What is the Usefulness of Reciprocal Space in Classical Physics?

[Azurite]

how do you describe the speed of an object in momentum space (energy, momentum as the 2 axes) where there is no distance or time? Can you give an example?

 

[Dale]

how do you describe the speed of an object in momentum space

Divide the object’s “position” in momentum space by its mass.

 

[Azurite]

Divide the object’s “position” in momentum space by its mass.

But how could an object have a position in the energy and momentum axis of momentum space which has no position or time? Can anyone shows any graph contrasting the distance/time plot versus energy/momentum part of any motion in motion like speeding train or ball thrown in the sky, etc. so I have visual idea of it? Thank you.

 

[Nugatory]

But how could an object have a position in the energy and momentum axis of momentum space which has no position or time? Can anyone shows any graph contrasting the distance/time plot versus energy/momentum part of any motion in motion like speeding train or ball thrown in the sky, etc. so I have visual idea of it? Thank you.

As in your other thread, you are confused about the meaning of the word "space". In this context, "space" is being used in the mathematical sense: https://en.wikipedia.org/wiki/Space_(mathematics)

 

[Azurite]

As in your other thread, you are confused about the meaning of the word "space". In this context, "space" is being used in the mathematical sense: https://en.wikipedia.org/wiki/Space_(mathematics)

I'm aware of it. Maybe to rephrase my questions;

"But how could an object have a position in the energy and momentum axis of which has no position or time? Can anyone shows any graph contrasting the distance/time plot versus energy/momentum part of any object in motion like speeding train or ball thrown in the sky, etc. so I have visual idea of it? Thank you."

 

[PeroK]

I'm aware of it. Maybe to rephrase my questions;

"But how could an object have a position in the energy and momentum axis of which has no position or time? Can anyone shows any graph contrasting the distance/time plot versus energy/momentum part of any object in motion like speeding train or ball thrown in the sky, etc. so I have visual idea of it? Thank you."

If you had a car, you could plot its "position" on a colour/make graph. The x-axis could be the make of the car and the y-axis the colour.

This is still a "position" (mathematical terminology) on a graph, but has no relation to physical position.

Another example might be a graph of economic data: your "position" on that graph might represent your socio-ecomomic status.

 

[Azurite]

If you had a car, you could plot its "position" on a colour/make graph. The x-axis could be the make of the car and the y-axis the colour.

This is still a "position" (mathematical terminology) on a graph, but has no relation to physical position.

Another example might be a graph of economic data: your "position" on that graph might represent your socio-ecomomic status.

Oh. I guess my question is. "But how could an object have a distance in the energy and momentum axis which has no distance or time axis?"

 

[PeroK]

Oh. I guess my question is. "But how could an object have a distance in the energy and momentum axis which has no distance or time axis?"

That's a different question; which I don't undestand. Distance from where?

 

[Azurite]

That's a different question; which I don't undestand. Distance from where?

Wait. Let me organize my thoughts.

The inverse of distance is number per unit distance which is a spatial frequency.
The inverse of time is number per unit time which is a temporal frequency.

Which of the above has the energy and momentum part? I guess wave number space is same as momentum space? (I know space doesn't mean our space but just word to denote graphs of mathematical object so no problem or confusion about this).

 

[PeroK]

Wait. Let me organize my thoughts.

The inverse of distance is number per unit distance which is a spatial frequency.
The inverse of time is number per unit time which is a temporal frequency.

Which of the above has the energy and momentum part? I guess wave number space is same as momentum space? (I know space doesn't mean our space but just word to denote graphs of mathematical object so no problem or confusion about this).

You've lost me. I would tend to say that the inverse of wavelength is spatial frequency and the inverse of period (time interval) is freqency. I thought we were talking about particles, not waves.

 

[Azurite]

You've lost me. I would tend to say that the inverse of wavelength is spatial frequency and the inverse of period (time interval) is freqency. I thought we were talking about particles, not waves.

Is it true reciprocal space is the same as momentum space? According to: https://en.wikipedia.org/wiki/Reciprocal_lattice

"In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, this first lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial function in real-space and is also known as the direct lattice. While the direct lattice exists in real-space and is what one would commonly understand as a physical lattice, the reciprocal lattice exists in reciprocal space (also known as momentum space or less commonly as K-space, due to the relationship between the Pontryagin duals momentum and position.) The reciprocal lattice of a reciprocal lattice, then, is the original direct lattice again, since the two lattices are Fourier transforms of each other."

