The radial probability density for an electron is r2R2(r). That means that the probability of finding an electron at a certain radius r within a radial thickness dr is dr* r2R2(r) for an infinitely thin shell and approximately r* r2avg R2(ravg) for a shell of finite thickness r.
The quantity ravg is some average radius within the shell...
(a) Estimate the probability that an electron in the n=1,l=0 state will be found in the region from r=0m to r= 10-15m.
(b) Repeat the calculation for n=2, l=1.
(d) Consider the state n=2, l=0 in hydrogen. What are the values of r for which the probability density is zero?
Sketch the probability density as a function of r for this state.
The solution examines a hydrogen atom and the radial angular momentum. The probability that an electron in the n=1 and I=0 state will be found in the region from r=0m to r=10-15m is determined.
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.