A particle is confined to a one-dimensional box (an infinite well) on the x-axis between x = 0 and x = L. The potential height of the walls of the box is infinite. The normalized wave function of the particle, which is in the ground state, is given by with 0 <= x <= L a) What is the probability per unit length of finding the particle? (Correct ans 2/L might have to do with sin(0)=0) b) What is the probability of finding the particle between x = 0 and x = L/3 (I got .196 or .20 by basic integral)

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The wave functions of a particle confined to an infinite potential well between x = 0 and x = L are

`Psi_n (x) = sqrt(2/L)sin((npix)/L)` , where n is an integer (n = 1, 2, 3...). These wave functions are normalized so that the probability of finding the particle in the well is 1 and the probability of finding the particle outside of the well is 0.

Since the particle in this problem is in the ground state, n = 1 and its wave function is

`Psi_1(x) = sqrt(2/L)sin(pix/L)` .

b) The probability of finding the particle between x = 0 and x = L/3 is then

`P = int_0 ^ (L/3) |Psi_1|^2 dx`

Let's work with the integrand first and rewrite it using a trigonometric half-angle identity:

`|Psi_1|^2 = 2/Lsin^2(pix/L) = 2/L*1/2*(1 -...

(The entire section contains 2 answers and 382 words.)

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