How does reciprocal space differ to momentum space (if the above is wrong that they are identical)?

 

[PeroK]

Is it true reciprocal space is the same as momentum space? According to: https://en.wikipedia.org/wiki/Reciprocal_lattice

"In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, this first lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial function in real-space and is also known as the direct lattice. While the direct lattice exists in real-space and is what one would commonly understand as a physical lattice, the reciprocal lattice exists in reciprocal space (also known as momentum space or less commonly as K-space, due to the relationship between the Pontryagin duals momentum and position.) The reciprocal lattice of a reciprocal lattice, then, is the original direct lattice again, since the two lattices are Fourier transforms of each other."

How does reciprocal space differ to momentum space (if the above is wrong that they are identical)?

Well, this started as a "B" level thread, but that has gone beyond my knowledge. Perhaps someone else can help.

 

[Dale]

But how could an object have a position in the energy and momentum axis of momentum space which has no position or time?

That is why I put the word position in quotes. It isn’t really position, it is just the numerical values of the coordinates of the object in energy-momentum space. Take the numerical values of the coordinates of the object in energy-momentum space and divide by the mass and you get the four-velocity.

If the object has a definite energy and momentum then it has a definite “position” in energy momentum space, ie definite numerical values of its coordinates in energy momentum space.

Can anyone shows any graph contrasting the distance/time plot versus energy/momentum part of any motion in motion like speeding train or ball thrown in the sky, etc. so I have visual idea of it?

Why don’t you do this yourself. It is a good exercise, but it will take more time than I have. Plot the graph of an object moving along the x-axis accelerating under a uniform force. Then calculate the energy and the momentum, write a parametric equation describing the energy and momentum as a function of time. Plot that parametric equation and look at the graph.

 

[Azurite]

That is why I put the word position in quotes. It isn’t really position, it is just the numerical values of the coordinates of the object in energy-momentum space. Take the numerical values of the coordinates of the object in energy-momentum space and divide by the mass and you get the four-velocity.

If the object has a definite energy and momentum then it has a definite “position” in energy momentum space, ie definite numerical values of its coordinates in energy momentum space.

Why don’t you do this yourself. It is a good exercise, but it will take more time than I have. Plot the graph of an object moving along the x-axis accelerating under a uniform force. Then calculate the energy and the momentum, write a parametric equation describing the energy and momentum as a function of time. Plot that parametric equation and look at the graph.

What is the relationship of energy-momentum space to reciprocal space kx,ky,kz in wave number? how do you convert between these two?

 

[Dale]

What is the relationship of energy-momentum space to reciprocal space kx,ky,kz in wave number? how do you convert between these two?

I don’t know QM. Why don’t you focus on the classical mechanics aspect first. Once you have that down then you could ask about that aspect of the question in the QM forum and get good feedback.

 

[Azurite]

I don’t know QM. Why don’t you focus on the classical mechanics aspect first. Once you have that down then you could ask about that aspect of the question in the QM forum and get good feedback.

what I was asking was not QM.. I just want to know the difference between reciprocal space and momentum space.. some say they are the same.. some say not.. so I want to know the exact difference of them in classical mechanics...

 

[Dale]

what I was asking was not QM.. I just want to know the difference between reciprocal space and momentum space.

I was not aware that reciprocal space was even used in classical mechanics. Sorry, I cannot help you with this aspect of your question.

 

[ZapperZ]

what I was asking was not QM.. I just want to know the difference between reciprocal space and momentum space.. some say they are the same.. some say not.. so I want to know the exact difference of them in classical mechanics...

Where is this used in classical mechanics? And can you properly cite these "some" people who are saying this to you?

Zz.

 

[Azurite]

Where is this used in classical mechanics? And can you properly cite these "some" people who are saying this to you?

Zz.

It is used in Raman spectroscopy... where instead of wavelength.. they used 1/wavelength.. is this not classical physics?

I learned from wiki folks they used this in crystal and deBroglie pilot wave... where momentum is defined as p=ℏk. Hence reciprocal space is interchangeable to momentum space. But not in QM.

how does this differ to momentum of classical mechanics and vintage QM?

And if you have space.. and you get reciprocal space.. it's 1/space.. so I thought since space was classical physics.. then reciprocal space is classical physics... just inverse of spacetime.. this is when you want to describe waves.. how do you describe waves in classical physics.. is it not you put this in wave space or reciprocal space...?

