Recent content by redtree

  1. redtree

    I The domain of the Fourier transform

    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}})...
  2. redtree

    I What is the Fourier conjugate of spin?

    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...
  3. redtree

    I Deceleration parameter sign

    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 +...
  4. redtree

    I What is the Fourier conjugate of spin?

    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?
  5. redtree

    I Chain rule for denominator in second order derivatives

    Great. Thanks so much!
  6. redtree

    I Chain rule for denominator in second order derivatives

    Given ## \frac{d^2x}{dy^2} ##, what is the chain rule for transforming to ##\frac{d^2 x}{dz^2} ##? (This is not a homework question)
  7. redtree

    I Fourier conjugates and inverse units

    Why do Fourier conjugates take inverse units?
  8. redtree

    I Covariance of Fourier conjugates for Gaussian distributions

    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...
  9. redtree

    I Covariance of Fourier conjugates for Gaussian distributions

    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...
  10. redtree

    I Covariance of Fourier conjugates for Gaussian distributions

    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...
  11. redtree

    I Covariance of Fourier conjugates for Gaussian distributions

    ##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...
  12. redtree

    I Covariance of Fourier conjugates for Gaussian distributions

    ##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}##?
  13. redtree

    I Covariance of Fourier conjugates for Gaussian distributions

    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...
  14. redtree

    I Covariance of Fourier conjugates for Gaussian distributions

    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...
  15. redtree

    I Fourier transform on manifolds

    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?
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