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Chapter 5: Q31E (page 188)
Verify that solution (5-19) satisfies the Schrodinger equation in form (5.18).
The solution of the Schrodinger equation is
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An electron is trapped in a quantum well (practically infinite). If the lowest-energy transition is to produce a photon ofwavelength. Whatshould be the well鈥檚 width?
Because protons and neutrons are similar in mass, size, and certain other characteristics, a collective term, nucleons, has been coined that encompasses both of these constituents of the atomic nucleus. In many nuclei, nucleons are confined (by the strong force, discussed in Chapter) to dimensions of roughfemtometers. Photons emitted by nuclei as the nucleons drop to lower energy levels are known as gamma particles. Their energies are typically in the Merange. Why does this make sense?
Quantum-mechanical stationary states are of the general form . For the basic plane wave (Chapter 4), this is , and for a particle in a box it is . Although both are sinusoidal, we claim that the plane wave alone is the prototype function whose momentum is pure-a well-defined value in one direction. Reinforcing the claim is the fact that the plane wave alone lacks features that we expect to see only when, effectively, waves are moving in both directions. What features are these, and, considering the probability densities, are they indeed present for a particle in a box and absent for a plane wave?
What is the probability that a particle in the first excited state of an infinite well would be found in the nuddle third of the well? How does this compare with the classical expectation? Why?
Consider a particle of mass mand energy E in a region where the potential energy is constant U0. Greater than E and the region extends to
(a) Guess a physically acceptable solution of the Schrodinger equation in this region and demonstrate that it is solution,
(b) The region noted in part extends from x = + 1 nm to . To the left of x = 1nm. The particle鈥檚 wave function is Dcos (109m-1 x). Is also greater than Ehere?
(c) The particle鈥檚 mass m is 10-3 kg. By how much (in eV) doesthe potential energy prevailing from x=1 nm to U0. Exceed the particle鈥檚 energy?
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