Section 10 Classical Limit Page 511 Lecture 28: Classical Limit Date Given: 2008/12/05 Date Revised: 2008/12/05 Page 512 Ehrenfest’s Theorem Ehrenfest’s Theorem How do expectation values evolve in time? We expect that, as quantum effects becomes small, the fractional uncertainty in a physical observable Ω, given by q h(∆Ω)2i/hΩi, for a state |ψ i, becomes small and we need only consider the expectation value hΩi, not the full state |ψ i. So, the evolution of expectation values should approach classical equations of motion as ~ → 0. To check this, we must first calculate how expectation values time-evolve, which is easy to do: d „ d « „ d « „ dΩ « hΩi = hψ | Ω |ψ i + hψ |Ω |ψ i + hψ | |ψ i dt dt dt dt i „ dΩ « = − [− (hψ |H)Ω |ψ i + hψ |Ω(H|ψ i)] + hψ | |ψ i ~ dt i „ dΩ « = − hψ |[Ω, H]|ψ i + hψ | |ψ i ~ dt i fi dΩ fl = − h[Ω, H]i + (10.1) ~ dt Section 10.1 Classical Limit: Ehrenfest’s Theorem Page 513 Ehrenfest’s Theorem (cont.) If the observable Ω has no explicit time-dependence, this reduces to d i hΩi = − h[Ω, H]i (10.2) dt ~ Equations 10.1 and 10.2 are known as the Ehrenfest Theorem. Section 10.1 Classical Limit: Ehrenfest’s Theorem Page 514 Ehrenfest’s Theorem (cont.) Applications of the Ehrenfest Theorem To make use of it, let’s consider some examples. For Ω = P, we have d i hPi = − h[P, H]i dt ~ We know P commutes with P2/2 m, so the only interesting term will be [P, V (X )]. We can see what that is via Taylor expansion of V (X ): " ∞ # ∞ X n X n [P, V (X )] = P, VnX = Vn [P, X ] n=0 n=0 ∞ X ` n−1´ dV (X ) = Vn −i n X = −i ~ ~ dX n=0 n n−1 dV (X ) where [P, X ] = −i ~ n X can easily be proven via induction and dX should be interpreted as “take the derivative of V (x) and replace x with X everywhere.” Section 10.1 Classical Limit: Ehrenfest’s Theorem Page 515 Ehrenfest’s Theorem (cont.) So we have d fi dV fl hPi = − dt dX which is the standard classical result for Newton’s second law in the presence of a potential V (x). Note that this result holds for arbitrary ~; it only becomes a good description of the state, though, when h(∆X )2i/hX i2 1 and h(∆P)2i/hPi2 1. Another classical result is found by setting Ω = X : d i fi P2 fl hX i = − [X , ] dt ~ 2 m 2 By the same kind of calculation as applied above, [X , P ] = 2 i ~ P, so d hPi hX i = dt m which is the classical relation between velocity and linear momentum. Section 10.1 Classical Limit: Ehrenfest’s Theorem Page 516 Ehrenfest’s Theorem (cont.) The above two results can be rewritten as d fi dH fl d fi dH fl hPi = − hX i = dt dX dt dP because, for the first equation, the derivative of the kinetic term in H with respect to X vanishes, and, for the second equation, the derivative of the potential term with respect to P vanishes. These we recognize as Hamilton’s equations. Section 10.1 Classical Limit: Ehrenfest’s Theorem Page 517 Correspondences between Classical and Quantum Mechanics Poisson Brackets and Commutators Recall in classical mechanics that the Poisson Bracket of two functions f (x, p) and g(x, p) of the the state variables x and p is ∂f ∂g ∂g ∂f {f , g}q,p = − ∂x ∂p ∂x ∂p We have in classical mechanics the generic result for the time evolution of a function F of the state variables x, p df ∂f = {f , H} + dt ∂t where H(x, p) is the classical Hamiltonian. Notice how similar this is to Ehrenfest’s Theorem; if one replaces f (x, p) with the expectation value of the analogous quantum operator F (X , P) and the Poisson bracket with − i [F , H], one obtains Ehrenfest’s Theorem. ~ Section 10.2 Classical Limit: Correspondences