# On a consequence of quantanization of energy

## Main Question or Discussion Point

Since energy units are J = kg (m/sec)^2, and this quantaty is discrete, if one of the units is continious, then it can be either irrational or rational number and could make J any irrational or rational number as well. In other words for every real number n of a unit in continuum there is a number J such that J is also real. That is contarary to the supposition, for then J is also going to be in continuum . Thus mass, distance and time all have to be discrete.

## Answers and Replies

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ZapperZ
Staff Emeritus
Eppur si muove said:
Since energy units are J = kg (m/sec)^2, and this quantaty is discrete, if one of the units is continious, then it can be either irrational or rational number and could make J any irrational or rational number as well. In other words for every real number n of a unit in continuum there is a number J such that J is also real. That is contarary to the supposition, for then J is also going to be in continuum . Thus mass, distance and time all have to be discrete.
What the.....?

Who said E HAS or MUST be discrete? You should not make such statement without understand the ORIGIN of discrete or quantized values in QM. Look at the energy of a free particle in QM. Is that quantized or discrete? Or what about the energy of the conduction band in a metal? Is that discrete?

Pay attention to the BOUNDARY CONDITIONS. These are a significant reason on why certain things have quantized properties while others don't. Do not misuse the results of QM this way.

Zz.

I think there is couter examples to what you said.

Think of a body in which its mass has a constant value in time and equals to 12,3 kg.

Assume you have a spring the body is attached at. Now supose you can only start from initial positions which constitutes a set of discrete points in the x direction (x = 1, 2, 3, 4....for example). Supose also that the initial velocity is always zero (or constitutes also a discrete set like v_0 = 2, 3.7, 8.45, 1000.0 ,...).

In this scenario, the mechanical energy will always have discrete values, and therefore will not be continuous, although space and time is.

Observe that potential and kinetics energies go through continuous transformations taken separatelly.

Best Regards,

DaTario

Space and time do not have to be continuus. And how can energy be either continuus or discrete under different conditions?The conditions are defined by the units of mass, distance and time and if one of them is allowed to be continuus, then it should be possible to have continuus energy values. Yet in most cases, like in the case of a photon only discrete values are allowed. Can you not see the contradiction? They all have to be discrete.

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Doc Al
Mentor
Eppur si muove said:
And how can energy be either continuus or discrete under different conditions? The conditions are defined by the units of mass, distance and time and if one of them is allowed to be continuus, then it should be possible to have continuus energy values.
Sorry, but you don't know what you are talking about. Take an electron for example. For an electron bound to a nucleus, the system is allowed only discrete energy states. But for a free electron, the allowed values of energy form a continuum.

Hi,

In the Bohr atom it's the energy levels that are integers, not the value of the quantized energy of the electrons themselves.

For photons, each photon has a packet of energy, but this energy does not have to be in the form of an integer.

juju

ZapperZ
Staff Emeritus
Eppur si muove said:
Space and time do not have to be continuus. And how can energy be either continuus or discrete under different conditions?The conditions are defined by the units of mass, distance and time and if one of them is allowed to be continuus, then it should be possible to have continuus energy values. Yet in most cases, like in the case of a photon only discrete values are allowed. Can you not see the contradiction? They all have to be discrete.
Yet, you completely ignored a COUNTER example that I gave you. Can you not see the contradiction to what you are saying? Experiments trump beliefs everytime!

Zz.

Experiments? No experiment can ever show that allowed values for a free electron form a continuum. The Uncertainty Principle will make it impossible for an error to me as small as desired thus resulting in error intervals and that makes it impossible to verify a continuum of a variable. Again, what I ask of you are not the accepted 'thruths' but a solution to a logical argument which I have shown above. The argument itself is flawless, but if you claim that its conclusion is false, show me where I made a wrong proposition. I neither assume that energy is discrete or that it is not, but rather consider both possibilities and it turns out that since having any of the units that make up a joule results in the possibility of having an energy continuum under any conditions, that conclusion being contradicted by the energy values of photons and bound electrons, it follows by converse that neither a time unit nor a distance unit or a mass unit can be in continuum. Surely you don't think that space or time or both would change their properties depending on whether an electron is bound or not.

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ZapperZ
Staff Emeritus
Eppur si muove said:
Experiments? No experiment can ever show that allowed values for a free electron form a continuum. The Uncertainty Principle will make it impossible for an error to me as small as desired thus resulting in error intervals and that makes it impossible to verify a continuum of a variable.
And THINK about it. If what you say is true, then it is IMPOSSIBLE to also determine that energy must always come quantized also! Get it? I have to make an experimental determination of quantization ALSO! And since my energy measurement is spread out due to the uncertainty principle, there's ZERO ABILITY to determine that it came in discrete quantized values!

