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Chapter 5: Q23E (page 187)
Write out the total wave function.For an electron in the n=3 state of a 10nm wide infinite well. Other than the symbols a and t, the function should include only numerical values?
The total wave function for n=3 state is given as,
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Under what circumstance does the integral diverge? Use this to argue that a physically acceptable wave function must fall to 0 faster than does as gets large.
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.
Simple models are very useful. Consider the twin finite wells shown in the figure, at First with a tiny separation. Then with increasingly distant separations, In all case, the four lowest allowed wave functions are planned on axes proportional to their energies. We see that they pass through the classically forbidden region between the wells, and we also see a trend. When the wells are very close, the four functions and energies are what we might expect of a single finite well, but as they move apart, pairs of functions converge to intermediate energies.
(a) The energies of the second and fourth states decrease. Based on changing wavelength alone, argue that is reasonable.
(b) The energies of the first and third states increase. Why? (Hint: Study bow the behaviour required in the classically forbidden region affects these two relative to the others.)
(c) The distant wells case might represent two distant atoms. If each atom had one electron, what advantage is there in bringing the atoms closer to form a molecule? (Note: Two electrons can have the same wave function.)
Figure 5.15 shows that the allowed wave functions for a finite well whose depth was chosen to be.
(a) Insert this value in equation (5-23), then using a calculator or computer, solve for the allowed value of , of which there are four.
(b) Usingfind corresponding values of E. Do they appear to agree with figure 5.15?
(c) Show that the chosenimplies that .
(d) Definingandto be 1 for convenience, plug your and values into the wave function given in exercise 46, then plot the results. ( Note: Your first and third values should correspond to even function of z, thus using the form with, while the second and forth correspond to odd functions. Do the plots also agree with Figure 5.15?
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.
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