Tagged Questions

A unitary linear operator which resolves a function on $\mathbb{R}^N$ into a linear superposition of "plane wave functions". Most often used in physics for calcalating the response of a time shift invariant linear system as the sum of its response to time harmonic excitation or for transforming a ...

425 views

Normalized wave functions in position and momentum space

Using the following expression for the Dirac delta function: $$\delta(k-k')=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(k-k')x}\mathrm{d}x$$ show that if $\Psi(x,t)$ is normalized at time $t=0$, ...
29 views

How to mathematically model a realistic aperture illumination?

I want to know a mathematical expression that I can use to model a realistic aperture illumination to produce the primary beam of an antenna so that the radial distribution of this aperture ...
30 views

Having trouble understanding the proof for Fourier Transform Scaling Property [closed]

Starting from Plancherel's Theorem: $$f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F(k)e^{ikx}dk ...(1)$$ $$F(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-ikx}dx ...(2)$$ I need to ...
58 views

Characteristics of an Optical System in the Fourier Domain

An imaging system can be characterized by its point spread function (PSF), which in most cases is space-variant. The final image is the result of the convolution of the PSF with the object (2d ...
54 views

98 views

A question about Fourier Transformation [closed]

Recently I try to evaluate a integral in a paper: $$\Gamma(x,y)=\int_{-\infty}^{\infty} \frac{dk}{2\pi} \sqrt{k^2+m^2} e^{ikx}$$ This is the Fourier Transform of: $$f(k)=\sqrt{k^2+m^2}$$ The ...
138 views

Wavefunction interpretations in QM

From two-slit electron-interference experiment we can infer that there is a wave $\psi(x,t)$ that can be associated with electron. The amplitude at some point is the sum of amplitudes reaching that ...
38 views

Estimation of the autocorrelation for data on finite size interval

Let's consider we have a continuous random signal ${ t \in ] - \infty \,;\, + \infty [ \mapsto b (t)}$. We assume this signal to be stationary, so that when ensemble-averaged, one may introduce the ...
27 views

Units of a fourier transform end energy density of the time dependent force

I have a signal, which is time dependent force F(t) (obtained from the Atomic Force Microscope) I wanted to estimate the energy content in the signal for given frequency band. I have calculated Energy ...
83 views

Fourier transform & asymptotic expansion of Klein-Gordon equation

I am looking for an approximate analytical solution to the generalized Klein-Gordon equation \frac{\partial^2{\phi}}{\partial{t^2}}+\frac{\partial^2{\phi}}{\partial{x^2}}+\phi=0 \end{...
248 views

Why can you only measure velocity or location in a particle?

I was talking to a family friend in the field of optics at a quantum scale (not sure the proper name for this) and he was explaining to me why you can only determine either the velocity or location of ...
114 views

29 views