Quantum mechanical approaches to the virial

Quantum mechanical approaches to the virial S.LeBohec Department of Physics an Astronomy, University of Utah, Salt Lae City, UT 84112, USA Date: June 30 th 2015 In this note, we approach the virial from a stanar quantum mechanics point of view. Section 1 reviews the classical virial theorem. Section 2 reviews the Ehrenfest theorem as we will use it in the quantuscussion. Then the virial is consiere quantum mechanically in two ifferent ways. In section 3, the expectation values of the position an momentum observables consiere inepenently are use to construct a classical virial for which we erive the virial theorem an thereby establish a property of wave function. In section 4, we efine a quantum virial observable an establish the quantum virial theorem. Section 5 is a brief iscussion an conclusion highlighting the fact the quantum virial theores in irect corresponence with the classical virial theorem while the classical virial consiere from a quantum mechanical approach merely correspons to an integral property of the wave functions. I. VIRIAL THEOREM IN CLASSICAL MECHANICS The virial is a quantity that arises from consiering the time erivative of the moment of inertia I about the origin for a system of particles. Tae a system of N particles, each with a mass, position r i an momentum p i with i running from 1 to N. Then I = N r i 2 an where G = N r i p i efines the virial. I N t = 2 r i r N i t = 2 r i p i = 2G =1 We can then loo at the time erivative of the virial: G t = N r i t p i + N r i p N i t = 2T + r i F i where we introuce T the inetic energy of the entire system an the force F i acting on the i th particle. The virial theorem erives from consiering the time average of the time erivative of the virial. We enote the average over a time perio τ as τ. N In a reference frame where the the systes globally at rest, p i = 0, a finite an stable boun system woul be such that r i an p i are boun, so, in the limit of infinite times, the time average G t N 2T = r i F i = 0. Consequently, which express the virial theorem [5] in its general form. We may then consier each particle to only be uner the influence of a superposition of pairwise interactions with every other particle. The force exerte by all the particles of the system on the i th particle is F i = F ij is the force exerte by particle j on particle i. With this, we can write j<i where we use [6] the fact that F ij = F ji. j>i N j=1;j i 1 F ij where N r i F i = r i F ij + r i F ij = r i r j F ij 2 j<i

We now further restrict ourselves to the cases in which the force between any two particles erives from a central potential V r : Using this in the above result, we obtain: 2T = j<i r ij V r r ij. F ij = i V r i r j = V r N r i F i = N j<i r i r j r i r j = V r i r j r r ij V r ij r. So we have, for the virial theorem, the form: Furthermore, if the interaction potential energy is proportional to a power law of the istance V r = V 0 r ν, then, introucing the total potential energy of the system V T ot = N V r ij, the virial theorem taes its most usual form: j<i 2T = νv T ot 3 This can be applie with ν = 1 in the case in the case of a Keplerian potential or with ν = 2 in the case of a networ of harmonic oscillators. 2 II. THE EHRENFEST THEOREM The virial theorescusse in the previous section concerns time averaging in the limit of infinite times. In the quantuscussion of the properties of the virial, this nee to be combine with the statistical nature of the outcome of measurements in quantum mechanics. As a consequence, we are going to concentrate on the time evolution of expectation values. Consiering an observable A, the expectation value of this observable is enote A = φt A φt when the consiere systes in the quantum state φt. The Ehrenfest theorem provies an expression for the time erivative of expectation values. In orer to establish this expression, we can procee irectly from the efinition of A : t A = t φt A φt + φt A t φt + φt A t φt We assume A inclues an explicit time epenence. If the time evolution of the state of the systes governe by the Schröinger equation with a Hamiltonian H, then, i t φ = H φ an, since H is hermitian, i t φ = φ H. So we obtain the expression of the Ehrenfest theorem: t A = 1 φt AH HA φt + φt A i t φt or, using the usual notations for the commutator an the expectation value, t A = 1 [A, H] + A i t Particularly interesting applications of the Ehrenfest theorem appear when consiering position A = R an momentum A = P operators. Consier a particle of mass m whose evolution is governe by a Hamiltonian H = P 2 /2m + V where V is the potential energy. In orer to apply the Ehrenfest theorem, we nee to express the commutator [R, H] an [P, H]. This can be one using [R, P ] = i : [R, H] = 1 2m [R, P 2 ] = 1 RP 2 P 2 R 2m = 1 2m i + P R P P RP i = i P m We also nee to express [P, H] = [P, V ], which can be one in position representation with P = i R = i : [P, H] = [P, V ] = i V V = i V + V V = i V

