10 votes

Can we quantify the pitch of a sound that is a mixture of many frequencies?

Pitch can be described as a subjective perception of an auditory stimulus which cannot be objectively, unambiguously quantified. It is strongly related to the objective physical property of frequency ...
user avatar
7 votes
Accepted

Integration of Laplacian by parts

Integrating by parts once gives: $\int d^3 r e^{-i\vec k\cdot \vec r}\nabla^2\phi(\vec r) = \int d^3r e^{-i\vec k\cdot \vec r}\vec \nabla\cdot \vec \nabla\phi(\vec r) = \int d^3r \vec \nabla\cdot\left(...
user avatar
  • 7,614
5 votes

Can we quantify the pitch of a sound that is a mixture of many frequencies?

Short Answer: Use the lowest of your frequencies, also known as the base tone, to represent the pitch. Full Answer: First off, it seems worth noting that many sounds that consist of many frequencies ...
user avatar
4 votes

Can we quantify the pitch of a sound that is a mixture of many frequencies?

The technical, quantitative definition of pitch only applies to "single sounds" such as music notes. Here, the pitch is the fundamental frequency. However, when applied to "multiple ...
user avatar
  • 7,174
3 votes
Accepted

Propagators in Frequency Space and Dyson's Expansion

From RHS of your last equation, firstly, you can take out the factor involving $t_1$, i.e. $\int_{-\infty}^{\infty}dt_1e^{-i(\omega'-\omega)t_1}=2\pi \delta(\omega'-\omega)$. Then the integral over $\...
user avatar
  • 441
3 votes
Accepted

Fourier transform of propagator in Lehmann spectral representation

Rewriting the labels in your last equation, we have $$iG(k,t)={i \over 2\pi}\int_{-\infty}^{+\infty}{d\omega A(k,\omega)\int_{-\infty}^{+\infty}{d\omega' {e^{-i\omega't} \over \omega' - (\omega - i\...
user avatar
  • 448
3 votes
Accepted

Doubt in position operator acting on fourier transform, $\tilde{f}(k)$ of $f(x)$

I'll try my best to answer each of your questions in the following: Yes, it should be $dx$ not $dk$ (it's a typo) No, you can still take the position operator inside the integral. This is probably a ...
user avatar
  • 116
3 votes
Accepted

Question on quantum field theory

You substitute the Fourier mode expression $$ \phi(x)= \int \frac{d^3k}{2E_k (2\pi)^3}( a_k e^{-ikx}+ a^* e^{ikx}) $$ where $kx= -{\bf k}\cdot {\bf x}+ E_k t$ and $$ [a_k, a^*_{k'}]= 2E_k (2\pi)^3 \...
user avatar
  • 41.8k
2 votes

How to predict the time domain of pulse from an amplitude mask in frequency domain?

First of all, I think the OP has some misconceptions, so I will start by addressing them "These pulses often contain multiple frequencies (i.e. polychromatic pulse) that are generated by pulse ...
user avatar
  • 99
2 votes
Accepted

Derivation of Fourier Transform in Quantum Mechanics

Hint: start with basic Fourier transform $$ F(x)=\frac{1}{\sqrt{2\pi}}\int dk f(k)e^{ikx}dk $$ and perform a change of integration variable: $$p=\hbar k.$$
user avatar
  • 39.2k
2 votes

Can we quantify the pitch of a sound that is a mixture of many frequencies?

In physics, high pitch is translated to high frequencies and low pitch is lower frequencies in sound frequency spectrum. for example in the spectrum chart below: the signal has most of its power ...
user avatar
  • 35
1 vote

Can we quantify the pitch of a sound that is a mixture of many frequencies?

The description of pitch consisting of multiple frequencies that are harmonics of the fundamental frequency is called timbre. It is the timbre of a note that gives the difference in tone between a ...
user avatar
1 vote

Propagators in Frequency Space and Dyson's Expansion

It seems to me that there is a typo in Holstein's book. The operator $V$ is not supposed to be time-dependent (except, of course, in the interaction picture). So e.g. in Eq. (4.18), it is supposed to ...
user avatar
  • 6,271
1 vote

Propagators in Frequency Space and Dyson's Expansion

Just factorize all t1 values, and integrate over the integrand. You then replace the substituent value from the delta function and you will get your formal equation
user avatar
  • 35
1 vote
Accepted

Vlasov equation in Fourier space with $k=0$

I think I have some answers to your questions. To begin with, note that, according to the text you have to take the spatial average of the equation you are referring to. Namely, for some volume $V$ $$\...
user avatar
  • 844
1 vote
Accepted

Fourier Component And Resonance

It means that the driving force is decomposed into sines and cosines and if one of this sinusoidal functions oscillates with frequency equal or close to the natural frequency, then the system ...
user avatar
  • 16.4k
1 vote
Accepted

What is the form of the wave packet in terms of momentum?

In your equation 2 you implicitly make the following definition of $\phi(p)$ \begin{equation} A\left(\frac{p}{\hbar}\right) = \phi(p) \end{equation} Let's think a bit more about how $\phi(p)$ should ...
user avatar
  • 34.4k
1 vote
Accepted

D'Alembert operator interaction term in QFT Lagrangian

I think he has a typo in $(2.21)$. In the first line the $m^2$ at the numerator of the second integral should actually be a $p^2$. You can see that in the second line a $p^2$ appears out of nowhere. ...
user avatar
  • 1,944
1 vote
Accepted

Interpreting a normalized Power Spectral Density (PSD)

You can interpret the PSD in units of dB the same way as you interpret it in more common units of dBm/Hz. The time integral squared amplitude of the voltage signal in the time domain can be thought of ...
user avatar
1 vote
Accepted

Explicit formulas for eigenstates of field operators

Yes, $\phi(x)$ is an arbitrary function of $x$ (maybe with some regularity and boundary conditions if we want to be completely precise) This second point is a generalization of the formula expressing ...
user avatar
  • 3,946
1 vote
Accepted

Complex $\Phi$ to prove the group velocity of a wavepacket

Suppose that we have waves with dispersion equation $\omega =\omega(k)$. A right-going wave-packet of finite extent, and with initial profile $\varphi(x)$, can be Fourier analyzed to give $$ \varphi(x)...
user avatar
  • 41.8k

Only top scored, non community-wiki answers of a minimum length are eligible