Physics 248 HW 1 Solutions

Note: I certainly don’t encourage you to do your solutions entirely in Mathematica unless you find you can do it more efficiently.  Normally I will write out my solutions by hand on paper and then attach whatever Mathematica calculations I did separately.  I did this one all in Mathematica so you can see some different ways to handle units, solve systems of equations, and plot.  I strongly recommend that you include units with numbers wherever you can in Mathematica, as that will allow you to quickly check for obvious errors when the units don’t cancel properly.

Problem 1

e/m: showed electrons are very light and negatively charged, so virtually all the mass in the atom must arise from the positive charges

Rutherford: showed that the positive charge is localized to a size of roughly (10^(-5))Å,  so the density of nuclear matter must be many orders of magnitude greater than the electron density in an atom

both the e/m and Rutherford experiments did not directly show quantum effects.  However, the improved understanding of the structure of the atom was key to Bohr being able to develop his first quantum model of the atom.

Hydrogen spectrum:  used by Bohr to originate the idea of energy levels, the Bohr relation hc/λ=(Subscript[E, i])-(Subscript[E, f])and the idea of quantum jumps.  Combined with Franck-Hertz, also gives independent measurement of h.

Photoelectric effect: the energy of electrons depends on the frequency of the light and not its intensity, in direct contradiction of classical theory.  Allows another measurement of h.

Problem 2

The Ba+ sits in a single energy level for a random amount of time, then “jumps” instantaneously to another level as it emits a photon.  The unpredictability of when the photons are emitted and the short duration of the emission time are key signatures of quantum phenomena.

Problem 3

Integrate[(E^(2π I (OverDot[Δ]) (t^2))),{t,-∞,∞},Assumptions->(OverDot[Δ]) >0]

Problem 4

With[{q=1.2},Table[Plot[Sum[Im[Sin[k π/8]Sinc[k π 0.8](E^(I k θ))],{k,-n,n}],{θ,0,2π},PlotRange->All],{n,{4,40,400}}]]

Show[%,AxesLabel->{"θ (rad)","f(θ)"}]

Problem 5 (1.1 in BD)

Emax[atom_]:=(hc/(λ[atom]))-ϕ(*defines a function*)

Part a&b, solved as a system of equations:

Solve[{Emax[Hg]==3.14 eV,Emax[Na]==0.36 eV},{hc,ϕ}]/.{λ[Hg]->2537 Å,λ[Na]->5890 Å}

h=(hc/(2.998 (10^18)Å/s))/.%

Part c:

λmax=(hc/ϕ)/.%%

Problem 6 (BD 1.2)

part a: Power/(energy per photon)=

(((10^3)J/s 2.998 (10^18)Å/s)/((12400 eV Å)(10^6)/s))((1 eV)/(1.6 (10^(-19)) J))

part b: assume pupil of 5 mm (square for simplicity).  Intensity X area/(energy per photon):

((1.6 (10^(-10))J/s/(m^2)(((5 (10^(-3))m))^2)eV)/(12400 eV Å/(5560 Å)1.6 (10^(-19)) J))

Problem 7 (BD 1.3)

λ=h/p=(h/(Sqrt[2m K]))=(hc/(Sqrt[2(mc^2) K]))
solve both parts of problem at once using list of rest energies (mc^2)={511000,939 (10^6)} eV and list of kinetic energies K={100,1/40} eV

(hc/(Sqrt[2(mc^2) KE]))/.{hc->12400 eV Å, (mc^2)->{511000,939 (10^6)} eV,KE->{100,1/40} eV}//PowerExpand//N

Problem 8 (BD 1.4)

λ = h/p=(hc/E)(since E=pc for relativistic electron)

(hc/e)/.{hc->12400. eV Å,e->100 (10^9)eV}