say,and when electrons are moving from left to right we can obtain the Ohm's law ,but if same particles are behaving like Quantum Particles,means obeying Schroedinger equation ,although it is Semiclassical or Quantum it is different issue,but i am assuming it like Quantum only.In this case we cant obtain Ohm's law.What my question is that when two scatterer are spaced by lambda/2 ,constructive interference occurs,but when they are spaced by lambda/4 destructive interference occurs,means Transmission will be more in case of destructive,so when we are plotting Energy Vs Transmission 4 or 5 sharp peaks are coming in my results,which i am thinking that it is due to the number of bound states are there 4 to 5 ,like what used to happen in Resonant Tunnelling Diode,which i am not sure whether i am right in this case or not?thats what my worry is or my doubt.

Ah! Thanks. I think I understand what you are getting at. If you treat electrons using quantum mechanics (whether you call them quantum particles or no), you will not be able to derive Ohm's law the way you can using classical physics (alternatively, you can't derive Ohm's law using the Schrödinger equation). To see this, simply try to use Ohm's law to calculate the total/overall resistance for an electron that "travels through" or "hops" over two scatters that can be treated as resistors: a double tunnel junction. Ohm's law requires the resistance to be computed additively. This yields the wrong answer. Why? Because there is no hard and fast rule for tunnel junctions or quantum transmission through any scatters as there is classically.

The reason for this is much like you describe: interference between the two scatterers changes depending upon how they are spaced. In fact, with the right configuration, we can get a transmission that exceeds that of the individual scatterers, and if we space them differently we can get a decrease in the transmission. Destructive interference does, as you say, lead to a peak or an increase.

Also, the fact that we can't derive Ohm’s law by treating electrons as “quantum particles’ doesn’t mean we can’t derive Ohm’s law using quantum mechanics: average over the phase shifts (over χ). By averaging over χ, we are basically using a quantum version of the classical method because we are summing probabilities (not amplitudes). 

So for single nanostructures, averaging over phase shifts “cancels out” (so to speak) the quantum interference.

Of course, this isn't sufficient for all models (again, no hard and fast rules).. When you take into account mixed channels, for example, you need to use the scattering matrix and Ohm’s law is obtained by averaging over all possible phase shifts. 

We still aren’t done. However,

1) I’ve already over-simplified

2) The main point is that, although there are no hard and fast rules and Ohm’s law is not derived the way it is classically (and cannot be), it can be derived.. Perhaps most importantly, the ways different situations result in an expression equivalent to Ohm’s law (in other words, the ways in which we can derive Ohm’s law) are independent from one another in that we arrive at Ohm’s law using different methods. However, we still get there.

Thanks again sir for your nice description,but still this confusion is thre that why number of peaks are increasing ,as i think there should be one peak when we are drawing Energy Vs Transmission,but i am getting 4 or 5.As Goodnick and Ferry  has explained in their book that if we have RTD(resonant tunneling diode)peaks will depend upon number of bound states or levels between two scatterer ,if three then 3 peaks etc--but this is the reason also in my cas also,I have referred Theory of Optics by Paul Drude,he has not mentioned in his book about this issue any where,so i am sure it is not there in Optics as i think.

Regards

suren

I'll take a look at Goodnick and Ferry's book (assuming you are talking about Transport in Nanostructures (Cambridge Studies in Physics and Microelectronic Engineering Vol. 6), but Paul Drude died in 1906. His Theory of Optics was published before Einstein explained the photoelectric effect, let alone the formulation of quantum mechanics (see attached link).

If you could point to the section of Goodnick & Ferry you are referring to that would be great as it would help the language barrier.

https://archive.org/stream/theooptics00drudrich#page/n7/mode/2up

Thanks a lot sir,you are absolutely right.I am talking about the same book.You can see S.Datta's book Quantum Transport also sir.

Thanks 

best wishes

Some questions that might help with the language barrier:

1) Do you have Mizuta, H., & Tanoue, T. (1995). The Physics and Applications of Resonant Tunnelling Diodes (Cambridge Studies in Semiconductor Physics and Microelectronic Engineering 2). Cambridge University Press. ?

2) Are you actually doing an experiment here and getting results that are problematic or is this an issue with the subject matter?

