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Chapter 5: Q67E (page 191)
Prove that the transitional-state wave function (5.33) does not have a well-defined energy.
The obtained solution for the energy still hasa wave function, so the original wave function does not have a defined energy.
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The potential energy shared by two atoms in a diatomic molecule, depicted in Figure 17, is often approximated by the fairly simple function where constants a and b depend on the atoms involved. In Section 7, it is said that near its minimum value, it can be approximated by an even simpler function鈥攊t should 鈥渓ook like鈥� a parabola. (a) In terms ofa and b, find the minimum potential energy U (x0) and the separation x0 at which it occurs. (b) The parabolic approximation has the same minimum value at x0 and the same first derivative there (i.e., 0). Its second derivative is k , the spring constant of this Hooke鈥檚 law potential energy. In terms of a and b, what is the spring constant of U (x)?
Consider a particle bound in a infinite well, where the potential inside is not constant but a linearly varying function. Suppose the particle is in a fairly high energy state, so that its wave function stretches across the entire well; that is isn鈥檛 caught in the 鈥渓ow spot鈥�. Decide how ,if at all, its wavelength should vary. Then sketch a plausible wave function.
Outline a procedure for predicting how the quantum-mechanically allowed energies for a harmonic oscillator should depend on a quantum number. In essence, allowed kinetic energies are the particle-in-a box energies, except the length Lis replaced by the distance between classical tuning points. Expressed in terms of E. Apply this procedure to a potential energy of the form U(x) = - b/x where b is a constant. Assume that at the origin there is an infinitely high wall, making it one turning point, and determine the other numing point in terms of E. For the average potential energy, use its value at half way between the tuning points. Again in terms of E. Find and expression for the allowed energies in terms of m, b, and n. (Although three dimensional, the hydrogen atom potential energy is of this form. and the allowed energy levels depend on a quantum number exactly as this simple model predicts.)
To show that the potential energy of finite well is
There are mathematical solutions to the Schr枚dinger equation for the finite well for any energy, and in fact. They can be made smooth everywhere. Guided by A Closer Look: Solving the Finite Well. Show this as follows:
(a) Don't throw out any mathematical solutions. That is in region Il , assume that , and in region III , assume that. Write the smoothness conditions.
(b) In Section 5.6. the smoothness conditions were combined to eliminate in favor of . In the remaining equation. canceled. leaving an equation involving only and , solvable for only certain values of . Why can't this be done here?
(c) Our solution is smooth. What is still wrong with it physically?
(d) Show that
localid="1660137122940"
and that setting these offending coefficients to 0 reproduces quantization condition (5-22).
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