In the hydrogen atom, the proton is not really a point charge but has a finite size. Assume that the proton behaves as a uniformly-charged sphere of radius R=10^(-15) m. Calculate the shift this produces in the ground-state energy of hydrogen.© BrainMass Inc. brainmass.com October 25, 2018, 3:16 am ad1c9bdddf
From perturbation theory, we know that an energy eigenvalue shifts to first order by:
were |psi> is the unperturbed state and V is the perturbation term in the Hamiltonian. In this problem, the unperturbed Hamilonian H_0 is:
H_0 = p^2/(2m) - e^2/(4 pi epsilon_0 r)
If we treat the proton as a uniformly charged sphere of radius R, the potential energy term in H_0 is not correct inside this sphere. So, let's evaluate the correct potential energy function.
Consider a sphere of radius R with a volume charge of rho. Then by Gauss' law, the radial component of the electric field at distance r will be:
E(r) 4 pi r^2 = Q(r)/epsilon_0 (1)
where Q(r) is the charge enclosed within a distance r from the center of the sphere. By symmetry, the electric fleld only has a radial component, so E(r) is also the magnitude of the electric field. And it then also follows that the potential only depends on r. If r < R, we have
Q(r) = 4/3 pi r^3 rho,
inserting this in (1) gives:
E(r) = rho/(3 epsilon_0) r (2)
If r > R, then Q(r) is the total charge ...
We explain how the finite size of the proton leads to a shift in the ground state energy of hydrogen. We work out this shift to first order in perturbation theory.
1. The wavelength spectrum of the radiation energy emitted from a system in thermal equilibrium is observes to have a maximum value which decreases with increasing temperature. Outline briefly the significance of this observation for quantum physics.
2. The “stopping potential” in a photoelectric cell depends only on the frequency v of the incident electromagnetic radiation and not on its intensity. Explain how the assumption that each photoelectron is emitted following the absorption of a single quantum of energy hv is consistent with this observation.
3. Write down the de Broglie equations relating the momentum and energy of free particle to, respectively, the wave number k and angular frequency w of the wave-function which describes the particle.
4. Write down the Heisenberg uncertainty Principle as it applies to the position x and momentum p of a particle moving in one dimension.
5. Estimate the minimum range of the momentum of a quark confined inside a proton size 10 ^ -15 m.
6. Explain briefly how the concept of wave-particle duality and the introduction of a wave packet for a particle satisfies the Uncertainty Principle.