3) Do you have the peak to valley ratio (PVR) value?

4) What is your (electron) transmission coefficient?

1) I dont have this book need to see in our library.

2)I am theoretical Physics student,or we can say issue with subject matter.

3)i dont have ,just tried to solve this problem with Quantum mechanics.

4)Just my transmission is increasing due to two scatterer than that of any one,like we can understand in a better way that like Transmission increases in a lens,when there is anti reflection coating on the top of lens ,similarly first scatterer acts as anti reflection coating kind of thing for second one.

Regards

Most of what I own are papers. However, I do have

Nanoelectronic Circuit Design by Jha & Chen (Eds.)

Nanoelectronics and Nanosystems: From Transistors to Molecular and Quantum Devices by Goser, Glösekötter, & Dienstuhl

Quantum Transport: An Introduction to Nanoscience by Nazarov & Blanter

Quantum Transport: Atom to Transistor by Datta

Quantum Theory of Conducting Matter: Newtonian Equations of Motion for a Bloch Electron by Fujita & Ito

Quantum Theory of Conducting Matter: Superconductivity by Fujita, Ito, & Godoy

Transport in Nanostructures by Ferry & Goodnick

Transport in Nanostructures (2nd Ed.) by Ferry, Goodnick, & Bird

The Physics and Applications of Resonant Tunnelling Diodes (Cambridge Studies in Semiconductor Physics and Microelectronic Engineering 2) by Mizuta & Tanoue

so if you can point to a section in any of these to help me understand exactly what you are trying to ask and to tell me, that would be helpful.

Oh Thanks sir,yes Please go to S.Datta :atom to transistor ,chapter number 11atom to transistor.11.2.2 coherent transport(one subband) sir.Please see it and please tell me how peaks are increasing sir.

Thanks

https://nanohub.org/resources/8679/download/2009.10.07-L25b-Klimeck-PISA.pdf

even sir it shows that ,peaks depends upon number of bound states between two scatterer sir

regards

It would be much more helpful if you could point to the sections you find relevant in the first or second edition of Transport in Nanostructures. For one thing, the link you state "shows that peaks depend upon the number of bound states..." doesn't mention bound states anywhere, nor does it refer to scatterers. It does refer to the density of states and how they show "the spatial and energetic 'location' of possible states" as well the following: "Transmission: shows spikes where the DOS is strong in the central RTD", describing these kind of results as "very unintuitive" and making similar remarks elsewhere such as "surprising energy crossings" and clear variances in the number of various ways in which different spiking reactions can occur, not that "there should be one" in any case. Moreover, they do use terms I am much more familiar with that I raised previously: transmission coefficients, transport coefficients, "dispersion 'complicated'", "transmission dramatically altered", "transmission coefficient", etc. What it doesn't say is anything about Ohm's law, energy vs. transmission (which is unfortunately vague and could mean many different things but none of can I seem to equate to your statement, and in general the power-point slides seem familiar to me in nature but don't seem to reflect what you have said, from your initial question to other details.

It would be very helpful if, when you want to e.g., point to a source that "shows that peaks depend upon the number of bound states between two scatterers", you use a source that actually uses the word "scatters", the term "bound states", or similar statements you've made previously. It is very hard to compare your statements about a source when what you describe the source as saying doesn't even appear in the source. That's why I was hoping you would use  the textbook you mentioned or something that you could use to point to something that actually reflects what you state (rather than a pdf slideshow that you describe using terms and words that do not appear anywhere in the file). Thanks!

Thanks sir for your kind reply.Sir as i have mentioned earlier message that it is in 11th chapter ,11.2.2 section

precisely Coherent Tansport (one subband)

So please see in eleventh chapter of S.Datta's book sir

I shall be highly thankful to you.

with regards

Book name is Quantum Transport atom to transistor,chapter 11 ;section i have mentioned sir.11.2.2 coherent transport (one sun and)

dear sir,it is not in the bok told you,but it is in Quantum Transport

This link might be clearer than the section from Datta's book: "Electron Transport as a Transmission Problem" (http://web.stanford.edu/~rsasaki/AP388/slide8)

Thanks for your answer.Need to think same thing in terms of Field Theory approach as well,like hopping and site energies.

Regards