Question

5 = 10 marks ] Question 1 [3 2 (a) Use the Fourier transform, -) / Ф(Р) e'pr/h d3p 27TH and the inverse transform 1 b(FeipF/hd3r Ф(Р) 2тh to prove the Fourier Integral Theorem: 1 ') ei(F'-p)/h d3p' d*r. Ф(р) - 2тh (b) Explain why the Dirac-ô may be represented via - ih)/ 1 8(F- F') (c) Show that for arbitrary wave functions /a,b(f) that / -/ Фа (р)" фь (р) d'р, Va(r = where ba and da (and /, and di) are related by Fourier transformation in the usual way. State the significance of this result for: The relationship between orthonormality conditions in position and momentum space. The physical interpretation of momentum-space wave functions (explain what special case is relevant for this).

Answer 1

CF) (PePAe F) eF- F). 5/s f d'e )eFdp b) Simitarly, ) : nkSeF p- 8(F-F') snch that c) orhonermad ,or S)た()ai: Sa6 If ta() (7) )&() are So is

The physical interpretation of momentum space wavefunction :: the modulus squared of the momentum space wavefunction is the the probability density to find the particle with momentum p.