spectra for FS and FT

Hi

steveB said:

I'll start with questions Q2 and Q3.

Where did you get the idea about needing a symmetric function when the DC component is subtracted? It's not even clear what your definitions of symmetric and symmetrize are.

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I have re-read my queries, Q2, and Q3, and think that I stated my confusion as clearly as possible. So, I would request you that you have another look on what I said there, if possible, so that you can know what goes inside my mind and how to guide me in right direction. Actually in Q2 I was telling you how I view the DC component, and in Q3 I was trying to show you that why my so-called understanding of the DC component is flawed. Okay, let me state it otherwise. What is the DC component in FS and what does it tell us? When is the DC component zero, and when it's zero, what does it tell us? Thank you.

steveB said:

Q5: Tricky question!. I view the mathematics as correct and exact. But I view all physics theories we know as an approximation. Even quantum mechanics is well known to be only approximately correct. The interesting thing is that non-relativistic quantum mechanics and electromagnetic field theory in vacuum are truly linear theories. That is, they are exactly linear in the mathematical sense, and hence the Fourier theory is an exactly correct and equivalent viewpoint within the theory. However, we know those theories are only approximate representations or models of reality. So, you have to make your own interpretation here. I think the question is not provable one way of the other, and it is a viewpoint we might hold on to. I don't have a strong opinion one way or the other.

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If you don't mind, I would say that I'm not satisfied with this reply, or you can say I'm missing your point. For the sake of continuity, the following was my question.

Is mathematical FS or FT representation of some function an exact replicate of that function? For example, if you have connected a wire to a source supplying an electric current in form of a square wave, then do you think that electric current is flowing in form of pulses (i.e. square wave) or in form of odd harmonics of sine wave? Putting it differently, do you think electrons are moving form of pulses (i.e. square wave) or odd harmonics of a sine wave?

I don't think it's matter of viewpoint we hold on to because FS or FT is mathematics and not metaphysics. Let me elaborate on this. Light having wavelength range of roughly 630-760 nm produces sensation of red color in a human brain. But how can I know that the sensation that I see as red doesn't produce in you the sensation that I see as green? We both might agree that the color is red, but how can I know that you see the color red in the same way as I see it? It's a metaphysical situation. But wavelength range of 630-760 nm is an exact mathematical formulation and it can be confirmed scientifically. So, if you say that it's just a matter of viewpoint that whether 'square wave electric current' exists as pulses or as odd harmonics of a sine function then I don't get it because physically either current can exist as pulses or odd harmonics and not both. I hope you can see where I'm having trouble. Thank you.

Regards
PG

PG1995 said:

I have re-read my queries, Q2, and Q3, and think that I stated my confusion as clearly as possible. So, I would request you that you have another look on what I said there, if possible, so that you can know what goes inside my mind and how to guide me in right direction. Actually in Q2 I was telling you how I view the DC component, and in Q3 I was trying to show you that why my so-called understanding of the DC component is flawed. Okay, let me state it otherwise. What is the DC component in FS and what does it tell us? When is the DC component zero, and when it's zero, what does it tell us?

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I still am confused by the original questions, but I can address these last question you stated because they are quite clear, even if not easy to answer.

The DC component of FS is simply the average value of the function over any time period that is an integer multiple of the period T. It tells us the overall offset of the function and probably some other things depending on the situation. Obviously in circuits, the DC value is something that we are concerned with and we often measure it and talk about it. The DC component is zero when there is no overall offset or when the average is zero over a period. With thought, I'm sure we can come up with more comments, but I view these questions as somewhat useless. If you have to ask the meaning of something after you defined it, then you don't probably don't need the meaning. When you need the meaning, the meaning will be obvious and you don't even need to think about it. It's like asking what a rock tells us. At first it tells us nothing. A rock is defined to be an object made of particular materials. However, a geologist can probably tell us many more things that the rock tells us.

PG1995 said:

Is mathematical FS or FT representation of some function an exact replicate of that function?

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I would say yes. Within the context, assumti0ns and foundations of the mathematics, the representation is an exact one.

