Homework 2

constants and unit definitions

This is a block of code I use to define my constants. The internal units for calculations are SI, but the numbers can be entered in other units.

s=1;ms=(10^(-3))s;us=(10^(-6))s;ns=(10^(-9))s;ps=(10^(-12))s; fs=(10^(-15))s;
Hz=1;GHz=(10^9)Hz;THz=(10^12)Hz;
m=1;cm=(10^(-2))m;mm=(10^(-3))m;um=(10^(-6))m;nm=(10^(-9))m;angs=(10^(-10))m;
kg=1;g=(10^(-3));
J=1;mJ=(10^(-3))J;eV=e J;keV=(10^3)eV;MeV=(10^6)eV;
W=1;kW=(10^3)W;GW=(10^9)W;mW=(10^(-3))W;
Torr=1/760;
K=1;

c=299792458 m/s;      
eps0=8.854187817 (10^(-12));   (*permittivity of free space*)
re=2.81794092 (10^(-15))m;
e=1.60217733 (10^(-19));		(*Coul electron charge*)
Natm=760 3.3 (10^16)/(cm^3);  (*no.particles in 1 atm*)
me=9.1093897 (10^(-31))kg;
mp=me/0.000544617013;
hbar=1.05457266 (10^(-34))J s;
h=2π hbar;
kB=1.3806503 (10^(-23)) J/K;

Problem 1: mean number of photons per mode.

note: you can look up or use Mathematica for your sums. Show your work though.

a. Referring to the class notes, the probability (Subscript[P, n]) of having an excitation energy (Subscript[E, n ]) is given by the sum  (Subscript[P, n])=((e^(−(Subscript[E, n])/(Subscript[k, B])T))/((∑_(j=0)^∞) (e^(−(Subscript[E, j])/(Subscript[k, B])T)))).

Show that if β=((h ν)/((Subscript[k, B])T)), we can rewrite this as
(Subscript[P, n])=((e^(−n β))/((∑_(j=0)^∞)(e^(−j β)))), 
then using the fact that this is a geometric progression, show that
(Subscript[P, n])=(e^(−n β))(1-(e^(-β)))

b. The mean number of photons in the mode is (Overscript[n, ¯])=(∑_(n=0)^∞)n  (Subscript[P, n]). Show that 
(Overscript[n, ¯])=(1/((e^β)-1))=(1/((e^((h ν)/((Subscript[k, B])T)))-1))

Problem 2: Blackbody emission

a. Read problems 2.2 and 2.3 and their solutions in the textbook by Svelto. Use the FindRoot[ ] function in Mathematica to numerically solve for the wavelength maximum of the blackbody spectrum, to give Wien’s law as expressed in problem 2.4.

b. problem 2.4

Problem 3: thermal camera calibration

A thermal camera measures the total integrated power in the wavelength range 8-14 μm. 
a. Define a function ρλ[λ_,T_] for the blackbody spectral energy density as a function of wavelength and temperature given in the solution to 2.2.

b. Make a plot of ρλ[λ,T] for 3 temperatures (0C, 100C, 200C over the wavelength range 1μm to 30μm.

c. Numerically calculate the area under the curve in this wavelength range as a function of temperature. Hint: define a function like thermalCamSig[ T_] := NIntegrate[ ]...

d. Make a plot of the signal vs temperature over a temperature range 0C to 200C.  This a simplified version of the calibration curve of the thermal camera.

Problem 4. Practice with QM wavefunctions
Consider a mixture of two states of a particle in a one-dimensional infinite square well with width L, so that ψ(x,t)=C (3 (Subscript[ψ, 2])(x,t)+4 (Subscript[ψ, 3])(x,t)).

a. Using Mathematica or calculating analytically, calculate the normalization constant C. Assume the eigenfunctions are normalized. Hint: using Dirac notation, so that (Subscript[u, n])(x)→|n⟩, makes this calculation much easier.

b. Calculate an expression for the charge density and make a plot of it at 3 representative times to illustrate the motion of the charge.

c. Calculate an expression for the time-dependent dipole moment.

d. Let the width of the well be L = 0.1nm. Calculate the following quantities: 
- From the infinite well energy levels, calculate the resonance frequency (Subscript[ω, 32]) - Calculate the corresponding transition energy in eV. 
- Calculate the spontaneous lifetime.