Since R an P o not have any explicit time epenence, the Ehrenfest theorem then irectly gives the two following relations for the i th particle of the quantum analog of the classical system consiere in our iscussion of the virial theoren section I: t R i = P i m 4 t P i = i V T ot = F i 5 These are Hamilton s equations in which we re-introuce F i, the force acting on particle i. The time erivative of the expecte values of the positions are equal to the expectation values of momenta ivie by the mass. The time erivatives of the expectation values momenta are equal to the expectation values of the forces. This is an important result as it provies a brige between the quantum an classical regimes. It establishes that the time evolution of expectation values in Born s probabilistic interpretation of quantum mechanics matches the prescriptions of classical mechanics. 3 III. CLASSICAL VIRIAL THEOREM IN THE QUANTUM REGIME The state of a system of N istinguishable particles can be escribe by the irect prouct of the wave functions of its iniviual constituents. We can then classically efine the virial as G C = N P i R i with R i an P i the position an momentum operators for particle i. This virial can be regare as classical since, following the Ehrenfest theorem, R i an P i in the quantum system will evolve with time in exactly the same way as r i an p i in the classical system for which we have establishe the virial theoren Section I. In particular, we alreay now that the classical virial theorem Equation 1 irectly applies: N N = R i F i = R i R j F ij m i j<i If the force F ij erives from a central potential of the form V 0 R i R j ν, this gives. N R i R j R i R j ν 2 R i R j j<i But still, let us follow the erivation as an exercise. Noting the components {x, y, z} of the position an momentum of the i th particle as Ri an P i respectively, we can tae the time erivative of G C : t G C = We can then apply the Ehrenfest theorem: t G C = 1 i t P i Ri + Pi t R i [P i, H] Ri + Pi [Ri, H] An, using the expressions we foun for the commutators [R, H] an [P, H] in Section II 6 G C t = Ri V N T ot + Where we mae use of the total potential energy operator, V T ot = V R l R j = 1 2 l=1 j<l V R l R j l=1 j l

an in the case of the power law central potential: V T ot R i R i R j ν 2 Ri Rj j i Time averaging for a boun systen the reference frame where it is at rest G C t = 0, we obtain N Ri m R i R j ν 2 Ri R j i j i Using the same manipulation as in Section I see Equation 2, we fin this is equivalent to Equation 8 an we have complete a quantum mechanical erivation of the classical virial theorem. N R i R j R i R j m ν 2 R i R j 8 i j<i It shoul be note that the lefthan sie is not the time average expectation value of the inetic energy. It is the time average inetic energy for the expectation value of the momenta. In the same way, the right han sie can not be written simply in terms of the total potential energy so the classical virial theorem can not be expresse quantum-mechanically in a form similar to Equation 3. In this equation, it shoul be highlighte that the expectation values are to be unerstoo as calculate for the many particle quantum state φt of the systen the course of its evolution following Schröinger s equation. This quantum form of the classical virial theorem therefore stans as a non-trivial property of the solution of the Schröinger equation. 4 7 IV. QUANTUM VIRIAL THEOREM We coul also consier the expectation value of the quantum virial G Q = N Pi R i. This is the usual approach to the virial theoren quantum mechanics as originally investigate by Vlaimir Foc [3] in 1930. Alternatively, we coul consier G Q = N Ri P i. However, as long as we are intereste only in the time erivative of the expectation value of G Q or G Q, this maes no ifference. Inee: G Q t = t Pi Ri = t R i Pi i = G Q t Since there are no explicit time epenences, the Ehrenfest theorem gives: G Q t = 1 i [Pi R i, H] Consiering the same Hamiltonian as in the previous section, we see that we nee to calculate [G Q, P 2 ] [Pi Ri, Pi 2 ] = P i Ri Pi 2 P 3 i R i = Pi Ri Pi 2 P 2 i R i = 2i Pi an [G Q, V ] can be obtaine in position representation: [Pi Ri, V T ot ] = i Ri V T ot V T ot Ri = V V T ot + Ri i + Ri V T ot V T ot V T ot Ri = V i R i R i 2 R i

