D12-2 - Quantum Mechanics - January D12-2 - Quantum Mechanics - due January 20, 2000

I. Path Integrals for Quantum Amplitudes



  1. For a classical particle moving under conservative forces show that the momentum of a particle at the endpoint xb is given by p = Sclass/xb where S = tatb dt L(x,[x\dot];t) is the action functional. Also show that the energy of the particle is given by E = Sclass/tb. Recall that the energy of a classical particle is given by E = L-p[x\dot].

  2. The path integral for the amplitude a particle to be at xb at time tb given the particle was at xa at ta is
    f(xa,ta;xb,tb) =
    b

    a 
     D[x(t)] e[i/((h/2p) )]tatb L(x,[x\dot];t) dt .
    (1)
  3. Evaluate the Gaussian integral - eiax2 dx for Re a > 0. Hint: Change variables to t = -iax2 and consider the closed contour in the complex t-plane: {t| 0 R,Rexp-ij [0 < j < p/2], -iR 0} and R . Consider amplitude for a particle whose motion is derived from a Lagrangian of the form L = a(t)[x\dot]2+b(t)x2+c(t)x+d(t).

    1. Show that f(xa,ta;xb,tb) = A(ta,tb) eiSclass[a,b]/(h/2p) for the general case. Hint: re-write the path integral in terms of the deviations from the classical path, [`x](t).
    2. For a free particle of mass m show that A(ta,tb) = A(tb-ta) = {[m/(2pi(h/2p) (tb-ta))]}.

  4. The amplitude for a free particle to be at x at time t given the particle was at x = 0 at t = 0 is f(x,t;0,0) = {[m/(2pi(h/2p) t)]}eimx2/2(h/2p) t.

    1. Consider the spatial variation of f(x,t;0,0) at fixed t and large x. Show that the amplitude has a wavelength l given by the de Broglie relation, p = h/l, for a particle of momentum p.
    2. Similarly, consider the temporal variation of f at fixed x and show that the amplitude has a frequency w given by the Einstein relation E = (h/2p)w for a particle with energy E.

  5. Calculate the amplitude for a particle of mass m moving in a harmonic potential with natural frequency w. Express the result in terms of the initial and final positions, xa and xb, and the time interval t = tb-ta.


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On 17 Jan 2000, 16:10.