physiCs Q-IV: An electron in a wire can be considered as if in one-dimensional i
ID: 1394744 • Letter: P
Question
physiCsQ-IV: An electron in a wire can be considered as if in one-dimensional infinite potential well. Consider L 0.1 mm long wire; (a) Calculate the ground-sate energy of an electron in such a wire. (b) If the electron's kinetic energy is 30 meV (almost equal to energy of a gas molecule at T-300 K), what will be the electron's quantum number n corresponding to that energy? (c) What will be probability of finding electronic within the interval Ax -o.001L at middle of the wire ie. xaL/2. Given that wave function yn(x) where sin n 1,2,3,4...
Explanation / Answer
L = 0.1 mm = 0.1 x 10^-3 m
a) E = n^2 h^2 / ( 8 m L^2 )
here n = 1
h = 6.63 x 10^-34
m = 9.09 x 01^-31
L = 0.1 x 10^-3
E = 1^2 x (6.63 x 10^-34)^2 / ( 8* 9.09 * 10^-31 *( 0.1*10^-3)^2 ) = 6.045 x 10^-30 J
b) 30 meV = 30 x 10^-3 x 1.6 x 10^-16 = 4.8 x 10^-21 J
n = sqrt(4.8 x 10^-21 / ( 6.045 x 10^-30)) = 28178.79?
Quantum number = 28179
c) Probability = (2/L). integral (Sin^2(n.Pi.x/L) ) from x = 0 to x = L/2
= (2/L) * (L/4) = 2/4 = 0.5
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