A function is nonzero only in the region of width centered at
where C is a constant.
(a) Find and plot versus the Fourier transform of this function.
(b) The function ) might represent a pulse occupying either finite distance (localid="1659781367200" position) or finite time (time). Comment on the wave number if is position and on the frequency spectrum if is time. Specifically address the dependence of the width of the spectrum on .
(a). The graph plot betweenand
(b). There is inverse relation between the width of the pulse , and the wave-number or frequency range.
The generalization of the Fourier series is known as Fourier transform and it can also refer to both the frequency domain representation and the mathematical function used. The Fourier transform facilitates the application of the Fourier series to non-periodic functions, allowing every function to be viewed as a sum of simple sinusoids.
The equation of the Fourier transform as,
The Euler formulas will also be used
The equations (2) separated into two integrals
When taking the integral of an odd function with equal but opposite limits of integration, the result will be zero, due to the symmetry of odd functions about the origin and since sine is an odd function, that integral will be zero:
Since the cosine is an even function with equal but opposite limits of integration, it can be written as just twice the integral from zero to the upper limit of integration (due to the symmetry of the even function about y-axis).
And then just integrate that from zero to
So, the Fourier transform ofis
The graph plot betweenand
Well, as or the interval gets smaller, which means less uncertainty in space or time, the more the uncertainty in the wave-numbers or frequencies respectively (more spreading). The width of the pulse in the frequency domain is found to be(inversely proportional with ).
Therefore, there is inverse relation between the width of the pulse , and the wave-number or frequency range.
With reckless disregard for safety and the law, you set your high-performance rocket cycle on course to streak through an intersection at top speed . Approaching the intersection, you observe green (540 nm) light from the traffic signal. After passing through, you look back to observe red (650 nm) light. Actually, the traffic signal never changed color-it didn't have time! What is the top speed of your rocket cycle, and what was the color of the traffic signal (according to an appalled bystander)?
Here we investigate the link between n and l, reflected in equation (7-33). (a) Show that if a classical point charge were held in a circular orbit about a fixed point charge by the Coulomb force, its kinetic energy would be given by (b) According to equation (7-30), the rotational kinetic energy in hydrogen is . Of course, r is not well defined for a “cloud”, but by using argue that the condition that l not exceed n is reasonable.
The well-known sodium doublet is two yellow spectral lines of very close wavelength. and It is caused by splitting of the energy level. due to the spin-orbit interaction. In its ground state, sodium's single valence electron is in the level. It may be excited to the next higher level. the 3p , then emit a photon as it drops back to the 3s . However. the 3p is actually two levels. in which L and S are aligned and anti-aligned. (In the notation of Section these are. respectively. the and the because the (transitions Stan from slightly different initial energies yet have identical final energies(the having no orbital angular momentum to lead to spin-orbit interaction), there are two different wavelengths possible for the emitted photon. Calculate the difference in energy between the two photons. From this, obtain a rough value of the average strength of the internal magnetic field experienced by sodium's valence electron.
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