Issued Wed, Mar 6 Due Wed, Mar 13

Required reading: through F+T 3.10 (Beiser: through 5.7)

1 F+T 3-7

2 F+T 3-9

3. F+T 3-10

In addition: 3c) Find the expectation value <x>

(Hint: Do you really have to do any integration at all?)

4a) In class, I made a handwavy defense of the correspondence principle: namely, that as n -> infinity, the quantum particle-in-a-box gives the same results as a classical particle-in-a-box.

Sharpen this up, by showing that as n -> infinity, the probability of finding a quantum particle (in a box of size L) between the positions goes to , independent of x., no matter if is big or small. (which is precisely the classical expectation.)

b) An important property of eigenfunctions of a system is that they are orthogonal to one another, i.e. . (Here, is the wavefunction correesponding to the nth energy eigenvalue)

Verify this orthogonality relation, for the specific case of the eigenfunctions of a particle in a one-dimensional box.

5a) A particle in a box that runs from 0 to L should have by symmetry. Verify that this is correct no matter what n is. Then, calculate for the nth wavefunction.

b) Show that in general, given some distribution of positions x, . (This quantity represents the deviation from the average, and can be used as a more rigorous definitition of what we mean by the uncertainty in the position, or .

c) Combine a and b, to find the uncertainty in the position of a particle in a box of size L, when n=1. What is the limit as n-> infinity?