I am trying to visualize the time dependence of a free particle given an initial wave-function using Python and I just wanted to know if I could use the in built FFT implementation from NumPy to find the coefficients of the integral

$$ \Psi(x,\,t)=\frac{1}{2\pi}\int_{-\infty}^\infty\phi(k)\exp\left[i\left(kx-\frac{\hbar k^2}{2m}t\right)\right]\,\mathrm{d}k $$

In the textbook I am basing this off of, they solve for $\phi(k)$ by finding the fourier transform analytically given an initial function $\Psi(x, 0)$. Logically, it seems like doing a discrete fourier transform to simulate a fourier transform on an array of values representing $\Psi(x, 0)$ is okay, but I'm wondering if there are some subtleties I might be missing that actually changes the interpretation of my results when using a discrete fourier transform to find $\phi(k)$ instead of a continuous one. More generally, I guess, is other than the fact one is applied to a discrete rather than continuous set of variables, what is the difference between a Discrete Fourier Transform and a Continuous Fourier Transform.


1 Answer 1


Welcome to scicomp!

Do you want to plot or do you want to do a simulation?

There is no problem with doing numerical simulation within a discretized fourier space. It comes with a couple of advantages and disadvantages.

Once you fourier transformed your profile, which in your case is a wave-amplitude, then calculating derivatives is very easy. You can simply multiply with $i~k_j$, with $k_j$ being your wavenumber for that particular coefficient. So calculating derivatives is very convenient in a DFT-code. (a laplace operator is basically $(i k_j)^2$)!)

The transformation from real space to fourier space can be done via the FFT algorithm and takes $O(N LOG(N))$ operations if i remember correctly. There are performant libraries to take care of this.

One problem of course is that you can not calculate nonlinearities directly. If you have an array of coefficients $r_i$ in real space, and you want to calculate a nonlinearity, then you can simply calculate $(r_i)^2$ etc. In fourier space that is not possible. Therefore in navier stokes codes for example the nonlinearity is calculated in real space, and then may be re-transformed back for the rest of the numerical treatment.

Another constraint is that you are limited to periodic or quasi-periodic problems. Since the very basis for your computation are essentially sine- and cosine functions, there is no good way to resolve any boundary conditions or discontinuities. If you want to look at one free particle within a large empty domain you should be fine though.

What kind of time-dependence do you mean? You are stating the initial condition, but not really the dynamics of what you want to do!

  • $\begingroup$ Thanks, that's really helpful! I guess I'm actually plotting, not simulating. By time-dependence, I mean I just want to plot the wavefunction at different values of t. $\endgroup$ Commented Oct 11, 2019 at 19:49
  • $\begingroup$ Well then you just create an array with the wavenumbers $\psi(k)$, multiply it with the correct time dependend phase, as your equation states, and transform to real space. Then just plot the profile $\endgroup$
    – MPIchael
    Commented Oct 12, 2019 at 7:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.