Problem 1 refers to: A standing electric field wave (lots of photons) in a quant

Question

Problem 1 refers to: A standing electric field wave (lots of photons) in a quantum
wire stretching between x = 0 and x = L is described by
E (x,t ) =Emax sin(3·x / L)cos(3·ct / L) . Let L = 900 nm.

1.(a) What is the wavelength of the wave? What is the ordinary frequency? In
which part of the electromagnetic spectrum (i.e., x-ray, UV, IR, light, etc.) is this
wave? Explain.

(b) How many nodes does the wave have for 0 x L? Where are they

(c) Calculate uE /t , where uE = 0E2 is the electric field energy density. Is the
electric field energy density constant in time at different points along the wire?
located?

Explanation / Answer

general way of representation of a wave is E = Emax*sin(kx)*cos(wt)

k is the wave number = 2*pi/lamda

lamda is the wavelength =

then k = 3*pi/L

then wavelength lamda = 2*pi/k = 2*pi/(3*pi/L)= (2L/3)

lamda = 2*900*10^-9/3 = 600 nm

frequency f = w/(2*pi) = (3*pi*c/(L*2*pi)) = 3*c/(2*L)

c is the speed of light = 3*10^8 m/s

f = (3*3*10^8)/(2*900*10^-9) = 5*10^14 Hz

visible spectrum

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