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Chapter 5: Q34E (page 188)
A 50 electron is trapped between electrostatic walls 200eV high. How far does its wave function extend beyond the walls?
Electron wave extent beyond the walls .
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An electron in the n=4 state of a 5 nm wide infinite well makes a transition to the ground state, giving off energy in the form of photon. What is the photon’s wavelength?
We say that the ground state for the particle in a box has nonzero energy. What goes wrong with in equation 5.16 if n = 0 ?
In several bound systems, the quantum-mechanically allowed energies depend on a single quantum number we found in section 5.5 that the energy levels in an infinite well are given by, whereandis a constant. (Actually, we known whatis but it would only distract us here.) section 5.7 showed that for a harmonic oscillator, they are, where(using anwith n strictly positive is equivalent towith n non negative.) finally, for a hydrogen atom, a bound system that we study in chapter 7,, whereconsider particles making downwards transition between the quantized energy levels, each transition producing a photon, for each of these three systems, is there a minimum photon wavelength? A maximum ? it might be helpful to make sketches of the relative heights of the energy levels in each case.
We learned that to be normalizable, a wave function must not itself diverge and must fall to 0 faster than as x get large. Nevertheless. We find two functions that slightly violate these requirements very useful. Consider the quantum mechanical plane wave Aei(kx-ax)and the weird functionpictured which we here call by its proper name. the Dirac delta function.
(a) Which of the two normalizability requirements is violated by the plane wave, and which by Dirac delta function?
(b) Normalization of the plane wave could be accomplished if it were simply truncated, restricted to the region -b<x<+bbeing identically 0 outside. What would then be the relationship between b and A, and what would happen to A as b approaches infinity?
(c) Rather than an infinitely tall and narrow spike like the Dirac delta function. Consider a function that is 0 everywhere except the narrow region where its value is a constant B. This too could be normalized, What would be the relationship between s and B, and what would happen to B as s approaches 0? (What we get is not exactly the Dirac delta function, but the distinction involves comparing infinities, a dangerous business that we will avoid.)
(d) As we see, the two "exceptional" functions may be viewed as limits of normalizable ones. In those limits, they are also complementary to each other in terms of their position and momentum uncertainties. Without getting into calculations, describe how they are complementary.
Using equation (23), find the energy of a particle confined to a finite well whose walls are half the height of the ground-state infinite well energy, . (A calculator or computer able to solve equations numerically may be used, but this happens to be a case where an exact answer can be deduced without too much trouble.)
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