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Chapter 5: Q69E (page 191)
does the wave function have a well-defined momentum? Explain.
The constant in the front multiplies the initial function and two opposing moving plane waves are added, so the momentum is not clearly defined.
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To determine the two bound state energies for the well.
The product of uncertainties in particle's momentum and position.
A half-infinite well has an infinitely high wall at the origin and one of finite height U0 at x= L . Like the finite well, the number of allowed states is limited. Assume that it has two states, of energy E1 and E2 , where E2 is not much below U0. Make a sketch of the potential energy, then add plausible sketches of the two allowed wave functions on separate horizontal axes whose heights are E1 and E2 .
Equation (5 - 16) gives infinite well energies. Because equation (5 - 22) cannot be solved in closed form, there is no similar compact formula for finite well energies. Still many conclusions can be drawn without one. Argue on largely qualitative grounds that if the walls of a finite well are moved close together but not changed in height, then the well must progressively hold fewer bound states. (Make a clear connection between the width of the well and the height of the walls.)
The figure shows a potential energy function.
(a) How much energy could a classical particle have and still be bound?
(b) Where would an unbound particle have its maximum kinetic energy?
(c) For what range of energies might a classical particle be bound in either of two different regions?
(d) Do you think that a quantum mechanical particle with energy in the range referred to in part?
(e) Would be bound in one region or the other? Explain.
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