I Orthogonal Basis of Periodic Functions: Beyond Sines and Cosines

[berkeman]

3) A mentor shuts it down as a waste of bandwidth.

Long thead is closed for Moderation...

 

[berkeman]

Thread is reopened provisionally.

 

[renormalize]

Thanks, is there a proof that it is the only solution anywhere?
Also, is it the same problem as the one I said in #109? Is e^+-ikx the only plane-wave solution for the wave equation?

May I ask the level of your knowledge about differential equations and their solutions? In post #114 I stated that the solution $\exp\left(-i\left(E/\hbar\right)t\right)\Psi_{0}$ to the time-dependent Schrödinger equation for constant E is unique. I know this because mathematicians long ago proved the existence and uniqueness of solutions to linear ODE initial-value problems. For example, see this page from the lecture notes at https://personalpages.manchester.ac...tYearODEs/Material/ExistenceAndUniqueness.pdf:

The substitutions:$$t\rightarrow x,\:\left|\Psi\left(t\right)\right\rangle \rightarrow y\left(x\right),\:i\frac{E}{\hbar}\rightarrow p\left(x\right),\:q\left(x\right)\rightarrow0$$along with the initial-value statement:$$\left|\Psi\left(0\right)\right\rangle =\Psi_{0}\rightarrow y\left(0\right)=\Psi_{0}$$ maps the Schrödinger equation precisely into the form of eqs.(3),(4), so the theorem applies and the unique solution is the exponential one found in post #114. And same is true for your plane-wave solution to the 2nd-order ODE it satisfies. The solutions must be exponentials and only exponentials. It's proven mathematics!

 

[anuttarasammyak]

As the titled mathematical question of "Orthogonal Basis of Periodic Functions: Beyond Sines and Cosines" there is nothing more to to talk with, I think. No periodic function case and continuity/discontinuity of base were also considered.

Beyond the titled question, @QuantumCuriosity42 seems to extend the question how to deduct photon energy of $n\hbar\omega$ from the mathematics. Photons have parameter of $(\omega,\mathbf{k})$ with relation of $\omega=c|\mathbf{k}|$. If we need another parameter for quantization, it would be the physics of another world.

My short attempt to reply the question.
Q: Why parameter is $\omega$ of vivbrational character $e^{i\omega t}$ ?
A: It comes from harmonic dynamics in quantization of em field. Harminics are common in light, sound, etc. But it is not universal. Non harmonic or non linear phenomena are also popular.

 

[QuantumCuriosity42]

May I ask the level of your knowledge about differential equations and their solutions? In post #114 I stated that the solution $\exp\left(-i\left(E/\hbar\right)t\right)\Psi_{0}$ to the time-dependent Schrödinger equation for constant E is unique. I know this because mathematicians long ago proved the existence and uniqueness of solutions to linear ODE initial-value problems. For example, see this page from the lecture notes at https://personalpages.manchester.ac...tYearODEs/Material/ExistenceAndUniqueness.pdf:
View attachment 335710
The substitutions:$$t\rightarrow x,\:\left|\Psi\left(t\right)\right\rangle \rightarrow y\left(x\right),\:i\frac{E}{\hbar}\rightarrow p\left(x\right),\:q\left(x\right)\rightarrow0$$along with the initial-value statement:$$\left|\Psi\left(0\right)\right\rangle =\Psi_{0}\rightarrow y\left(0\right)=\Psi_{0}$$ maps the Schrödinger equation precisely into the form of eqs.(3),(4), so the theorem applies and the unique solution is the exponential one found in post #114. And same is true for your plane-wave solution to the 2nd-order ODE it satisfies. The solutions must be exponentials and only exponentials. It's proven mathematics!

Thank you so much! Your post was insightful. (At least if Schrodinger equation was derived without supposing the premise you arrive at using it for a state of constant energy. If not it would be underwhelming.)
But then, what really means an "state of constant energy"? Is that a photon?
Also, the reasoning we use now to derive E=h*f as @PeterDonis told me in another thread (see screenshot) is that one you showed me @renormalize ?

 

[QuantumCuriosity42]

As the titled mathematical question of "Orthogonal Basis of Periodic Functions: Beyond Sines and Cosines" there is nothing left to to talk, I think. No periodic function case and continuity-iscontinuity of base were also considered.

Beyond the title @QuantumCuriosity42 seems to extend the question how to deduct photon energy of $n\hbar\omega$ from the mathematics. Photons have parameter of $(\omega,\mathbf{k})$ with relation of $\omega=c|\mathbf{k}|$. If we need another parameter for quantization, it would be the physics of another world.

My short attempt to reply the question.
Q: Why parameter is $\omega$ of vivbrational character $e^{i\omega t}$ ?
A: It comes from harmonic dynamics of quantization of em field. Harminics are common in light, sound, etc. But it is not universal. Non harmonic or non linear phenomena are also popular.

What do you mean by: "No periodic function case and continuity of base were already considered."?
I think that as of yet nobody told another valid basis of periodic orthogonal functions with different frequencies.
And what do you mean by: "If we need another parameter for quantization, it is the physics of another world."?
How do you know that?

 

[anuttarasammyak]

As the titled mathematical question of "Orthogonal Basis of Periodic Functions: Beyond Sines and Cosines" there is nothing more to to talk with, I think. No periodic function case and continuity/discontinuity of base were also considered.

Addition with some digression:

１
For function f of finete |f(x)|^2 in $(-\infty,+\infty)$, $\{\delta(x-t)\}$ is a complete set of basis.
Integral tarnsform
f(x)=\int f(t)\delta(t-x) dt
This identical transformation is obvious and not so interesting. Anyway it is a transform formally.
Orthogonality ＆Normalization
\int \delta(t-a)\delta(t-b) dt = \delta(a-b)
It shows also "normalization" for contimuous variable also to $\delta(0)$ as well as Fourier transform case.
Continuity
It can be regarded continuous in Schwartz method of continuous function sequence. Even derivatives exist which works as
\int f(t) \delta'(t-x) dt = - f'(x)
Periodicity
Not periodic.

