(b) On the same diagram show the variation of ¥(n=1), ‘¥(n=2) and the product
Y(n=1):¥(n=2) across the length of the box. Comment on the physical
significance of the product ¥(n=1):'¥(n=2).
(5)
(c) For the five wavefunctions (n = 1 through n = 5) for a particle-in-a-box, state whether
each of the following statements is TRUE or FALSE about the probability of finding the
particle near x = :
(5)
(i) Least forn=1
(ii) The same (and non-zero) for n = 1, 2, 3,4 and 5
(iii) Zero forn=1, 2,3,4and5
(iv) Least forn=5
(v) Least forn=2 andn=4
QUESTION 3
[19]
(a) With reference to a free particle moving in the x-direction whose wave function is
WY =Ae™, derive expressions of the eigenvalue of the momentum operator,
Py = “inex and the expectation value of the momentum of an observable.
(9)
(b) The normalised wave function for a particle-in-a-box is of the form
¥(x)-(2) sin( “x
Show that the particle-in-a-box wavefunctions are not eigenfunctions of the
as7
momentum operator, Py , but they are for Px.
(6)
(c) Show that the position operator, x, and momentum operator, P,, do not commute.
What does this indicate about the measurement of position and momentum?
(4)