Find the probability of finding an electron - calculate an

Question

Question 1.

(a) Photons with energy of 1 keV are Compton scattered through an angle of 90° A 0

Calculate Δλ/λ - for the scattered photons.

(b) Use the Figure below showing transitions between energy levels in the Hydrogen atom to help answer this question. What are the shortest and longest wavelengths in the Paschen series?

(c) If ψ(x,t) describes the wavefunction of an electron what is the interpretation of ψ^.ψ ?

(d) Calculate an estimate for the kinetic energy of a particle of mass 10^-9 kg confined to a 10^-6 m box. Hint: use (Δp)² = (p²)_average

(e) Is ψ(t) = Acos(ωt) + iAsin(ωt) an eigenfunction of the time dependent Hamiltonian? Explain your answer using less than half a page.

Question 2.

(a) A particle has a wavefunctioh given by ψ(x) = Ae^-x2/2L2 and energy h²/2mL². Find the potential energy as a function of x.

(b) For an electron in an asymmetric infinite potential well V = 0 for 0 < x < L and V = ∞ elsewhere.

The eigenfunctions of the time-independent energy operator are given by:

ψ_n(x) = √2/L sin(nΠx/L)     n = 1,2,3

Find the probability of finding an electron between x = 0.2L and x = 0.4L.

(c) Sketch the wavefunction for the n = 2 state of a finite square potential well.

(d) A hydrogen atom in a water molecule can be modelled as a simple harmonic oscillator. If the zero-point energy is 0.15 eV then what is the wavelength of a photon that can be absorbed by the molecule in undergoing a transition between, adjacent energy levels?

(e) A particle is incident on a step potential of height V_o. Sketch the wavefunction for the incident and transmitted wave for the case where E < V_o and the case where E >V_o.

(f) What is the degeneracy of the energy state containing the quantum numbers n_x = 2, n_y = 2, n_z = 4 for a particle in a cubic 3D infinite potential well?

(g) Explain, in no more than half a page, how the observation of a split line for the ground state of silver in the Stern-Gerlach experiment is explained by the requirement that the electron has a spin quantum number s = 1/2.

(h) Normalise the time-independent wave function ψ(x) = Acos(Πx/a) to show, for a particle confined in an infinite potential well of width a that A = √2/a.

Part B - Special Relativity

Question 3.

Assume the train shown in the figure below is 1 km long between B' and A' as measured by the observer at C and is moving at 0.5 c.

(a) Draw a two-observer diagram corresponding to the frames S' and S showing the wordlines of the points B', A', B and A.

(b) Indicate the length of the train on the diagram in each frame.

(c) Which length is the proper length?

(d) Indicate a proper time interval for both frames on the diagram.

(e) According to an observer at B, what is the length of the train?

(f) According to the observer at C a lightening bolt simultaneously strikes the points B' and A'. According to an observer at C, midway between A' and B', what is the time interval between the arrival of the wavefronts of light from the bolts?

A bullet is fired from B' towards A' at t' = 0 with speed u'_x = 0.9c. According to an observer atB:

(g) What is the speed of the bullet?

(h) What is the time of flight of the bullet?

(i) What is the distance travelled by the bullet?

(j) How long after the bullet hits A' does the observer see the event (i.e. when does the wavefront from the event reach B)?

(k) A signal light ahead of the train is red (650 nm) (according to the observer at B).

How fast must the train travel for the observer at B' to see the light as green (525nm)?

Question 4.

(a) A piece of space junk with mass 10^-3 kg attains, through a series of near collisions, a speed of 0.01c with respect to an Earth observer. An alien spacecraft passing by with y = 10^ collides with the space junk. Assuming all of the energy of the space junk is released in the collision what is the energy released in the resulting explosion - expressed in units of Megatons (comparable to a powerful nuclear weapon)?

(b) The K⁰ particle decays according to:

K⁰ -> Π⁺ + Π^- decays when at rest in the lab frame, what are the kinetic energies of the pions (Π⁺ + Π^-)?

(c) Two particles, one with rest mass n\ and velocity y, and the other with rest mass m and velocity (-v), collide and stick together. What is the mass of the resulting object?

(d) Write down the 4-momentum vector and define each component.

(e) What is the momentum of a proton that has total energy three times its rest energy?