Chapter IV - Quantum Bound States

4.4) An electron in the n=4 state of a 5-nm wide infinite square well makes a transition to the ground state, giving off energy in the form of a photon. What is the photon's wavelength?

4.9) Classically, if a particle is not observed, the probability of finding it in a box is a constant 1/L per unit length along the entire length of the box. With this, show that the "classical expectation value" of the position is L/2, that the expectation value of the square of the position is L²/3, and that the uncertainty in position is .

4.10) Show that the uncertainty in the position of a particle in an infinite well in the general case of arbitrary n is given by
.
Discuss the dependence; in what circumstances does it agree with the "classical uncertainty" of ?

4.11) Show that the uncertainty in the momentum of a particle in an infinite well in the general case of arbitrary n is given by nh/2L.

4.12) What is the product of the position and momentum uncertainties for a particle in an infinite well? Discuss the result.

The following two questions refer to the half-infinite well defined by:

4.18) Advance an argument based upon p = h/l that there is no bound state in a half-infinite well unless U₀ is at least h²/32mL². (Hint: What is the maximum wavelength possible within the well?)

4.20) Write solutions to the Schrodinger equation appropriate in the various regions, impose required continuity conditions, and obtain the energy quantization condition:

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