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374chw11

# 374chw11 - HW#11 —Phys374—Spring 2008 Prof Ted Jacobson...

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Unformatted text preview: HW#11 —Phys374—Spring 2008 Prof. Ted Jacobson Due before 5pm, Tuesday, May 13, 2008 Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/374c/ [email protected] 1. Evaluate R b a δ ( x 2- 3) dx for (i) [ a,b ] = [- 1 , 1], (ii) [ a,b ] = [0 , 2], (iii) [ a,b ] = [- 2 , 0], (ii) [ a,b ] = [- 2 , 2]. (See Chapter 14 for Dirac delta functions. Section 14.3 discusses delta function of a function, which was also explained in class.) 2. Consider the integral I = Z Z f ( x,y ) δ ( x 2 + y 2- R 2 ) δ (( x- a ) 2 + y 2- R 2 ) dxdy, taken over the entire xy plane. (a) Make sketches in the ( x,y ) plane showing geometrically where the two delta functions in in the integrand are non-zero, for a/R = 0 , 1 , 2 , 3. (b) Evaluate I . ( Suggestion : First do the y integral, using the first delta function to identify the relevant y values.) (c) Explain the qualitative behavior the dependence of I on a/R in terms of your sketch in part 2a. In particular explain why it diverges where it diverges, and where it is zero. ( Guidance : Imagine the delta functions as having a small width, before taking the limit as the width goes to zero and the height to infinity, so each of their regions of nonzero support forms a ring. Consider how the area of the region in which both delta functions are non-zero depends on a/R . The idea behind this was explained in class.) 3. Wavepackets and group velocity for a relativistic quantum particle In relativistic quantum mechanics, the (complex) wave function Φ for a spinless par- ticle of mass m satisfies the partial differential equation ∂ 2 t Φ = c 2 ∂ 2 x Φ- ( m 2 c 4 / ¯ h 2 )Φ , (1) where c is the speed of light and ¯ h is Planck’s constant. This is called the Klein- Gordon equation. For simplicity it is assumed here that the wave function depends on only one space coordinate...
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374chw11 - HW#11 —Phys374—Spring 2008 Prof Ted Jacobson...

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