Download PHYS-2100 Introduction to Methods of Theoretical Physics Fall 1998 1)

PHYS-2100 Introduction to Methods of Theoretical Physics
Fall 1998
Homework Assignment Due Tuesday, Nov.10
1) Nettel, Exercise 6-1. You might find it useful to use the conversion factor hc = 200 MeV-fm .
The term “resolving power” (needed in part b) just refers to the ability to distinguish images
separated by a certain angular distance, due to wave diffraction. For the relative resolving
powers, just compare the angles θ = λ ⁄ d where λ is the wavelength and d is the size of the
aperture, for the electron and optical microscopes. (You can assume a wavelength of 550nm
for visible light.) You’ll find more discussion in HRW5e, Sec.37-5. Also, the term “10 keV”
electron refers to the electron’s kinetic energy.
2) Nettel, Exercise 6-2. I think he means “(6.2e)” instead of “(6.2a)”. Follow our derivation of
normal modes on a string, class notes on Sept.17, or Nettel pages 56-57.
3) Prove Eq.6.24b in Nettel. That is, show that for an observable Q , the uncertainty ∆Q in Q ,
1⁄2
defined to be ∆Q = [ ( Q – Q ) 2 ]
(in “bar” notation) or ∆Q = 〈 ( Q – 〈 Q〉 ) 2〉 1 ⁄ 2 (in
“bracket” notation) can also be written as
2
2 1⁄2
∆Q = ( 〈 Q 〉 – 〈 Q〉 )
4) This problem is all based on the “particle in a one-dimensional box” (with impenetrable walls)
that we discussed in class. The relevant wave functions and other discussion are in Nettel,
Sec.6.2, except that our box has width L = 2a and extends from x = – a to x = a .
a) In class we found that 〈 x〉 , the expectation value of position, was zero. What value would
you get for the problem as set up by Nettel in Fig. 6.1, i.e. a box from x = 0 to x = L ?
b) For our box, centered at x = 0 , show that the expectation value of momentum 〈 p〉 is also
hd
zero. Recall that the momentum operator is p = ---- ------ . What value would you get for the
i dx
problem as set up by Nettel in Fig.6.1?
c) Find the expectation value 〈 x 2〉 for the eigenfunction with label n , and thereby show that
a n2π2 – 6 1 ⁄ 2
the uncertainty in position is given by ∆x = ------  --------------------
.
3 
πn 
hπ
d) Show that the uncertainty in momentum is given by ∆p = -------n .
2a
e) Discuss the product ∆x ⋅ ∆p of the position and momentum uncertainties. What does it
mean that this value is never zero? For what value of n is it the smallest? What happens
to the product as n → ∞ ? Is the behavior at large n due to the behavior of ∆x , ∆p , or
both? Does this behavior make sense physically?