- #1
leehufford
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Homework Statement
A particle is described by the wave function psi(x) = b(a2-x2) for -a < x < +a and psi(x) = 0 for x < -a and x > +a, where a and b are positive real number constants.
a) Using the normalization condition, find b in terms of a.
b) What is the probability to find the particle at x = +a/2 in a small interval of width 0.010a?
c) What is the probability for the particle to be found between x = +a/2 and x = +a?
Homework Equations
Psi(x) = b(a2-x2)
The integral of Psi(x)2 from negative infinity to infinity must equal one, so that way it can exist.
P(x) dx = (Psi(x))2
The Attempt at a Solution
I have an answer, but I was hoping someone could confirm I did this right, this concept is brand new to me and my answer from A looks weird to me.
Part A:
b2* Integral of (a2-x2)(a2-x2) dx = 1 (From -a to a, since everywhere else Psi is zero).
b2[a4x - (2/3)a2x2 + (1/5)x5] dx = 1 evaluated from -a to a...
b2[(a5-(2/3)a5+(1/5)a5) - (-a5 + (2/3)a5-(1/3)a5) = 1
b2(16a5/15) = 1, so b = sqrt(15/16a5).
Part B:
P(x)dx = (Psi(x))2 dx
= (15/16a5)(a2-x2)2 dx
= (15/16a5)(a2 - (1/4)a2)2(0.010a)
= (15/16a5)(3/4)a4(0.010a)
=0.007, or 0.7 percent chance of finding the particle there.
Part C: I didn't attempt part C because I wanted to hopefully get some feedback that I was on the right track before I started part C. Looks like I will actually have to integrate for part C since the interval is bigger. Thanks for reading.
Lee
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