PS Walsh-Fourier tranform basis is periodic.
ref. https://www.stat.pitt.edu/stoffer/dss_files/walshapps.pdf
It is discrete in popular sense, but we can regard them continuous as well as $\delta$ funciton is.
Actually derivative of Walsh functions are sum of periodical chain of $\delta$ functions.

２
In the last century wavelet transform of both discrete and continuous was found ref. https://en.wikipedia.org/wiki/Wavelet_transform
Scaling and shifting of a chosen mother wavelet seems a new idea apart from FT.
FT basis has infinite length which does not decay at all. It is mathematical ideal body but many physical phenomena take place locally. Wavelet transform with farther decaying mother wavelet might be convenient in such cases, I suspect.

 

[QuantumCuriosity42]

Addition with some digression:

１
For function f of finete |f(x)|^2 in $(-\infty,+\infty)$, $\{\delta(x-t)\}$ is a complete set of basis.
Integral tarnsform
f(x)=\int f(t)\delta(t-x) dt
This identical transformation is obvious and not so interesting. Anyway it is a transform formally.
Orthogonality ＆Normalization
\int \delta(t-a)\delta(t-b) dt = \delta(a-b)
It shows also "normalization" for contimuous variable also to $\delta(0)$ as well as Fourier transform case.
Continuity
It can be regarded continuous in Schwartz method of continuous function sequence. Even derivatives exist which works as
\int f(t) \delta'(t-x) dt = - f'(x)
Periodicity
Not periodic.

PS Walsh-Fourier tranform basis is periodic.
ref. https://www.stat.pitt.edu/stoffer/dss_files/walshapps.pdf
It is discrete in popular sense, but we can regard them continuous as well as $\delta$ funciton is.
Actually derivative of Walsh functions are sum of periodical chain of $\delta$ functions.

２
In the last century wavelet transform of both discrete and continuous was found ref. https://en.wikipedia.org/wiki/Wavelet_transform
Scaling and shifting of a chosen mother wavelet seems a new idea apart from FT.
FT basis has infinite length which does not decay at all. It is mathematical ideal body but many physical phenomena take place locally. Wavelet transform with farther decaying mother wavelet might be convenient in such cases, I suspect.

So the only other basis of periodic functions apart from sines/cosines, are walsh functions?
Wavelets are generic waves I think.

 

[anuttarasammyak]

So the only other basis of periodic functions apart from sines/cosines, are walsh functions?

Yes, as far as I know now. I know I do not know very much.

 

[Haborix]

Or the Mathieu functions already mentioned in this thread. Or the various Jacobi elliptic functions... But I'm not sure what we're getting at anymore.

 

[QuantumCuriosity42]

Or the Mathieu functions already mentioned in this thread. Or the various Jacobi elliptic functions... But I'm not sure what we're getting at anymore.

Mathie functions are not periodic. I think Jacobi elliptic functions are not orthogonal.

 

[QuantumCuriosity42]

Thank you so much! Your post was insightful. (At least if Schrodinger equation was derived without supposing the premise you arrive at using it for a state of constant energy. If not it would be underwhelming.)
But then, what really means an "state of constant energy"? Is that a photon?
Also, the reasoning we use now to derive E=h*f as @PeterDonis told me in another thread (see screenshot) is that one you showed me @renormalize ?
View attachment 335721

Does anyone know what is the other derivation "using different reasoning from the reasoning Placnk originally used" that he mentions on that screenshot?

 

[vanhees71]

The modern derivation of Planck's Law is of course to use QED at finite temperature. The result is the same.

 

[Haborix]

Mathie functions are not periodic. I think Jacobi elliptic functions are not orthogonal.

An infinite subset of Mathieu functions are periodic. You may be right about elliptic functions not being orthogonal, but if I recall they are complete so a basis could be constructed from them.

 

[QuantumCuriosity42]

The modern derivation of Planck's Law is of course to use QED at finite temperature. The result is the same.

Could you point me to the derivation please?

 

[QuantumCuriosity42]

An infinite subset of Mathieu functions are periodic. You may be right about elliptic functions not being orthogonal, but if I recall they are complete so a basis could be constructed from them.

I want a periodic and orthogonal basis, because if the only one satisfying both conditions are sine/cosine then that would explain why they are the fundamental frequencies.

 

[vanhees71]

We have repeatedly quoted other periodic orthornomal function systems. Why should sine and cosine the only periodic orthonormal functions?

 

[QuantumCuriosity42]

We have repeatedly quoted other periodic orthornomal function systems. Why should sine and cosine the only periodic orthonormal functions?

Ok, then that is set. And could you tell me where can I find the derivation of Planck law you told me before?

 

[QuantumCuriosity42]

This is the best explanation I found to my question as of yet, I don't know if it is correct or if it can be improved.
https://physics.stackexchange.com/q...r-frequencies-in-the-basis-of-sine-waves?rq=1
I want an explanation like that, but for E=h*f (and light colors) instead of for sound. :(
Is the EM radiation of an electron energy state transition a pure sinusoid? Why?

 

[QuantumCuriosity42]

These two newly released videos (today) are the best I've ever seen on the topic.




In the first video he ignores any other light pulse and goes with a monochromatic one without taking into account any other or explaining why.
But yeah, at the end of the second video she says we still don't know why light slows down in water with a non monochromatic pulse. Looks like my question was unexpectedly at the limit of today's current physics knowledge (She's a theoretical physicst).