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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?