Consider spinless non-relativistic free particles of mass M
moving in a two dimensional square box. Find an expression for the density of
Remember ρ(E)dE is defined as the number of energy levels per unit volume between E and E + dE.
The area of a circle of radius k in 2D k-space is A = πk2.
The number of states in the circle is Nstates = (A/4)/(π/L)2 = k2L2/(4π).
Only positive values of kx and ky are allowed.
The density of states in k-space is n(k) = dN/dk = kL2/(2π).
E = k2ħ2/(2M). dE/dk = kħ2/M.
n(E) = dN/dE = (dN/dk)(dk/dE) = ML2/(2πħ2).
Consider a particle in the ground state of a one-dimensional square well of
with a and depth U0.
Assume that the well is very deep and
k/k0 = (2m(E + U0)/(2mU0))½ << 1
for the ground state, so that the ground state wave function is nearly identical to that of the infinite square well. At t = 0, a time dependent perturbation W(t) = Wcosωt is turned on.
(a) What is the minimum frequency necessary to free the particle from the well?
(b) For frequencies greater than this minimum frequency, use perturbation theory to find the transition rate.
You can assume that the free particles will be in a box of size L >> a.
A hypothetical particle of mass 2m can decay into two different final states
under the action of a perturbation H’.
(a) 2 m --> m + γ,
(b) 2 m --> 1.5 m + γ.
Here m and 1.5 m represent particles of mass m and 1.5 m respectively. If the matrix elements of H’ are equal, i.e. <2 m|H'|m> = <2 m|H'|1.5 m>, what will be the relative likelihood of decay to m as opposed to 1.5 m? You may assume the recoil fragment (m or 1.5 m) receives negligible kinetic energy.
A uniform periodic electric field acts upon a hydrogen atom, which at t = 0 is in its ground state. Determine the minimum frequency of the field necessary to ionize the atom and use perturbation theory to evaluate the probability for ionization per unit time. For the sake of simplicity, assume the electron in the final state to be free.
To each allowed knml
there corresponds a wave function ψnml(x,y,z).
The tips of the vectors knml divide k-space into elementary cubes of edge length 2π/L. We have one vector per (2π/L)3 volume of k-space.
The number of vectors in a volume d3k of k-space therefore is dN = d3k/(2π/L)3.
d3k = k2dkdΩ,
dN = k2dkdΩ/(2π/L)3 = k2dkVdΩ/(2π)3 = number of state with a wave vector of magnitude between k and k + dk in a solid angle dΩ.
dN/dk = k2dΩ/(2π/L)3 = k2VdΩ/(2π)3. (V = L3.)
The density of states for the electron is dN/dE = (dN/dk)(dk/dE), with E = ħ2k2/(2m).
dN/dE = ρ(β,E) = 2½m3/2VE½dΩ/(2πħ))3.
(We do not multiply by 2 since the interaction cannot flip the spin.)
The ground state wave function of hydrogen is (πa02)-½exp(-r/a0).
We therefore have WE1 = (Vπa02)-½∫exp(-ik∙r-r/a0)(-er∙E)d3r.
To evaluate this integral let k point into the z-direction and use spherical coordinates.
r = r(r,θ,φ), E = E(r0,θ0,φ0).
Then E∙r = E0r[cosθcosθ0 + sinθsinθ0cos(φ-φ0)] and
WE1 = -(Vπa02)-½eE0∫∫∫exp(-ik∙r-r/a0)[cosθcosθ0 + sinθsinθ0cos(φ-φ0)]r3drsinθdθdφ
= 2πcosθ0(Vπa02)-½eE0∫-11dcosθ[∫0∞exp(-ikrcosθ - r/a0)r3dr]cosθ.
WE1 = -2πcosθ0(Vπa02)-½eE0∫-11dx (3! x)/(a0-1 + ikx)4 = -i2πcosθ0(Vπa02)-½eE0 16ka05/(1 + k2a02)3.
We have EI = ħω0 = μe4/(2ħ2), a0 = ħ2/(me2), so we may write a02 = ħ4/(m2e4) = ħ/(2mω0).
Energy conservation requires that E = ħω - ħω0.
Therefore k2 = 2mE/ħ2 = 2m(ω - ω0)/ħ, k2a02 = (ω - ω0)/ω0, 1 + k2a02 = ω/ω0.
The probability that an electron is ejected into a solid angle dΩ with energy E per unit time therefore is
w(1,dΩ E) = ( π/2ħ)2½m3/2VE½dΩ/(2πħ))3|-i2πcosθ0(Vπa02)-½eE0 16ka05/(ω/ω0)3|2
= E02(cosθ0)2(ω/ω0)6[(ω - ω0)/ω0]3/2a03dΩ 28/(4πħ)
To find the probability per unit time that an electron is ejected with energy E we must integrate over all possible angles between the direction of the field and the direction of k.
∫02π∫-11(cosθ0)2dcosθ0dφ0 = 4π/3.
w(1,dΩ E) = (256/3)(a03/ħ) E02(ω/ω0)6[(ω - ω0)/ω0]3/2.
Near the threshold for ionization the probability increases from zero as (ω - ω0)3/2. If ω >> ω0 it decreases as ω-9/2. It has a maximum for ω = 4ω0/3.
Nuclei sometimes decay from excited states to the ground state by internal conversion, a process in which an atomic electron is emitted instead of a photon. Let the initial and final nuclear states have wave functions Ψi(r1,r2,..,rz) and Ψf(r1,r2,..,rz), respectively, where ri describes the protons. The perturbation giving rise to the transition is the proton-electron interaction,
W = -∑i=1z e2/|r - ri|,
where r is the electron coordinate.
(a) Write down the matrix element for the process in lowest-order perturbation theory, assuming that the electron is initially in a state characterized by the quantum numbers nlm, and that its energy, after it has been emitted, is large enough so that its final state may be described by a plane wave. Neglect spin.
(b) Write down an expression for the internal conversion rate.
(c) For light nuclei, the nuclear radius is much smaller than the Bohr radius for the given Z, and we can use the expansion
1/|r - ri| ≈ 1/r + r∙ri/r3.
Use this approximation to express the transition rate in terms of the dipole matrix element
d = <Ψf|∑i=1z ri|Ψi>.