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 x_b is given by p = ¶S_class/¶x_b where S = ò_ta^t_b dt L(x,[x\dot];t) is the action functional. Also show that the energy of the particle is given by E = ¶S_class/¶t_b. 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 x_b at time t_b given the particle was at x_a at t_a is

   f(xa,ta;xb,tb) = ó õ b a   D[x(t)] e[i/((h/2p) )]òtatb L(x,[x\dot];t) dt .

   (1)

3. Evaluate the Gaussian integral ò_-¥^¥ e^iax2 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(x_a,t_a;x_b,t_b) = A(t_a,t_b) e^{iS_class[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(t_a,t_b) = A(t_b-t_a) = Ö{[m/(2pi(^h/_2p) (t_b-t_a))]}.

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) = Ö{[m/(2pi(^h/_2p) t)]}e^{imx2/2(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, x_a and x_b, and the time interval t = t_b-t_a.