PG1995 said:

For example, if you have connected a wire to a source supplying an electric current in form of a square wave, then do you think that electric current is flowing in form of pulses (i.e. square wave) or in form of odd harmonics of sine wave? Putting it differently, do you think electrons are moving form of pulses (i.e. square wave) or odd harmonics of a sine wave?

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I think it is both at the same time. You can see the pulse, but you can also filter the pulse and see the harmonics if you want.

PG1995 said:

I don't think it's matter of viewpoint we hold on to because FS or FT is mathematics and not metaphysics.

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FS and FT is indeed mathematics and not metaphysics, but your question seems to be more a matter of physics, not mathematics and not metaphysics. In physics we deal with viewpoints and interpretations all the time. The viewpoints and interpretations don't really hold much value, other than to make us feel comfortable. I don't mean that to say they are not important or worthy of consideration, but the interpretation does not change the answer to the physics calculations. Consider the quantum mechanics interpretations discussed here. https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics

PG1995 said:

Let me elaborate on this. Light having wavelength range of roughly 630-760 nm produces sensation of red color in a human brain. But how can I know that the sensation that I see as red doesn't produce in you the sensation that I see as green? We both might agree that the color is red, but how can I know that you see the color red in the same way as I see it? It's a metaphysical situation. But wavelength range of 630-760 nm is an exact mathematical formulation and it can be confirmed scientifically.

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Color blind people don't see color the same as we do, but if we take a person with normal vision and show him red light with a scientifically verified frequency of 632 nm, then that person always says the color is red. So, color is a definition by association. We are born and are told the names of colors we see. We then associate those colors with those names. We know that colors can combine to produce other colors, so a green color we see may actually have no green wavelengths in it. Blue and yellow can make green, for example. So, this is a very confusing thing to talk about and you could spend many hours discussing it.

PG1995 said:

So, if you say that it's just a matter of viewpoint that whether 'square wave electric current' exists as pulses or as odd harmonics of a sine function then I don't get it because physically either current can exist as pulses or odd harmonics and not both. I hope you can see where I'm having trouble. Thank you.

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What I was trying to say is that the pulses and the harmonics are the same thing within a linear theory, but if the physics is not truly linear and we are using an approximate linear theory, then they may not be the same, or it may be hard to prove they are the same. In this case, I think we do cross over into viewpoints. Why do you say that an physical reality can not be both a pulse and a set of harmonics? Duality is an recurring theme in physics. There is a duality in almost all mathematics used in physics. We have wave/particle duality, frequency/time duality, position/momentum duality, covariant/contravariant duality, holographic duality etc. Electrons exhibit properties of both particles and waves. Is an electron really a particle or really a wave? I doubt it, but we always choose one viewpoint in a particular situation.

Thank you.

steveB said:

I still am confused by the original questions, but I can address these last question you stated because they are quite clear, even if not easy to answer.

The DC component of FS is simply the average value of the function over any time period that is an integer multiple of the period T. It tells us the overall offset of the function and probably some other things depending on the situation. Obviously in circuits, the DC value is something that we are concerned with and we often measure it and talk about it. The DC component is zero when there is no overall offset or when the average is zero over a period.

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I hope you don't mind me asking few more questions. I believe I'm not able to know what is really confusing me. The DC component of FS has a value equivalent to average value of a function over its period. Why do we need DC component in FS? This is the reason I can think of. The FS of a function uses sinusoids the interference of which generates the given function. The average value of a sinusoid over its period is zero therefore average value of the all the sinusoids of a FS is zero. Therefore, to be an exact representation of the function, the FS should have an offset so that the combined average value of all the sinusoids is equivalent to that of the function. Do I make sense?

So far I have seen that DC component always has positive value, can't it have negative value? Can't be the offset for a sinusoid negative?

steveB said:

FS and FT is indeed mathematics and not metaphysics, but your question seems to be more a matter of physics, not mathematics and not metaphysics. In physics we deal with viewpoints and interpretations all the time. The viewpoints and interpretations don't really hold much value, other than to make us feel comfortable. I don't mean that to say they are not important or worthy of consideration, but the interpretation does not change the answer to the physics calculations.

