2. General Physics

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2.4 Problems

2.4.5  Modern Physics

Problem 1:  The Bohr Atom

Part (a)  Using Bohr's model of quatization, derive the energy levels as a function of the orbit number n.  Plot the energy levels.

coulomb = Z e^2/r^2

centrifugal = m v^2/r

eq1 = coulomb == centrifugal

angmom = m v r

eq2 = angmom  ==  n ℏ

rule1 = Solve[{eq1, eq2}, {v, r}] // Flatten

kinEn  =  1/2 m v^2

potEn  =  -Z e^2 /r

totEn  =  kinEn + potEn

energy[n_] = totEn /. rule1

\(values = {e -> ElectronCharge
		\ \ \ \ \ \ \ \ \ \ \ \ , 
        m -> ElectronMass
		\ \ \ \ \ \ \ \ \ \ \ \ , 
        Z -> 1/\((4\ π\ ϵ\_0)\)
			\ \ \ \ \ \ \ \ \ \ , 
        ϵ\_0 -> 8.854\ 10^\(-12\)\ Coulomb^2/\((Newton\ Meter^2)\)

        				\ \ \ \ \ \ \ \ , ℏ -> PlanckConstantReduced}\)

energyEV[n_] := Convert[energy[n] //. values, ElectronVolt]

energyEV[1]

\(\(Plot[
      Evaluate[Table[energyEV[n], {n, 1, 10}]/ElectronVolt]
, {x, 0, 1}, 
      Axes -> {False, True}, PlotRange -> All]; \)\)

Part (b)  Derive the frequency of radiation produced in a trasition from Bohr orbit n1 to Bohr orbit n2.

eq3 =  ℏ ω  == energy[ni] - energy[nf] // Simplify

Thread[eq3/ℏ, Equal]

freq[ni_, nf_]  :=  
      Convert[(energy[ni] - energy[nf])/(2 π ℏ) //. 
          values, Hertz]

freq[3, 1]

freq[Infinity, 1]