6) According to quantum mechanics, the position (x) of a particle in a one dimen

ID: 2133268 • Letter: 6

Question

6) According to quantum mechanics, the position (x) of a particle in a one dimensional box with

dimensions - L/2 ? x ? L/2 (L constant) can be described by the following probability

distribution function p(x):

p(x) = Acos2[?x/L] for -L/2 ? x ? L/2, and 0 for all other x.

a) Find the normalization constant A in terms of L.

b) Find the mean, mode, and median position of the particle in the box.

c) Show that the variance (?^2) of x is given by:

?^2=((L/pi)^2)*((pi^2-6)/12)

d) What is the probability of finding the particle in the region: L/4 ? x ? L/2?

PART (b)

Plot the "p(x) = Acos2[?x/L]"; for -L/2 ? x ? L/2

According to this diagram, we find:

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x_mode = 0

(x=0 is most probable point)

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x_median = 0

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