Chemistry 593: The Semi-Classical Limit David Ronis McGill University

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1 Chemistry 593: The Semi-Classical Limit David Ronis McGill University. The Semi-Classical Limit: Quantum Corrections Here we will work out what the leading order contribution to the canonical partition function in the so-called Boltzmann Statistics approximation to account for quantum exchange symmetries. For concreteness, and in order to not make a cumbersome notation even worse, we ll consider an interacting set of structureless particles. Our starting point is a general, albeit formal, expression for the canonical partition function, i.e., Q = Tr e β H, () where H is the Hamiltonian and β /k B T. The assumption that the particles are structureless allows us to consider a Hamiltonian of the form H h m + V (r ). () The trace is invariant under unitary transformations and so we can use any representation in evaluating it; here we will use the momentum eigenfunctions as a basis. That is ψ p (r ) h 3 eip r /h, (3) where p are the momentum eigenvalues. It follows that Q = h 3! dx e ip r /h e β H e ip r /h, (4) where dx dp...dp dr...dr and the factor of! approximately accounts for the exchange symmetries in what is known as Boltzmann Statistics. The calculation simplifies slightly if we consider the β Laplace transform of Q; i.e., or, byusing Eq. (4), where and Q(z) 0 d β e β z Q(β ), (5) h 3! Q(z) = dx e ip r /h B eip r /h, (6) z + H classical B m h + p p. * Our approach follows that of J.-P. Hansen and I.R. McDonald, Theory of Simple Liquids (cademic Press, Y, 976) Sec (7a) (7b)

2 Chemistry The Semi-Classical Limit The classical Hamiltonian H classical p p /m + V (r )and B is the difference between the classical and quantum mechanical kinetic energies. Clearly B and must be small if a classical description is justifiable. The operator /( B) isknown as the resolvent operator and is easily shown to obey the following equality B = + B B (8a) = + B + B B +... (8b) where the second equality is obtained by iterating the first and is nothing more than an operator generalization of the geometric series. When Eq. (8b) is used in Eq. (6) we see that the first term results in the Laplace transform of the classical partition function, and thus, h 3![ Q(z) Q classical (z)] = dx e ip r /h By using Eqs. (7a) and (7b) we see that B eip r /h = eip r /h B + B B +... eip r /h. (9) m h + ih p ote that we got both st and nd (in h )order terms. The terms O(h 0 )cancel! When this result is used in the first term in Eq. (9) we obtain (0) h m dx, () where we have integrated by parts to move one of the s to the left. The purely imaginary terms linear in p vanish when integrated since the integrand is odd in the momenta. (Remember that the partition function is real and so we expect this to happen more generally). Unfortunately, we re not quite done, since the B terms in Eq. (9) also contribute at O(h ). By noting that B is Hermitian, it follows that dx e ip r /h B B eip r /h = dx By combining these terms we find that B r eip = h m dx p h 3![ Q(z) Q classical (z)] = h m dx + O(h 3 ). () m p + O(h 4 ). (3) We can now inv ert the Laplace transforms by making use of the convolution theorem, which states that the Laplace transform of a convolution integral,

3 Chemistry The Semi-Classical Limit f * g τ 0 f (t τ )g(τ ) is Thus, returning to Eq. (3), we see that h 3![Q Q classical ] = h m β 0 f * g = f g. m dx β 0 d β e β H (β β )β F F d β p F (β β )(β β ) + O(h 4 ), where F H is force acting on the particles. The various β integrals are easy to do and we find that h 3![Q Q classical ] = h β 3 m dx e β H F F β m (p F ). (4) Since momentum and coordinates are separable classically, and moreover the momentum distribution is just the Maxwell-Boltzmann distribution, we can carry out all the momentum integrations and find that and finally, where with Q Q classical π mk B T 3/ h β 3 h 4m! dr e βv F F, (5) Q Q classical h β 3 4m F F, (6) Q classical = π mk B T 3/ Z h! which is known as the configurational partition function. (7) Z dr e βv (8) This result is strange. If the particles are identical we can write F F = Σ F j F j = F F, (9) j where F is the total force exerted on particle. The correction is huge, unless F F O( )which seems unlikely. Fortunately, the explicit factor of is just what is needed to make the extensive thermodynamic functions extensive. Tosee this, remember that the Helmholtz Free Energy,, isrelated to the partition function as = k B T ln(q), which from Eq. (6), can be written as

4 Chemistry The Semi-Classical Limit = classical k B T ln h β 3 4m F F + O(h 4 ) The expansion we just carried out above assumes that h is small parameter and has O(h 4 )corrections that were dropped. The logarithm also has similar corrections; specifically, by noting that the Taylor expansion of ln( x) is ln( x) = x x... xn n..., which when applied to Eq. (0) shows that the x term is O(h 4 ). Given that we ve already dropped O(h 4 )terms in arriving at Eq. (0), consistency requires that we not keep them in the logarithm. Thus, h β 3 4m F F + O(h 4 ) Λ β (0) 48π F F + O(h 4 ), () where the last expression arises when we note that h =Λ mk B T /π,where Λ is the thermal de Broglie wavelength. Of course, what remains to be shown is that the O(h 4 )terms that arise both from the expansion of the trace and the logarithm have enough cancellations to give a result that is O(). This is the case, as we will see later in a more general context. We can make several observations about our result. First, note that the leading order free energy correction is positive, no matter what the interaction potential. Second, Eq. () can be rewritten as Λ β 48π dr Z F e βv (r ) = Λ β 48π dr e βv (r ) Z F, () where the last equality is obtained by integrating by parts. To go further, we ll assume that the interactions are pairwise additive; i.e., F i = Σ F i, j (r i, j ), (3) j i where r i, j r i r j and F i, j is the force particle j exerts on i. With this, given that the particles are identical, we can rewrite Eq. () as ( ) Λ β 48π dr F, dr r, dr... e βv (r ) 3 dr, (4) Z where the factor of ( ) is just the number of ways of choosing the particle interacting with particle. By introducing the so-called generic reduced pair correlation function, ρ () (r, r )and pair correlation function, g () (r, r ), i.e., ρ () (r, r ) ρ g () (r ) ( ) dr 3... dr e βv (r ) where ρ /V is the number density, wecan rewrite Eq. (4) as Λ ρ 48π dr F, dr r, g() (r, ) Z, (5)

5 Chemistry The Semi-Classical Limit = Λ βρ 48π dr, F, r, g() (r, ), (6) where the last equality is obtained by going to a coordinate system centered on r for the r integration and assuming that the system is translationally invariant. The subsequent integration in r gives afactor of volume. Finally, bywriting F, (r, ) =, u, (r, ), where u, (r, )isthe pair potential and only depends on the distance between particles and, Eq.(6) can be rewritten as Λ βρ 48π dr, [,u, (r, )]g () (r, ) (7a) Λ βρ dr r 0 u, (r) r + r u, (r) r g () (r), (7b) where, is the Laplacian for r,. The nd equality is obtained by changing the integration and Laplacian to polar coordinates and letting r, r. The energies that matter are those O(k B T ); hence, a crude estimate of size of our correction is Λ ρσ = ρλ 3 σ /Λ,where σ is the molecular interaction length scale. The use of Boltzmann Statistics already requires that ρλ 3 <<,and thus, unless σ /Λ >>,the semi-classical approximation should be valid. The proceeding estimates are crude. Hansen and Weis * performed monte carlo simulations for a model of neon near the triple point and showed that the corrections are about % of the Boltzmann statistic s classical result. * J.P. Hansen and J.J Weis, Phys. Rev. 88, 34 (969).