GR9677 #17


Problem


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Quantum Mechanics}Probability
The careless error here would be to just directly square the grids. When one remembers the significance of the meaning of the probability , one finds that one must square the wave function, and not the grids.
The total probability is,
The unnormalized probability from to is,
The normalized probability is thus , as in choice (E).


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Comments 
mhazelm 20080908 12:05:19  If you are familiar with probability theory, this amounts to computing the probability of event A, where A is the the particle's being between x =2 and x=4. From probability, this is given as
P(A) = # ways A can occur/# total possible outcomes.
Now # ways A can occur: if x is between 2 and 3 (an interval of 1) then Psi takes on the value 2 (so Psi^2 = 4). If x is between 3 and 4 (another interval of 1), then Psi^2 = 9. Thus # ways A can occur = 4+9 = 13.
Now for the total number of possibilities, we just do the same thing, but for all possibilities of x. Our intervals are all 1, so we get
1(1^2) + 1(1^2) +1(2^2) + 1(3^2) + 1(1^2) = 16. Now just use the fact that P(A) = 13/16 and you're done with answer E being correct.  

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