Given the domain of the integral for the Fourier transform is over the real numbers, how does the Fourier transform transform functions whose independent variable is complex?
For example, given
\begin{equation}
\begin{split}
\hat{f}(k_{\mathbb{C}}) &= \int_{\mathbb{R}} f(z_{\mathbb{C}})...
I note the general Taylor series for ##a(t)## as:
\begin{equation}
\begin{split}
a(t)&\approx a(t_0) + a'(t_0) (t-t_0) + \frac{1}{2!} a''(t_0) (t-t_0)^2 ....
\end{split}
\end{equation}
which I rewrite as:
\begin{equation}
\begin{split}
a(t)&\approx a(t_0)\left(1 +...
Momentum ##\vec{p}## and position ##\vec{x}## are Fourier conjugates, as are energy ##E## and time ##t##.
What is the Fourier conjugate of spin, i.e., intrinsic angular momentum? Angular position?
Given two variables ##x## and ##k##, the covariance between the variables is as follows, where ##E## denotes the expected value:
\begin{equation}
\begin{split}
COV(x,k)&= E[x k]-E[x]E[k]
\end{split}
\end{equation}
If ##x## and ##k## are Foureir conjugates and ##f(x)## and ##\hat{f}(k)## are...
Is there a generalized form of the Fourier transform applicable to all manifolds, such that the Fourier transform in Euclidean space is a special case?
I understand that the Uncertainty Principle relates the variances of Fourier conjugates. I am having trouble finding: 1) the mathematical relationship between the expectation values of Fourier conjugates generally; 2) and then specifically for a normalized Gaussian. Any suggestions or insights?
I apologize for the simplicity of the question. I have been reading a paper on the Legendre transform (https://arxiv.org/pdf/0806.1147.pdf), and I am not understanding a particular step in the discussion.
In the paper, Equation 16, where ##\mathcal{H} = \sqrt{\vec{p}^2 + m^2} ##...
I apologize ahead of time for the simplicity of the question, but this has really been bothering me.
Given the de Broglie relation, assuming natural units, where ##\hbar = 1##:
\begin{equation}
\begin{split}
\vec{k} &= M \vec{v}
\end{split}
\end{equation}
My question regards...
I have a question about a very specific step in the derivation of Euler-Lagrangian. Sorry if it seems simple and trivial. I present the question in the course of the derivation.
Given:
\begin{equation}
\begin{split}
F &=\int_{x_a}^{x_b} g(f,f_x,x) dx
\end{split}
\end{equation}...
My discussion of the Friedmann metric comes from the derivation presented in section 4.2.1 of the reference: https://www1.maths.leeds.ac.uk/~serguei/teaching/cosmology.pdf
I have a couple of simple questions on the derivation. The are placed at points during the derivation.
I note the...
Can anyone recommend papers that directly curve-fit redshift as a function of luminosity distance for type Ia supernova and gamma ray bursts? I am looking for papers that do not curve-fit the data via an assumed model, even one as simple as Friedmann–Lemaître–Robertson–Walker (FLRW) metric. I...
Given two probability amplitude wavefunctions, one in position space ##\psi(r,k)## and one in wavenumber space ##\phi(r,k)##, where ##r## and ##k## are Fourier conjugates, how is it possible for the modulus squared, i.e., probability density, of BOTH wavefunctions to be normalized? It seems...
I apologize for the simple question, but I am trying to understand the Mass Discrepancy-Acceleration Relation and its relationship to ##\mu(x)## (from https://arxiv.org/pdf/astro-ph/0403610.pdf).
The mass discrepancy, defined as the ratio of the gradients of the total to baryonic...
Given that position and momentum are Fourier conjugates, what is the derivation for the equation ##\hbar \vec{k} = m \vec{v}##, where momentum-space momentum is defined as ##\hbar \vec{k}## and position-space momentum is defined as ##m \vec{v}##?
I am studying phase and group velocity in non-dispersive and dispersive media. My question is the following: Is there any reason why a dispersive medium simply cannot be modeled as a type of field?
I apologize for the simplicity of the question (NOT homework). This is a statistical question (not necessarily a quantum mechanical one).
If I have an initial probability function with an associated expected value and then a second probability function is superimposed on the initial...
I apologize for the simple question, but I have not been able to find the answer.
For the inner portion of a galaxy rotation curve (where the outer portion is the part invariant to distance and the inner part is where rotational velocity increases with radius), how much is simply due to...
I note the following:
\begin{equation}
\begin{split}
\langle\vec{x}_n|e^{-i \frac{\mathcal{H}_n}{\hbar} (t_n-t_0)}|\vec{x}_{0}\rangle &=\delta(\vec{x}_n-\vec{x}_0)e^{-i \frac{\mathcal{H}_n}{\hbar} (t_n-t_0)}
\end{split}
\end{equation}
I divide the time interval as follows...
The Green's function is defined as follows, where ##\hat{L}_{\textbf{r}}## is a differential operator:
\begin{equation}
\begin{split}
\hat{L}_{\textbf{r}} \hat{G}(\textbf{r},\textbf{r}_0)&=\delta(\textbf{r}-\textbf{r}_0)
\end{split}
\end{equation}
However, I have seen the following...
I just want to make sure that I am understanding the Dirac Delta function properly. Is the following correct?:
For two variables ##x## and ##y##:
\begin{equation}
\begin{split}
\delta(x-y) f(x) &= f(y)
\end{split}
\end{equation}
And:
\begin{equation}
\begin{split}
\delta(x-x) f(x) &=...
Given a convolution:
\begin{equation}
\begin{split}
g(x) * h(x) &\doteq \int_{-\infty}^{\infty} g(z) h(x-z) dz
\end{split}
\end{equation}
Do ##z## and ##x## have to be independent? For example, can one set ##x=z+y## such that:
\begin{equation}
\begin{split}
\int_{-\infty}^{\infty} g(z)...
1. Given a Markov state density function:
## P((\textbf{r}_{n}| \textbf{r}_{n-1})) ##
##P## describes the probability of transitioning from a state at ## \textbf{r}_{n-1}## to a state at ##\textbf{r}_{n} ##. If ## \textbf{r}_{n-1} = \textbf{r}_{n}##, then ##P## describes the probability of...
Two questions regarding the completeness relation:
First: I understand that the completeness relation holds for basis vectors such that ## \sum_{j=1}^{m} | n_{j} \rangle \langle n_{j} | =\mathbb{I}##. Does it also hold for unit-normalized sets of state vectors as well, where ## | \phi_{j}...
I am familiar with the derivation of the resolution of the identity proof in Dirac notation. Where ## | \psi \rangle ## can be represented as a linear combination of basis vectors ## | n \rangle ## such that:
## | \psi \rangle = \sum_{n} c_n | n \rangle = \sum_{n} | n \rangle c_n ##
Assuming an...
In classical mechanics, the Hamiltonian and the Lagrangian are Legendre transforms of each other. By analogy, in quantum mechanics and quantum field theory, the relationship between the Hamiltonian and the Lagrangian seems to be preserved. Where can I find a derivation of the Lagrangian...
I am trying to find a derivation of gravitational redshift from a static metric that does not depend on the equivalence principle and is not a heuristic Newtonian derivation. Any suggestions?