Question: Starting with the assumption that a general wave function may be treated as an algebraic sum of sinusoidal functions of various wave numbers, explain concisely why there is an uncertainty principle.
The more sinusoidal wave functions are required to express a function, the less spreading it has in space, and vice versa. This follows immediately from the Fourier theorem and has a connection to the Heisenberg uncertainty principle.
Mathematically, the uncertainty relation can be stated as .
Fourier established this theory, which connects the function spreading in space to the range of wavenumber , long before the Heisenberg uncertainty principle. The relation states that .
In other words, if we require a function with a very small spatial bandwidth, we must cover a wide variety of wavelengths or wavenumbers; as a result, we must combine many sinusoidal functions.
If we only take note that the momentum is equal to , then this may be directly connected to the Heisenberg connection. Consequently, if we substitute that back in, we will reestablish our original relationship . So, the same conclusion is drawn: the more wavenumbers added together (greater uncertainty in momentum), the less uncertainty there is in location, and the opposite is also true.
Question: Analyzing crystal diffraction is intimately tied to the various different geometries in which the atoms can be arranged in three dimensions and upon their differing effectiveness in reflecting waves. To grasp some of the considerations without too much trouble, consider the simple square arrangement of identical atoms shown in the figure. In diagram (a), waves are incident at angle with the crystal face and are detected at the same angle with the atomic plane. In diagram (b), the crystal has been rotated 450 counterclockwise, and waves are now incident upon planes comprising different sets of atoms. If in the orientation of diagram (b), constructive interference is noted only at an angle, at what angle(s) will constructive interference be found in the orientation of diagram (a)? (Note: The spacing between atoms is the same in each diagram.)
In Example 4.2. neither nor are given units—only proportionalities are used. Here we verify that the results are unaffected. The actual values given in the example are particle detection rates, in particles/second, or . For this quantity, let us use the symbol R. It is true that the particle detection rate and the probability density will be proportional, so we may write = bR, where b is the proportionality constant. (b) What must be the units of b? (b) What is at the center detector (interference maximum) in terms of the example’s given detection rate and b? (c) What would be , and the detection rate R at the center detector with one of the slits blocked?
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