The semiclassical limit you're describing says that the amplitude for a particle to get from here to there in a set time is equal to the exponential of the classical action for the corresponding classical trajectory. In symbols this reads
⟨xb|U(T)|xa⟩=∫DφeiS[ϕ]/ℏ≈eiS[φcl(xa,xb,T)]/ℏ.
In a general quantum state, however, particles are not "here" and don't end up "there": they have an initial probability amplitude
⟨x|ψ(0)⟩ for being at each position
x at time
t=0 and will have a final probability amplitude
⟨x|ψ(T)⟩ for being at position
x at time
T. To apply the approximation, you pull out the propagator and insert a resolution of the identity:
⟨x|ψ(T)⟩=∫dy⟨x|U(T)|y⟩⟨y|ψ(0)⟩=∫dyeiS[φcl(y,x,T)]/ℏ⟨y|ψ(0)⟩.
To get a full semiclassical limit, you also need a semiclassical initial state (since otherwise you've obviously got no hope!). You take, then, a state with (relatively) sharply defined position and momentum (of course, the state will occupy some finite region of phase space but you usually can assume, in these circumstances, that it is small enough), and this will make the amplitudes for points outside the classical trajectory interfere destructively and vanish.
EDIT
So how does this happen? For one, y must be close to the initial position, y0 in order to contribute to the integral. For small displacements of the endpoints, then, the action along the classical trajectory varies as δS=pφ,xδx−pφ,yδy
(cf. Lanczos, The Variational Principles of Mechanics, 4th edition, Dover, eqs 53.3 and 68.1, or simply do the standard integration by parts and set
∫δLdx=0 along the classical trajectory). The main contribution of the initial state to the phase is of the form
eipcly, which means that the integral has more or less the form, up to a phase,
⟨x|ψ(T)⟩≈∫y0+Δx/2y0−Δx/2ei(pφ,y−pcl)y/ℏdy.
Here the momentum pφ,y is determined by x and (to leading order) y0, since there is a unique classical path that connects them. This momentum must match (to precision Δp≈ℏ/Δx, which we assume negligible in this semiclassical limit) the classical momentum of the initial state, pcl, and therefore only those x's on the trajectory determined by the initial state will have nonzero amplitudes.
This post imported from StackExchange Physics at 2014-03-12 15:43 (UCT), posted by SE-user Emilio Pisanty