... or are you stating that your idea requires no experimental observation?

You STILL have not addressed that, based on QM, the conduction BAND is a BAND, and not discrete energy level. This is the SAME Quantum Mechanics that produced the uncertainty principle that you so like to quote without understanding. Or have you ever looked at the spectral lines of hydrogen using a spectrometer, and THEN compare that with the spectrum from an incandescant light bulb? No? Yes?

Either you address the issue of the conduction band in solids (open a solids state text) which was DERIVED using quantum mechanics, or you refrain from speaking about "truths" since you have been avoiding them like a plague. If you continue with this line without addressing something that's is staring at you right in your face, then you are trying to introduce something that is OUTSIDE of the currently accepted ideas in physics. If that is the case, then I will point out to you PF's policy that you have explictly agreeded to and will refer you to the IR section for the submission of your "ideas". It no longer belongs in this part of PF.

Zz.

I think the misunderstanding comes from talking about energy per se. It is always the energy of something, e.g.
- a free electron: nothing there to forbid energy values, so its energy spectrum is continuous
- a bound electron: it is bound, and can't have any energy it likes -> its energy spectrum is discrete

In your argument, you don't consider these boundary conditions. But they are important for the distinction between the physical quantity "energy" and the values of this quantity that the system has access to.

I do understand the uncertainty principle quite well and agree with you on the fact that it is impossible to determine whether a variable is quantazed or continuus by means of a direct measurement. However, the photoelectric effect can only be explained with energy quantized. Now, how is it verified that in a conduction band electrons are allowed to have energy values in continuum? Surely not by an experiment, by what then - a hypothesis? And I do realize also that Incandescent Bulb Emission Spectrum looks much different from that an emisson spectrum of a hydrogen or helium. But is it not possible that the quanta of energy are so small that the intervals are small enough for our instruments to be unable to show that in fact the light intensity at 600.005nmon on this graph - Incandescent Bulb Emission Spectrum for example is zero? How is your statement that free electrons' energy values are in continuum or that the incandescent light bulb emission spectrum is continuus are derived from QM? What you speak of are hypotheses not truths. And you provide no argument in your defence except relience on those 'truths'. Again you haven't shown a flaw or a contradiction in my argument as it is. If one says that energy is discerete for bound electrons because they are the only values it has access to than one implies that the electron consequently has access only to certain values for distance, time and mass. Yet for a free electron time for example is allowed to be in continuum. And how can the nature time change depending on whether an electron is bound or not?

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ZapperZ
Staff Emeritus
Eppur si muove said:
I do understand the uncertainty principle quite well and agree with you on the fact that it is impossible to determine whether a variable is quantazed or continuus by means of a direct measurement. However, the photoelectric effect can only be explained with energy quantized. Now, how is it verified that in a conduction band electrons are allowed to have energy values in continuum?
Because we OBSERVE, via the photoemission experiment, that there is a CONTINUOUS E vs k dispersion curve! See http://arxiv.org/abs/cond-mat/0507653, especially the Mo(110) surface state, which is a "standard" metallic example. Or look at my avatar. That's PHOTOEMISSION experimental data of E vs k. It isn't discrete! And I did the experiment myself!

There's a difference in claiming that photon energy comes in "discrete clumps". NO ONE is disputing that. But you then assume that ALL energies are "discrete" or quantized. That's horribly wrong! Go open the 1st two chapters of Ashcroft and Mermin's Solid State Text. You'll see the dispersion curve for a metal and you'll see that for the Bloch wavefunction, the energy levels are CONTINUOUS!

So what's the difference between the quantized energy level in an atom, and the continuous energy BAND in solids? The BOUNDARY CONDITIONS, which is what I mentioned in the very first reply in this thread. For some reason, you either missed that, or did not realize the importance of that.

Photon energy comes in discrete quanta. But the RANGE of frequency that determine the energy in each photon depends entirely on the nature of the source! I can use a synchrotron facility and literally dial in ANY frequency I want, limited only to my ability to position the undulate/wiggler in the ring! It isn't quantized!

Zz.

The graphs that you have shown above are not much diffrent from the example I cited myself in that that they appear to our human eye to be continuus but this is by no means a verification or a proof that it is so. Refer to my previous reply. They might be suggestive enough to a human eye just like a tv images seem continuus from a distance. Have you proved that it is impossible to have the energy of a free electron quantized? I want you to consider my argument again more carefully. The units of distance and time cannot change their properties depending on whether an electron is bound or not. Since the Photoelectric Effect demonstrates and verifies without any doubt that energy is discrete or quantized it is thus the only possible solution. Hence, all energies must come in only certain discrete amounts even if the scale is so fine that it appears to be continuus based on our crude measurements. And of course your ability to postion the undulator/wiggler magnets in your synchroton facility is not even close to the accuracy needed to show that it is in fact possible to dial in every frequency resulting in photon energies over a continuum.

