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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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A particle is described by the wave function
(a) Show that the normalization constantis correct.
(b) A measurement of the position of the particle is to be made. At what location is it most probable that the particle would be found?
(c) What is the probability per unit length of finding the particle at this location?
Show that that is, verify that unless the wave function is an Eigen function of the momentum operator, there will be a nonzero uncertainty in the momentumstarts with showing that the quantity
Is . Then using the differential operator form ofand integration by parts, show that it is also,
Together these show that ifis. 0. then the preceding quantity must be 0. However, the Integral of the complex square of a function(the quantity in the brackets) can only be 0 if the function is identically 0, so the assertion is proved.
A tiny particle is in a 1 cm wide enclosure and take a yearto bounce from one end to the other and back(a) Haw many nodes are there in the enclosure (b) How would your answer change if the particle were more massive or moving faster.
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.)
A finite well always has at least one bound state. Why does the argument of Exercises fail in the case of a finite well?
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