The Ehrenfest Theorems

Transcription

1 The Ehrenfest Theorems Robert Gilmore Classical Preliminaries A classical system with n egrees of freeom is escribe by n secon orer orinary ifferential equations on the configuration space (n inepenent coorinates) in the Lagrangian representation L L = 0 () q q an 2n first orer orinary ifferential equations on the phase space (n inepenent coorinates an the conugate momentum of each coorinate) in the Hamiltonian representation q = + H p p = H q (2) The first set of equations is calle the Euler-Lagrange equations an the secon set is calle the Hamiltonian equations of motion. Functions escribing particles epen on time implicitly through the time epenence of the particle s coorinates an momenta; they may also epen explicitly on time: f = f(q(t), p(t), t). The time erivative of such functions has the form f = f + f q i q i + f p i p i = f + f H f H q i p i p i q i = f + {f, H} (3) The partial erivative provies information about the explicit time epenence of the function. The implicit time epenence, epening on the motion of the particle, is provie by the Poisson bracket: {f, g} f g f g (4) q p p q

2 2 Classical - Quantum Corresponence An elegant formulation of Quantum Theory is given in terms of a relation between the Poisson bracket of Classical Mechanics an the commutator (Lie) bracket of Quantum Mechanics: Classical Mechanics QuantumMechanics [ {f, g} ˆf, (5) ĝ] i h The hat ˆ on the right han sie inicates that the corresponents are operators. To see how this works, we observe irectly from Equ.(4) that {q, p k } = δ k. From this an the Classical Quantum mapping Equ.(5) we observe that [ˆq, ˆp k ] = i hδ k. Thiss relation can be satisfie in one of two obvious ways, one emphasizing the coorinates (the usual), the other emphasizing their conugate momenta: Representation ˆq ˆp k h Coorinate q i q k Momentum h p k i p A final piece of the puzzle that we nee is the time-epenent Schröinger equation. For a free particle with wavefunction ψ(x) = e ikx the momentum is hk by looking at the eigenvalue of the momentum operator ˆp = ( h/i) / x. If we woul like to represent the time-epenence of a free particle moving to the right with a momentum hk in the form e ikx ωt we must choose the time-epenent form of the Schröinger equation to be 3 Ehrenfest Theorems ψ(x, t) Hψ(x, t) = +i h The Ehrenfest Theorem comprises a whole class of results, all of which assume the same form: Classical Mechanics Quantum Mechanics (7) A = B Â = ˆB To show this, we write own the time erivative of an expectation value as follows: (6) 2

3  = ψ (x, t)âψ(x, t)v = ψ (x, t)  ψ(x, t)v + ψ (x, t) Âψ(x, t)v + In the expression above we can replace ψ(x,t) ψ (x,t) by H/i h. The result is ψ (x, t)â ψ(x, t) V (8) by H/i h an we can replace H]  =  + [Â, i h (9) Equation (9) for Quantum systems is ientical to Equ. (3) for Classical systems through the Quantum-Classical corresponence of Equ. (5). This is Ehrenfest s Theorem. 4 Simple Applications The following results are immeiate: x = p m p = V x = F L = r F (0) The Hamiltonian equations of motion are obtaine as follows. Set  = ˆq. In this case q = i h [q, H] = i h i h H = H () p p The replacement of the commutator [q, H] by the erivative i h p H is a irect application of Equ. (6). In a similar way we fin Ehrenfest s limit for the other of Hamilton s equations. The symmetry is q = + H q = + H (2) p p (3) p k = H p k = H (4) q k q k (5) 3

4 If there is any uncertainty about taking the partial erivative of the operator H with respect to a coorinate or a momentum, the commutator (over ±i h) can be taken instea. 5 Virial Theorem The Virial of Clausius is efine by G = p q = p q (6) The Virial has the imensions of Action (A = pq). Its time erivative has expressions in terms of useful physical quantities (kinetic energy, force) G = q p p + q = p m p + r F = 2T r V (7) an in terms of Euler-like operators on the Hamiltonian G = q p + q p = ( p q p q ) H (8) The Virial Theorem is ifferent from classical expressions given in previous sections in that it is a statistical statement. It is an expression of long time averages: τ G G = lim τ τ 0 τ = lim 2T τ τ 0 q V = 2T q V q (9) If the motion is perioic or, more generally it is boune, G(q(τ), p(τ)) G(q(0), p(0)) is boune so the limit above vanishes. If the potential is a homogeneous function of the coorinates, so that V (q) = α q n, then by Euler s theorem an the bouneness of the motion we fin 2T nv = 0 (20) This is the stanar equipartition of energy theorem for systems in thermoynamic equilibrium. For Coulomb potentials (n = ) this result tells us that the mean value of the potential energy is twice the mean value of the kinetic energy, an of opposite sign. The Quantum Mechanical statement of this theorem is also ifferent from expressions given previously, in that it must involve two averaging operations. Spatial averages are enote by an temporal averages are enote by. The Virial is efine as a symmetrize generalization of the classical expression: 4

5 Ĝ = 2 [ˆp, ˆq ] + = 2 (ˆp ˆq + ˆq ˆp ) (2) The time erivative is given by the usual expression Ĝ = [Ĝ, H] (22) The commutators are easily compute. They give 2T an r F as before. Integrating the time erivative, we fin for boun states The result for a homogeneous potential of egree n is 2T q V = 0 (23) 2T = n V (24) We observe that spatial expectation values are time-epenent in general, but expectation values in an eigenstate are time inepenent. In an eigenstate the statement above is true at all times, not only on average, so we fin for a boun eigenstate in a homogeneous potential 2T = n V (25) 5