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Chapter 7: Q12CQ (page 278)
Question: A particle is trapped in a spherical infinite well. The potential energy is 0for r < a and infinite for r > a . Which, if any, quantization conditions would you expect it to share with hydrogen, and why?
Answer:
The particle will have the same quantization which is found in hydrogen.
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Consider an electron in the ground state of a hydrogen atom. (a) Sketch plots of E and U(r) on the same axes (b) Show that, classically, an electron with this energy should not be able to get farther than from the proton. (c) What is the probability of the electron being found in the classically forbidden region?
Imagine two classical charges of -q, each bound to a central charge of. +q One -q charge is in a circular orbit of radius R about its +q charge. The other oscillates in an extreme ellipse, essentially a straight line from it’s +q charge out to a maximum distance .The two orbits have the same energy. (a) Show that. (b) Considering the time spent at each orbit radius, in which orbit is the -q charge farther from its +q charge on average?
Question: Consider an electron in the ground state of a hydrogen atom. (a) Calculate the expectation value of its potential energy. (b) What is the expectation value of its kinetic energy? (Hint: What is the expectation value of the total energy?)
When applying quantum mechanics, we often concentrate on states that qualify as “orthonormal”, The main point is this. If we evaluate a probability integral over all space of or of , we get 1 (unsurprisingly), but if we evaluate such an integral for we get 0. This happens to be true for all systems where we have tabulated or actually derived sets of wave functions (e.g., the particle in a box, the harmonic oscillator, and the hydrogen atom). By integrating overall space, show that expression (7-44) is not normalized unless a factor of is included with the probability.
Consider a vibrating molecule that behaves as a simple harmonic oscillator of mass , spring constant 103N/m and charge is +e , (a) Estimate the transition time from the first excited state to the ground state, assuming that it decays by electric dipole radiation. (b) What is the wavelength of the photon emitted?
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