5 so [P i R i, H] = i P i 2 i R i V T ot R i with this: G Q t = Pi 2 R i V T ot Here we recognize the first term as twice the expectation value of the systenetic energy T = N same time, consiering the time average over an infinite perio, a boun systen its rest frame satisfies N so that 2T = or or or R i V T ot R i In cases where the potential energy is a power law of inex ν: 2T = ν or, finally,. j i 2T = ν j i R i V R i R j R i R j V R i R j R i R j Ri R j R i R j R i 2 R i R j R i R j 2T = ν V R i R j R i 2 2R i R j + R j 2 R i R j R i R j j<i 2T = ν V R i R j j<i 2 T = ν V T ot P 2 i 2 GQ t. At the = 0 an we recover the virial theoren the exact same formulation as in classical mechanics except for the fact that the inetic an potential energies have to be replace by their expectation values. V. SUMMARY AND CONCLUSIONS We have efine the virial for a quantum systen two ifferent ways. In Section III, we consiere a classical virial, G C = N P i R i. Since P i an R i have the same equation of motion as the corresponing quantities in the classical system, we coul apply the classical virial theorerectly to the quantum form replacing r i, p i an F i by their expectation values. We were then able to erive the same relation quantum-mechanically: N R i R j R i R j ν 2 R i R j j<i

The classical virial theorem can be seen as an integral property of the solutions of the Schröingier equation. In Section IV, we consiere the more usual quantum virial, G Q = N P i R i. We obtaine a virial theorem with exactly the same form as in classical mechanics provie the inetic an potential energies are replace by their quantum mechanical expectation values: 2 T = ν V T ot This suggests that the operation of taing the expectation value can be regare as a continuation of the time averaging τ to reveal the contribution of a ynamics internal to the wave function. In fact, when consiering the system to be in a stationary state, the time averaging becomes superfluous an we obtain a relation between expectation values 2 T = ν V T ot which, when regare as the time averaging of an internal ynamics, is ientical to the classical form of the virial theorem: 2T = νv T ot. At the same time an in complementarity, the virial theorem for G C becomes egenerate since there are no ynamics other than those internal to the wave function, leaing to P i = 0. This consieration of the virial theoren quantum mechanics certainly oes not provie any proof for the necessity of an explicit continuation between the classical ynamic ominating at scales larger than the e Broglie wavelength an a particulate ynamic internal to the wave function. However, the fact that the classical Section I an quantum Section IV virial theorems are ientical in form, follows well from Nelson s stochastic quantization [2] an more recently from Nottale s scale relativity [4]. From these point of views, the wave function property reveale by the quantum consieration of the classical virial in Section III can be seen as a superfluous matching constraint appearing because of the artificial iscrimination between the ynamics at classical an quantum scales, with the wave function merely playing the role of a wrapper of the latter. 6 VI. ACKNOWLEDGEMENT In the preparation of this note, I have use the Wiipeia page on the virial theorem https://en.wiipeia.org/wii/virial theorem. I am grateful to Janvia Rou for her helpful comments an to Eugene Mishcheno for noticing an error of reasoning in an earlier version of this note. [1] L.F.Abbott & M.B.Wise, Dimension of a quantum-mechanical path, Am.J.Phys. 491, 37-39, 1981 [2] E.Nelson, Derivation of the Schringer Equation from Newtonian Mechanics, Phys. Rev. 150, 1079 1966 [3] V.Foc, Bemerung zum Virialsatz. Zeitschrift fr Physi A 63 11: 855858 1930 [4] L.Nottale, Scale Relativity An Fractal Space-Time: A New Approach to Unifying Relativity an Quantum Mechanics ; Worl Scientific Publishing Company; 1 eition, 2011; ISBN 978-1848166509 [5] The wor virial erives from latin vis which mean force. The wor an the theorem are both ue to Ruolf Clausius in 1870 in On a Mechanical Theorem Applicable to Heat, in Philosophical Magazine, Ser. 4, vol. 40, 1870, p. 122127 [6] Using the fact that F ij = F ji, we have: N r i F i = r i F ij + r i F ij = r i F ij r i F ji j<i j>i j<i We can write the last term regrouping those obtaine for the same values of j: an we see that j>i r i F ji = r 1 F 21 + r 1 F 31 + r 2 F 32 + r 1 F 41 + r 2 F 42 + r 3 F 43 + j>i r i F ji = r i F ji = r j F ij j>i Combining the two terms in the original expression: j=1 i<j j<i N r i F i = r i r j F ij j<i