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# QuantumMechanicsII PHYS517

Drexel

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This 5 page Class Notes was uploaded by Vernice Schuster on Wednesday September 23, 2015. The Class Notes belongs to PHYS517 at Drexel University taught by RobertGilmore in Fall. Since its upload, it has received 23 views. For similar materials see /class/212526/phys517-drexel-university in Physics 2 at Drexel University.

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Date Created: 09/23/15

The Ehrenfest Theorems Robert Gilmore 1 Classical Preliminaries A classical system With 71 degrees of freedom is described by 71 second order ordinary differential equations on the con guration space 71 independent coor dinates in the Lagrangian representation d 8L 8L 7 e 7 0 1 dt qu qu and Zn rst order ordinary differential equations on the phase space 71 inde pendent coordinates and the conjugate momentum of each coordinate in the Hamiltonian representation 2 dt Bpj dt T qu The rst set of equations is called the EulerLagrange equations and the second set is called the Hamiltonian equations of motion Functions describing particles depend on time implicitly through the time dependence of the particle s coordinates and momenta they may also depend explicitly on time f fqtpt t The time derivative of such functions has the form E WE dt at aqz dt 8p dt ggaH af 8H 8t Bqlapz 7 81018111 3f 7 EH H 3 The partial derivative provides information about the explicit time dependence of the function The implicit time dependence depending on the motion of the particle is provided by the Poisson bracket 3f 39 3f 39 ngziieii 4 j qu Bpj apj qu 2 Classical Quantum Correspondence An elegant formulation of Quantum Theory is given in terms of a relation be tween the Poisson bracket of Classical Mechanics and the commutator Lie bracket of Quantum Mechanics Classical Mechanics QuantumMechanics A A 5 f 9 9 H m The hat A on the right hand side indicates that the correspondents are operators To see how this works we observe directly from Equ4 that qjpk 61 From this and the Classical A Quantum mapping Equ5 we observe that liprk 0 16 Thiss relation can be satis ed in one of two obvious ways one emphasizing the coordinates the usual the other emphasizing their conjugate momenta Representation Ifj le Coordinate 39 i i q i 8111C 6 Momentum i i Pic 2 Bpj A nal piece of the puzzle that we need is the timedependent Schrodinger equation For a free particle with wavefunction all the momentum is 71k by looking at the eigenvalue of the momentum operator hi88x If we would like to represent the timedependence of a free particle moving to the right with a momentum 71k in the form 5 1 we must choose the timedependent form of the Schrodinger equation to be WW 13 sz t 2727 3 Ehrenfest Theorems The Ehrenfest Theorem77 comprises a whole class of results all of which assume the same form Classical Mechanics A Quantum Mechanics 1A B A i 7 dt dt To show this we write down the time derivative of an expectation value as follows gm g ltztgtA ltztgtdv 1Zx t z tdV WAN tdV1Zx MWW s BWJM 3t In the expression above we can replace W by iHihi The result is by Hz39h and we can replace d A 7 8A 21 H ltAgt7ltEgtltTgt 9 Equation 9 for Quantum systems is identical to Equi 3 for Classical systems through the QuantumClassical correspondence of Equi This is Ehrenfest s Theoremi77 4 Simple Applications The following results are immediate d p a ltEgt gm lt7gtltFgt gm ltergt 10 The Hamiltonian equations of motion are obtained as follows Set A if In this case 4 lt41Hlgt garagth lt8Hgt 11 The replacement of the commutator qb H by the derivative 27371 is a direct 3 application of Equi In a similar way we nd Ehrenfest s limit for the other of Hamilton s equations The symmetry is a 8H dltqj 8H dt may 4 dt 3 12 13 dpki 8H dltpkgt7 8H i aiqk A dt ltaiqkgt 14 15 If there is any uncertainty about taking the partial derivative of the operator H With respect to a l t the over iih can be taken instead 5 Virial Theorem The Virial of Clausius is de ned by Gij11jl339q 16 j The Virial has the dimensions of Action dA Epdq Its time derivative has expressions in terms of useful physical quantities kinetic energy force 516 7 dqj 51101713 E7 j Epjqj 7EprF72TirVV 17 and in terms of Eulerlike operators on the Hamiltonian dG dqj dpj 8 8 7 7 7 7 7 7 H 18 dt 10 dt 1 dt map qjaqj l The Virial Theorem is di erent from classical expressions given in previous sections in that it is a statistical statement It is an expression of long time averages i 1 T dG 1 T 8 7 7 G 771530 0 E dt771A Higqjiqjv dt72TiqVV 19 If the motion is periodic or more generally it is bounded Gq7 p7 7 Gq0p0 is bounded so the limit above vanishes If the potential is a ho mogeneous function of the coordinates so that Vq alql then by Euler s theorem and the boundedness of the motion we n 7n70 20 This is the standard equipartition of energy theorem for systems in thermody namic equilibrium For Coulomb potentials n 71 this result tells us that the mean value of the potential energy is tWice the mean value of the kinetic energy and of opposite sign The Quantum Mechanical statement of this theorem is also di erent from expressions given previously in that it must involve two averaging operations Spatial averages are denoted by lt gt and temporal averages are denoted by 7 The Virial is de ned as a symmetrized generalization of the classical expression A 1 A A 1 A A A A GZ lpja41l Z P1111111Pj 21 j j The time derivative is given by the usual expression d A A ltGgt l0 Hl 22 The commutators are easily computed They give 2T and r F as before lnte grating the time derivative we nd for bound states QT 7 q VV 0 23 The result for a homogeneous potential of degree n is W nlt gt 24gt We observe that spatial expectation values are tirnedependent in general but expectation values in an eigenstate are time independent In an eigenstate the statement above is true at all times not only on average so we nd for a bound eigenstate in a homogeneous potential lt2Tgt MW 25

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