Duality is an recurring theme in physics. There is a duality in almost all mathematics used in physics. We have wave/particle duality, frequency/time duality, position/momentum duality, covariant/contravariant duality, holographic duality etc. Electrons exhibit properties of both particles and waves. Is an electron really a particle or really a wave? I doubt it, but we always choose one viewpoint in a particular situation.

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I agree with you. I believe we have this duality principle just to facilitate ourselves because we don't understand the reality fully. Einstein also didn't accept this duality principle. But this does not mean Einstein cannot be wrong. So, an electron is either a particle, wave or something having features of both. But it cannot be a particle in one situation and a wave in another. Please correct me if I'm wrong. Thanks.

Regards
PG

PG1995 said:

I hope you don't mind me asking few more questions.

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Not at all.

PG1995 said:

I believe I'm not able to know what is really confusing me. The DC component of FS has a value equivalent to average value of a function over its period. Why do we need DC component in FS? This is the reason I can think of. The FS of a function uses sinusoids the interference of which generates the given function. The average value of a sinusoid over its period is zero therefore average value of the all the sinusoids of a FS is zero. Therefore, to be an exact representation of the function, the FS should have an offset so that the combined average value of all the sinusoids is equivalent to that of the function. Do I make sense?

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You make perfect sense. You answered your own question better than I could.

PG1995 said:

So far I have seen that DC component always has positive value, can't it have negative value? Can't be the offset for a sinusoid negative?

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Yes, of course. Offset can be positive or negative.

PG1995 said:

I agree with you. I believe we have this duality principle just to facilitate ourselves because we don't understand the reality fully. Einstein also didn't accept this duality principle. But this does not mean Einstein cannot be wrong.

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Yes, I agree. But, also don't forget that the mathematics shows much of the duality I mentioned. And, yes, of course Einstein can be and, in fact, was wrong about many things. A physicist's job forces him to be wrong much more often that he is right. Mistakes and errors are not any indication of the quality of a great physicist. It is the times that he gets it right, long before any one else has the proper tools to get it right, that makes him great, and by this standard Einstein was the greatest of all time.

PG1995 said:

So, an electron is either a particle, wave or something having features of both. But it cannot be a particle in one situation and a wave in another. Please correct me if I'm wrong. Thanks.

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As to, "what an electron can be" and "what an electron is", I have no ability to say. However, I can say that we treat the electron as a particle in one situation and as a wave in another.

Thank you, Steve.

Q1:

steveB said:

Yes, of course. Offset can be positive or negative.

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But so far I have never seen or solved a FS problem with negative DC component or offset. Could you please show me some function which has negative offset? Google didn't help me.

steveB said:

As to, "what an electron can be" and "what an electron is", I have no ability to say. However, I can say that we treat the electron as a particle in one situation and as a wave in another.

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I think a small discussion about wave-particle duality is in order here. I will be using this text. The wavelength for a particle of matter of momentum, p, is given as:

[LATEX]\lambda =\frac{h}{p}=\frac{h}{mv}[/LATEX]
The de Broglie wavelength is inversely proportional to both mass and velocity of the particle. The value of Planck's constant is 6.63 x 10^-34 Js or J/Hz. Also note that relativistic mass is given as:

[LATEX]m=\frac{m_{o}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}[/LATEX]
Q2:
A moving stone will have a very, very tiny wavelength. As the wavelength is inversely proportional to both mass and velocity, therefore decreasing any of these quantities will make wavelength longer and more easily observable, right? But the document says that the concept of matter having a wavelength only becomes significant at very high velocities, why is so? Why isn't it significant at lower velocities? This means that though one can reduce the mass to make the wavelength more observable, reducing the velocity won't help at all.

This begs another question that will increasing the velocity help in any way? It's obvious that the wavelength will become smaller if the velocity is increases. At this point, it should also be noted that relativistic mass becomes greater as velocity is increased.

Q3: Everywhere the de Broglie wavelength of matter particles is discussed, no one ever talks about amplitude or period of the particles. I think these two parameters are equally important. For example, when amplitude is very, very small then wave motion won't be easily apparent.

