Fourier Analysis

A real wavefunction can be decomposed into a sum of harmonic terms using the technique of Fourier analysis. With fixed boundaries this decomposition becomes

where N is the number of points on the space grid, L is once again the length of the medium, and n is referred to as the bin number. With periodic boundaries, both and terms are present in the decomposition.

For real the Fourier analysis graph will display either for fixed boundaries or for periodic boundaries.   Expansion coefficients for a function defined on a uniform grid can be obtained very efficiently using a numerical technique called the Fast Fourier Transform, FFT. It is discussed in detail in Chapter 2 and has been implemented in WAVE .

• Real FFT
  Load the WAVE program. Select [Graph-2]|[Fourier Analysis of Y(x)] to enable the Fast Fourier Transform, FFT, of the function . Notice that the height of the red bar follows the oscillations of the wave when the program is running; the blue bar records the red bar's maximum value. Since the program default uses N=128 points on the space grid, the bin scale on the abscissa may be too large. Zoom in on the FFT graph using the red button in the left-hand corner to display the scale inspector. Set xMax to 16. (Even though the abscissa measures the FFT bin number, we refer to the abscissa as ``x" since the inspector is generic to all graph objects.)

  a.

  The FFT graph almost shows the distribution of eigenfunctions that make up the E&M wave. If we write the solution of in terms of the PDE eigenfunctions, ,

  

  and compare this expansion to eq:sinFFT, we see that the FFT coefficients have absorbed the time dependence.

  

  Select a more complicated wavefunction and notice that the blue bar graph will show the eigenfunction distribution if the program runs through one oscillation of the lowest-frequency component.

  b.

  Change the boundary condition to Periodic . Since the eigenfunctions depend on boundary conditions, you must now reselect the eigenfunction using [Init]|[Modes] . The FFT has changed, too. First, the red bar remains constant when the program is run. Why? Second, the number of bins on the abscissa is . Why?

The choice of fixed or periodic boundary conditions selects between two different complete orthonormal sets of eigenfunctions. Both and terms are allowed for periodic boundaries while only terms are allowed for fixed boundaries. Fixed boundaries can, however, support half-integer wavelengths so the total number of functions is equal to the number of grid points in each set. We are interested in the frequency components and not in relative phases, so for periodic boundaries, the bar graph displays magnitudes of sin and cos of the same frequency in a single term of height and the   number of bins in the FFT graph is reduced to . A plot such as this, i.e. a plot of the magnitude of the Fourier components at a frequency, is called a power spectrum although we will continue to refer to our graphs as FFTs since this algorithm is the basis of our analysis.

• Time Series FFT
  It is also possible to perform a FFT analysis of time series data collected by a detector. You must first create a detector and then select [Graph-1]|[Fourier Analysis- Y(t)] . Select [Graph-2]|[Detector Readings] as the second graph and run the program. The analysis graph will appear after the detector has collected 256 data points and will again show the power spectrum, . Remember that analysis graphs only collect data during a screen redraw so the time between data points is .

  a.

  Calculate the total time interval during which the data was collected, T. Bin 1 contains the amplitude at . Bin 2 contains the amplitude at the , etc. Find the bin number with the maximum amplitude and calculate its frequency.

  b.

  Is the time series FFT located in a single bin? What happens if the length of the time series is extended using the blue attributes button?

  c.

  How does the time resolution (i.e., the animation speed) effect the time series FFT?

   

• Complex Eigenfunctions
  The solution to the Schrödinger equation can also be expanded in a Fourier series if the expansion coefficients, , are complex. Examination of the eigenfunctions shows that the expansion coefficients are phasors

  

  with angular frequency . Since the magnitude of any phasor is independent of time, we immediately obtain the eigenfunction distribution by plotting as red bars in the analysis graph.

  a.

  Set the EquationType to Schrödinger and the boundary to Fixed . Select modes 1 and 2 with amplitudes equal to . Select the FFT of last so that the scale is properly set. Run the program. Notice that the particle probability oscillates between the left-hand and right-hand sides of the box. The bar graph produced by the FFT is, however, constant.

  b.

  For a dramatic example of a constant FFT but a wildly varying wavefunction try an initial Gaussian with width of and average momentum (i.e., wavenumber) of 50. What you are seeing is called dispersion. Each mode has a different phase velocity so the modes making up a particular function, in this case a Gaussian, drift out of phase producing distortion of the original shape.

  c.

  Select mode 4 and notice the single FFT bin. Change the boundary to Periodic . Notice that there are now two nonzero bins and that the scale of the FFT graph has both positive and negative bin numbers. Explain the FFT.

  d.

  Select mode 4 while the boundary is set to Periodic and then switch the boundary condition to Fixed . Explain the FFT.

  The student of quantum mechanics should realize that each bin in the FFT analysis graph represents the probability amplitude of a momentum (kinetic energy) eigenstate when periodic (fixed) boundary conditions are selected.

• Coherent States
    An interesting case of a variable FFT and showing the correspondence between classical physics and quantum mechanics will be explored in this exercise. Enter into the text field of the [Medium]|[Parser]

  input screen to create a simple harmonic oscillator (SHO) potential of in atomic units. Select a Gaussian initial wavefunction having zero average momentum, i.e., k=0, and offset from the equilibrium point. Set the boundary condition to periodic in order to show momentum eigenstates. Run the program with analysis set to FFT of and contour plots. Notice and explain the following:

  a.

  The wavefunction remains a Gaussian as it propagates. Its FFT is also Gaussian.

  b.

  The oscillations of the center of the wavefunction and the FFT are out of phase.

  c.

  Try very narrow and very wide initial Gaussians. What do you observe about the width? The initial width that will propagate without changing shape is called a coherent state. Find this width. Hint: Try the width of the ground state.

    Any Schrödinger wavefunction can be expanded in terms of SHO eigenfunctions. An interesting property of these eigenfunctions is that their energy eigenvalues, , are equally spaced. The phase oscillations induced by the factor in the time-dependent Schrödinger equation eigenfunctions are therefore harmonic and any initial wavefunction will reoccur after one phase oscillation of the ground state. Investigate this behavior of the following two wavefunctions using parser input under [Init]|[User Defined Function] .

  d.

  A square pulse, . The Heaviside step function, , is recognized by the parser.

  e.

  A SHO wavefunction that has been offset from equilibrium. Enter the n -th SHO wavefunction offset from equilibrium, . The SHO wavefunctions can be found in most quantum mechanics texts.