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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 P=\int |\langle \psi | \psi \rangle|^2 dV, one finds that one must square the wave function, and not the grids.

The total probability is,

The un-normalized probability from x=2 to x=4 is,

The normalized probability is thus 13/16, as in choice (E).

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2008-09-08 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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