Q4:
In diffraction experiments such as the famous Young's double-slit experiment, the size of the slit shouldn't be much larger than the size of the wavelength of light under observation. If the size of slit is larger then interference pattern won't be sharp and only a few fringes are formed. If we are to perform diffraction experiment on electrons then should the size of the slit (I think in this case some crystal will work as a diffraction grating and distance between the molecules will determine the size of the grating) reasonably close to the wavelength size of electrons? What if the size of wavelength is larger than the diameter of electrons, this way electrons won't be able to pass through the slit? Thank you.

Regards
PG


Helpful links:
1: https://en.wikipedia.org/wiki/Double-slit_experiment
2: https://en.wikipedia.org/wiki/Davisson–Germer_experiment

PG1995 said:

But so far I have never seen or solved a FS problem with negative DC component or offset. Could you please show me some function which has negative offset? Google didn't help me.

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Take any function you have been dealing with lately and add -10737383 to it. This will obviously have a large negative offset.

Another example would be something like y=t^2-0.5 from t=0 to t=1 , and then make it periodic with period T=1.

PG1995 said:

A moving stone will have a very, very tiny wavelength. As the wavelength is inversely proportional to both mass and velocity, therefore decreasing any of these quantities will make wavelength longer and more easily observable, right? But the document says that the concept of matter having a wavelength only becomes significant at very high velocities, why is so? Why isn't it significant at lower velocities? This means that though one can reduce the mass to make the wavelength more observable, reducing the velocity won't help at all.

This begs another question that will increasing the velocity help in any way? It's obvious that the wavelength will become smaller if the velocity is increases. At this point, it should also be noted that relativistic mass becomes greater as velocity is increased.

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You are biting off a lot with this question. I can't answer this properly and be brief. And, you would probably need a good physicist to answer this really well. One of the issues is that you are mixing relativity and quantum mechanics, which strictly gets you into quantum field theory and is beyond basic non-relativistic quantum mechanics.

Basically, a large mass is not going to display quantum effects noticeably, so whatever the wavelength actually is, you arent' going to notice it. You will be able to localize both position and velocity (or momentum). When the mass is small, you can start to notice the uncertainty principle and you can't localize both position and velocity. Now, you have wave effects.

The other aspect is that if you did a diffraction experiment of electrons through crystals (as you mentioned below), a low speed will create a wavelength too long to interact with the small lattice spacing on the crystal. So this is very tricky stuff. You need to get the mass and speed in a range where the effects will be noticeable.

I know there is much more to your question, but you really need to study this in great depth to get a good handle on it.

PG1995 said:

Everywhere the de Broglie wavelength of matter particles is discussed, no one ever talks about amplitude or period of the particles. I think these two parameters are equally important. For example, when amplitude is very, very small then wave motion won't be easily apparent.

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Sure, those can be important if you are doing experiments or engineering that involve such things. But, most of us dont' need to worry about it. So, I wouldn't say no one ever talks about it. When was the last time you went to a physics conference specializing in a sub-field that cares about these things? (joking)

PG1995 said:

In diffraction experiments such as the famous Young's double-slit experiment, the size of the slit shouldn't be much larger than the size of the wavelength of light under observation. If the size of slit is larger then interference pattern won't be sharp and only a few fringes are formed. If we are to perform diffraction experiment on electrons then should the size of the slit (I think in this case some crystal will work as a diffraction grating and distance between the molecules will determine the size of the grating) reasonably close to the wavelength size of electrons? What if the size of wavelength is larger than the diameter of electrons, this way electrons won't be able to pass through the slit? Thank you.

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Yes, you are correct about these experimental demonstrations. The details of of how effectively high speed electrons can pass through matter are not something I know well. Obviously there is a range of wavelength, as compared to lattice constant, that needs to be maintained. I seem to recall that very thin pieces of metal are used in diffraction experiments so that the beam can get through (actually, i think this was for xrays, not electrons. I guess they just reflect off the surface for electrons).

Keep in mind that these types of questions can be researched in great depth by delving into the actual published papers, some of which are very famous. Anyway, good questions and poor answers from me. Maybe someone else here can give more details.