Quantum wells. (a) For an electron in an infinite square well potential, what mu
ID: 1787484 • Letter: Q
Question
Quantum wells.
(a) For an electron in an infinite square well potential, what must be the width L of the well in order that the energy difference between the n=2 and n=1 states corresponds to the energy of a photon with wavelength (i) 600 nm (red) and (ii) 490 nm (blue)?
(b) Use the uncertainty principle to get a “ball park” order of magnitude estimate of the magnitude of the electron's momentum in the n=1 state for the "red" well in (a).
(c) Suppose that you increase the width of the “red” potential well by a factor of 10. By what factor will the energy difference between the n=2 and n=1 states change?
Explanation / Answer
a. let width of well be L
then for n = 1, lambda = 2L
n = 2, lambda = L
energy of electron in nth energy level = En
now, lambda = h/p
2Em = p^2
E = p^2/2m = h^2/lambda^2*2m
E1 = h^2/8m*L^2
E2 = h^2/2m*L^2
E2 - E1 = 3h^2/8mL^2
for red wavelength a) lambda = 600 nm
hc/lambda = 3h^2/8mL^2
L^2 = 3h*lambda/8mc
L = 7.392*10^-10 m
for lambda = 490 nm
L^2 = 3h*lamnda/8mc
L = 6.680*10^-10 m
b. for red well, n = 1
momentum, = p = h/2L = 4.484*10^-25 kg m/s
c. for increasgin L by factor of 10, lambda increases by factor of sqrt(10) and ebergy difference decreaes by factor of 100