Please see the attached file for the fully formatted problems.
1. A particle is distributed along the positive x axis with a probability density
P(x) = C x2 exp(-(x/a)2). Determine it's average coordinate <x>, and the fluctuation <x2> ; where <x>= x-<x>
2. The coordinate of a particle is distributed along the WHOLE x-axis with a probability density
P(x) = C /(x2 +a2)3/2.
Determine C, <x>, and the fluctuation <x2>
The average coordinate of a particle is determined. The solution is detailed and well presented.
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.