Page 69 - Electrical Properties of Materials

52 The electron

3.12. Show that the differential equation for the electric The solutions for the four lowest states are as follows: ∗
field of a plane electromagnetic wave, assuming exp(–iωt) 1 2
ψ(ζ)= H n (ζ) exp(– ζ ),
time dependence has the same form as the time-independent 2
Schrödinger equation for constant potential. Show further that where
the expression for the Poynting vector of the electromag- 2 mω 0
ζ = αx, α = ,
netic wave is of the same functional form as that for the
quantum-mechanical current. 2
H 0 =1, H 1 =2ζ, H 2 =4ζ –2, and
3
H 3 =8ζ –12ζ.
3.13. The potential energy of a classical harmonic oscillator
is given as Find the corresponding energies. Compare them with the
1
2 2
V(x)= mω x . energies Planck postulated for photons.
2 0
We get the ‘quantum’ harmonic oscillator by putting the above *Here H n is the Hermite polynomial of order n, named after
potential function into Schrödinger’s equation. Charles Hermite (who studied them in 1864).