(Top graph) The wave function of a particle is shown below. At which location is the probability of finding the particle the highest? A) Location p, B) Location s, C) Location q, D) Location...

(Top graph) The wave function of a particle is shown below. At which location is the probability of finding the particle the highest?

A) Location p,

B) Location s,

C) Location q,

D) Location r<- most likely

(Bottom Graph) The wave function of a particle is shown below. At which location is the probability of finding the particle to the right of the origin is (blank) the probability of finding the particle to the left of the origin. 

A) Equal to <-my buddy says due to length and abosulte value they are equal

B) lower than

C) higher than

 

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Expert Answers
Borys Shumyatskiy eNotes educator| Certified Educator

Hello!

The probability of being at some interval is the integral of the square of a wave function (over that interval). I suppose "location" means "small interval around a given point", because for a continuous wave function the probability of being at the point itself is always zero.

For small intervals and continuous wave function the probability is about `Delta x*|Psi(x)|^2,` therefore it is greater at that locations where `|Psi(x)|` is maximal. Yes, the answer to the first question is D) r.

There is some confusion at the second question. Does it asks where the probability is greater, anywhere at the left or anywhere at the right? Or is asks about a small neighborhood of the origin?

If the first, then we have to compare two integrals. It is not so simple because we have no exact expressions for the left and right parts. We have to compare the areas under two halves of the graph, and not for the bottom graph itself or its absolute value, but its square.

If we suppose that the left part of the wave function is `(1-sin(4 pi x))` and the right is `-3sin(pi x),` then `int_(-1)^0 (1-sin(4pi x))^2 dx = 1.5` and `int_0^1 (-3sin(pi x))^2 dx = 9/2,` so the right half is much more probable.

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