Summarize the similarities are differences between the three simple bound cases considered in this chapter.
Similarly: All of the cases had a non-zero ground state Kinetic Energy and the form standing waves.
Differences: In case of infinite square well, the wave function doesn’t extend in to area outside walls, but in other cases it did.
The three simple bond cases considered were the infinite well, the harmonics oscillators and the finite square well. All of those have a non-zero ground-state Kinetic Energy from standing waves. The harmonics oscillators and the infinite well can take on infinite number of states while the finite square well has a finite limit to the number of states it can hold.
In the case of the infinite square well, the wave functions doesn’t extend into the area outside of the walls, but In the other two cases the wave function can extend into the classically forbidden area.
For the finite square well, the energy level becomes more spaced as more are added while the harmonic oscillators are evenly spaced.
Hence, the point of differences & similarities b/w three simple bound cases considered in this chapter are discussed above.
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 L is 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.)
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.
If a particle in a stationary state is bound, the expectation value of its momentum must be 0.
(a). In words, why?
(b) Prove it.
Starting from the general expression(5-31) with in the place of , integrate by parts, then argue that the result is identically 0. Be careful that your argument is somehow based on the particle being bound: a free particle certainly may have a non zero momentum. (Note: Without loss of generality, may be chosen to be real.)
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