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}})...
Spin is a measure of energy: intrinsic angular momentum. Other forms of energy are the Fourier conjugates of variables in position space. It seems theoretically problematic that no position space representation exists for spin. Nor do I understand why the observation that spin is measured as...
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?
I understand what you are saying, and I don't want to make this a debate about definitions. I note that the American Mathematical society considers the Fourier transform of the square root of the normalized Gaussian also to be a the square root of a p.d.f., i.e. both ##f(x)^2## and...
The mathematical uncertainty principle says a lot about the relationship between Fourier conjugates, specifically the relationship between their variances. Again, I am merely trying to see what has been discovered regarding the relationship between their expected values as represented by the...
The mathematical uncertainty principle is a description of the relationship between the variances of Fourier conjugates with normalized Gaussian distributions. In order to calculate both variances, one must interpret both ##f(x)## and ##\hat{f}(k)## as probability functions. See for example...
##f(x)## is not an arbitrary probability distribution. I have defined ##f(x)## as the probability amplitude of the normalized Gaussian. Furthermore, the Fourier transform of a Gaussian probability distribution is also a Gaussian probability distribution (and the Fourier transform of the...
##k## is the Fourier conjugate of ##x##.
My understanding is that ##k## is not completely independent of ##x##. Were ##k## and ##x## completely independent, i.e., ##Cov(x,k) = 0##, then why would ##E(x^2) E(k^2) = \frac{1}{16 \pi^2}##?
I apologize if I was unclear. I am concerned with probability, specifically probability amplitude functions and Fourier transforms. This is not a question about quantum mechanics.
I will restate the question more explicitly. Given the normalized Gaussian probability amplitude in ##x##-space...
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?