## 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 = ķSclass/ķxb where S = ōtatb dt L(x,[x\dot];t) is the action functional. Also show that the energy of the particle is given by E = ķSclass/ķtb. 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) = óõ b a D[x(t)] e[i/((h/2p) )]ōtatb L(x,[x\dot];t) dt .
(1)
3. Evaluate the Gaussian integral ō-ĨĨ eiax2 dx for Re a > 0. Hint: Change variables to t = -iax2 and consider the closed contour in the complex t-plane: {t| 0Ū R,Rexp-ij [0 < j < p/2], -iRŪ 0} and RŪ Ĩ. 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) = Ö{[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) = Ö{[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.