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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 a 2D infinite well whose sides are of unequal length.
(a) Sketch the probability density as density of shading for the ground state.
(b) There are two likely choices for the next lowest energy. Sketch the probability density and explain how you know that this must be the next lowest energy. (Focus on the qualitative idea, avoiding unnecessary reference to calculations.)
An electron is in the 3d state of a hydrogen atom. The most probable distance of the electron from the proton is. What is the probability that the electron would be found between?
An electron in a hydrogen atom is in the (n,l,ml) = (2,1,0) state.
(a) Calculate the probability that it would be found within 60 degrees of z-axis, irrespective of radius.
(b) Calculate the probability that it would be found between r = 2a0 and r = 6a0, irrespective of angle.
(c) What is the probability that it would be found within 60 degrees of the z-axis and between r = 2a0 and r = 6a0?
A wave function with a non-infinite wavelength-however approximate it might be- has nonzero momentum and thus nonzero kinetic energy. Even a single "bump" has kinetic energy. In either case, we can say that the function has kinetic energy because it has curvature- a second derivative. Indeed, the kinetic energy operator in any coordinate system involves a second derivative. The only function without kinetic energy would be a straight line. As a special case, this includes a constant, which may be thought of as a function with an infinite wavelength. By looking at the curvature in the appropriate dimension(s). answer the following: For a givenn,isthe kinetic energy solely
(a) radial in the state of lowest l- that is, l=0; and
(b) rotational in the state of highest l-that is, l=n-1?
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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