 

[ZapperZ]

Where in classical physics does ‘h” appear?

Zz.

 

[Azurite]

Where in classical physics does ‘h” appear?

Zz.

What was meant was in the case of crystal and de Broglie physical pilot wave (not the Bohm imaginary one located in configuration space), you can state reciprocal space as equivalent to momentum space.. but I guess not in others... because in orthodox quantum, the wave is not a real wave (?)

Also wiki mentioned:

https://en.wikipedia.org/wiki/Reciprocal_lattice
Reciprocal space (also called "k-space") is the space in which the Fourier transform of a spatial function is represented (similarly the frequency domain is the space in which the Fourier transform of a time dependent function is represented). A Fourier transform takes us from "real space" to reciprocal space or vice versa. Reciprocal space comes into play regarding wave-mechanics:

Does this mean without Planck constant h.. you can never equate reciprocal state to momentum space? But there seems to be a close relationship which is... a function in momentum space is the Fourier transform of a function in position space. Comment?

 

[Dale]

It is used in Raman spectroscopy

QM

used this in crystal and deBroglie pilot wave... where momentum is defined as p=ℏk.

QM

then reciprocal space is classical physics... just inverse of spacetime.

Sure, but relating it to momentum space is not classical physics.

But there seems to be a close relationship which is... a function in momentum space is the Fourier transform of a function in position space. Comment?

My comment is that it is pretty rude of you to be pushing this here. Your OP said nothing about reciprocal space or linking it to momentum space. I answered your OP to the best of my knowledge, and when you tried to extend the question beyond my knowledge instead of pretending to know the answer and possibly feeding you bad information I let you know my limitations and gave you a recommendation on where you could go for an answer.

Instead of thanking me and asking your question in the QM forum, you have continued to prod me about a topic I don’t know. It seems like you are more interested in making me repeatedly admit my ignorance on this topic than you are about getting an answer. It is rude. Why are you doing it instead of just asking in the QM forum?

 

[Azurite]

QM

QM

Sure, but relating it to momentum space is not classical physics.

My comment is that it is pretty rude of you to be pushing this here. Your OP said nothing about reciprocal space or linking it to momentum space. I answered your OP to the best of my knowledge, and when you tried to extend the question beyond my knowledge instead of pretending to know the answer and possibly feeding you bad information I let you know my limitations and gave you a recommendation on where you could go for an answer.

Instead of thanking me and asking your question in the QM forum, you have continued to prod me about a topic I don’t know. It seems like you are more interested in making me repeatedly admit my ignorance on this topic than you are about getting an answer. It is rude. Why are you doing it instead of just asking in the QM forum?

Sorry nothing personal. I really thought it was classical because I heard a person talking about spacetime having an actual opposite called inverse spacetime. Maybe he is wrong. I'm just confused. I'll ask this in the QM. Thanks really to all for sharing.

(added: good to know about this info that linking it to momentum space is not classical physics... thanks for this.. I continued asking in messages prior so I can be corrected.. I thought it can still be classical physics.. or maybe inverse spacetime is not momentum space at all)

 

[ZapperZ]

Sorry nothing personal. I really thought it was classical because I heard a person talking about spacetime having an actual opposite called inverse spacetime. Maybe he is wrong. I'm just confused. I'll ask this in the QM. Thanks really to all for sharing.

(added: good to know about this info that linking it to momentum space is not classical physics... thanks for this.. I continued asking in messages prior so I can be corrected.. I thought it can still be classical physics.. or maybe inverse spacetime is not momentum space at all)

There are two things that should be cleared off here:

1. My training is in condensed matter/solid state physics. And I've done XRD, LEED, RHEED techniques in which one maps the data to reciprocal space of the lattice. So yes, I am quite familiar with this area.

2. There ARE quantities in classical physics that have the dimension of 1/length. A very simple example of this is the wavenumber "k" that even intro general physics students have been introduced to. HOWEVER, this is different than what you are trying to imply, i.e. there is a "usefulness" of reciprocal space in classical mechanics at the same level as what we deal with, say, in solid state physics. If there is, I don't see it, or at least, it isn't a common methodology in a typical physics curriculum.

Note that at the undergraduate level solid state physics, we introduce the concept of reciprocal space very early in the course (in some text, it is even in Chap. 2!). So there is a very clear introduction to this concept, and it is upfront. Have you seen a similar level of usage and "publicity" on reciprocal space in classical physics?

Zz.