between Classical and Quantum Mechanics Page 518 Correspondences between Classical and Quantum Mechanics (cont.) The above general classical expression for time evolution of a function f (x, p) yields Hamilton’s equations when applied to f = x and f = p: dx ∂H dp ∂H = {x, H} = = {p, H} = − dt ∂p dt ∂x We saw earlier the quantum version of these equations, again with { , } replaced by − i [ , ]. ~ Now, recall that Postulate 2 can be rewritten to specify [X , P] rather than the matrix elements of X and P in the {|x i} basis. We showed that the two formulations were equivalent. But the latter formulation was a postulate, not derivable in any way. The analogies we’ve drawn between classical and quantum mechanics suggest a prescription that leads from classical mechanics to [X , P] = i ~. Specifically, we know in classical mechanics that {x, p}x,p = 1 (just calculate it). We’ve seen above that, in many cases, quantum mechanical Ehrenfest relations reduce to the equations of i classical mechanics if we make the correspondence − [ , ] ↔ { , }q,p where q and p are the variables that define the classical trajectory and~ that are replaced by the operators Q and P in quantum mechanics. This correspondence would give, for example, [X , P] = i ~. Morover, it provides a mean to convert any classical problem to a quantum mechanical problem. It’s not a derivation, but it is a prescription for what the Postulate 2 commutators should be; along with the other postulates, it defines quantum mechanics completely. Section 10.2 Classical Limit: Correspondences between Classical and Quantum Mechanics Page 519 Correspondences between Classical and Quantum Mechanics (cont.) The next interesting correspondence is a direct one between the Schr¨odingerEquation and the Hamilton-Jacobi equation of classical mechanics (don’t worry if you don’t know know what the latter is.) You of course know and love the Schr¨odinger Equation, which, projected onto the {|x i} basis, is 2 ∂2ψ ∂ψ − ~ + V (x) ψ = i 2 m ∂x2 ~ ∂t where V (x) is the classical potential in which the particle moves. We know from the WKB approximation that a reasonable approximate form for the wavefunction is p i S(x,t) ψ(x, t) = ρ(x, t) e ~ (10.3) Section 10.2 Classical Limit: Correspondences between Classical and Quantum Mechanics Page 520 Correspondences between Classical and Quantum Mechanics (cont.) Let’s insert the above form into the Schr¨odinger Equation: 2 ( 2√ √ 2 2 ) ∂ ρ 2 i ∂ ρ ∂S 1 √ „ ∂S « i √ ∂ S i √ i − ~ + − ρ + ρ e S + V (x) ρ e S 2 2 2 ~ ~ 2 m ∂x ~ ∂x ∂x ~ ∂x ~ ∂x √ ff ∂ ρ i √ ∂S i S = i ~ + ρ e ~ ∂t ~ ∂t The equation contains terms up to second order in ~; neglect all terms containing ~ or 2 ~ . (To do this correctly, one ought to really divide through by S(x) and discard all terms that have the dimensionless parameter ~/S(x) 1, but that will yield the same √ i S result.) This leaves (canceling the common ρ e ~ ) 1 „ ∂S «2 ∂S + V (x) + = 0 2 m ∂x ∂t This is the Hamilton-Jacobi Equation of classical mechanics. It is equivalent to (in fact, derived from) Hamilton’s Equations. It is not particularly remarkable that we obtain a classical equation of motion by taking the classical limit of the Schr¨odinger Equation; we’ve done the same thing with the Ehrenfest relations. The remarkable thing is the role the classical action plays in the quantum-mechanical problem in the semiclassical limit, as we saw in connection to the WKB approximation. Section 10.2 Classical Limit: Correspondences between Classical and Quantum Mechanics Page 521.