Eppur si muove said:
The units of distance and time cannot change their properties depending on whether an electron is bound or not.
Who says they do?

A VERY macroscopic analogy: Imagine a plane, i.e. a field of continuous space. Take your dog for a walk on this field.
If the dog is free, it can go wherever it wants, i.e. it can have every position in this continuous space.
If the dog is on a lead, it can only stay in a discrete part of the space, but the nature of space doesn't change, it is still continuous.

You can't detach energy from its carrier in a physical experiment!

ZapperZ
Staff Emeritus
Eppur si muove said:
The graphs that you have shown above are not much diffrent from the example I cited myself in that that they appear to our human eye to be continuus but this is by no means a verification or a proof that it is so. Refer to my previous reply. They might be suggestive enough to a human eye just like a tv images seem continuus from a distance. Have you proved that it is impossible to have the energy of a free electron quantized?

YES! Open a QM text! Look at the wavefunction of a free particle. Now find the discrete values of E or p if you can!

I want you to consider my argument again more carefully. The units of distance and time cannot change their properties depending on whether an electron is bound or not. Since the Photoelectric Effect demonstrates and verifies without any doubt that energy is discrete or quantized it is thus the only possible solution.
This is bogus! You looked at the photoemission data and you say I cannot make any claims about a continuous E vs. k. YET, you used the same experiment to prove that it is discrete or quantized? What kind of a hoax is this?

Take a bunch of tennis balls. Shoot them one at a time with a constant velocity at a detector that measure its kinetic energy. The detector will measure this energy in CLUMPS. This is what we mean by saying it comes in discrete quanta. HOWEVER, there's nothing to say that I cannot vary the VELOCITY continuously! It means that while it comes in clumps, the energy that each tennis ball carries can vary CONTINUOUSLY! So if I plot the energy the detector measure versus the velocity, it is a CONTINUOUS curve.

Hence, all energies must come in only certain discrete amounts even if the scale is so fine that it appears to be continuus based on our crude measurements.
And I can ask you the same thing that you asked me. Have you DERIVED such "discrete energies" for a free particle?

You are forgetting that the band structure of solids are DERIVED theoretically, and verified experimentally. Show theoretically that such bands are discrete. If you cannot show that in your NEXT posting, I will lock this thread because you will then be making speculative posting that violates the PF rules that you have explictly agreed to. You are then welcome to propose your idea to be considered for submission to the IR section of PF.

Good luck!

Zz.

Here is a purely theoretical argument that all energy no matter its boundary conditions should come in discrete amounts:

Let us consider the energy of a free electron, suppose that energy is continuus, or that J is a real number E in continuum. Since J = kg(m/sec)^2, and if kg, m and sec are associated with numbers K, M and S corrispondigly, then E = K(M/S)^2 and at least one of either K, M or S has to be a real number in continuum. Thus we have derived from the original supposition that at least one of the three - time, space(distance) or mass have to be continuus. And let us also say that the properties of space, time and mass should not change depending on which physical phenomenon being considered. Let us now take into account the case when an electron is bound, clearly energy now comes only in certain amounts and is discrete. Thus E is not in continuum, and therefore M,K or S are not in continuum either.This fact can be very easily shown algebraicly. If at least one of the S, K or M are in continuum, then the expression K(M/S)^2 can easily take on every single value in continuum. For example if K is in continuum, then let K=N/(M/S)^2 where N is a real number in continuum, then by substitution E=N and thus E is also in continuum. That is contrary to the supposition, and similar arguments can be easily shown for M and S. It is clear then that neither M,K, or S can be continuus if E is not. Thus we have verified that according to the second instance where energy is discrete, - space, time and mass all have to come in discrete amounts. That however contradicts our first derivation which states that at least one of the three - space, time or mass have to be contiuus. Since we assumed that the properties of space time or mass cannot change depending on which physical phenomenon is considered there should be only one unique solution thus one of the solution has to be wrong. Energy either comes in discrete amounts or it is continuus. Boundary conditions play no role here, for I consider the fundamental properties of space, time and mass which do not change whatever the boundary conditions.Experiments show without any doubt that there is undisputable evidence that energy has to come in discrete amounts in the case of photolectric effet, bound electrons and photons. That energy has to be contiuus in the case of free electrons is not verified by any experiements without doubt. No graph can ever serve as a proof. Neither can a textbook.

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The kinetic energy of a free particle is continuous. This can only be otherwise
if you assume that different intertial frames of refererence are constrained to certian
quantized velocities.

Quantized momentum is one thing, but quantized velocity? That's got problems...

ZapperZ
Staff Emeritus