G5.651: Statistical Mechanics Notes for Lecture 1 I. DERIVATION OF THE DISCRETIZED PATH INTEGRAL We begin our discussion of the Feynman path integral with the canonical ensemble. The epressions for the partition function and epectation value of an observable A are, respectively Q(N, V, T) = Tr(e βh ) A = 1 Q Tr(Ae βh ) It is clear that we need to be able to evaluate traces of the type appearing in these epressions. We have already derived epressions for these in the basis of eigenvectors of H. However, since the trace is basis independent, let us eplore carrying out these traces in the coordinate basis. We will begin with the partition function and treat epectation values later. Consider the ensemble of a one-particle system. The partition function evaluated as a trace in the coordinate basis is Q(β) = d e βh We see that the trace involves the diagonal density matri element e βh. Let us solve the more general problem of any density matri element e βh. If the Hamiltonian takes the form H = P m + U(X) K + U then we cannot evaluate the operator ep( βh) eplicitly because the operators for kinetic (T) and potential energies (U) do not commute with each other, being, respectively, functions of momentum and position, i.e., K, U] In this instance, we will make use of the Trotter theorem, which states that given two operators A and B, such that A, B], then for any number λ, Thus, for the Boltzmann operator, e λ(a+b) = lim e λb/p e λa/p e λb/p] P P e β(k+u) = lim e βu/p e βk/p e βu/p] P P and the partition function becomes Q(β) = lim P d e βu/p e βk/p e βu/p] P Define the operator in brackets to be Ω: Ω = e βu/p e βk/p e βu/p Then, Q(β) = lim P d Ω P 1
In between each of the P factors of Ω, the coordinate space identity operator I = d is inserted. Since there are P factors, there will be P 1 such insertions. the integration variables will be labeled,..., P. Thus, the epression for the matri element becomes Ω = d d P Ω Ω 3 3 P P Ω P = d d P i Ω i+1 1=, P+1= The net step clearly involves evaluating the matri element i Ω i+1 = i e βu(x)/p e βp /mp e βu(x)/p i+1 Note that in the above epression, the operators involving the potential U(X) act on their eigenvectors and can thus be replaced by the corresponding eigenvalues: i Ω i+1 = e β(u(i)+u(i+1)/ i e βp /mp i+1 In order to evaluate the remaining matri element, we introduce the momentum space identity operator I = dp p p Letting K = P /m, the matri remaining matri element becomes i e βk/p i+1 = dp i p p e βp /mp i+1 = dp i p p i+1 e βp /mp Using the fact that it follows that i e βk/p i+1 = 1 π h p = 1 π h e ip/ h dpe ip(i i+1)/ h e βp /mp The remaining integral over p can be performed by completing the square, leading to the result i e βk/p i+1 = ( ) 1/ ( mp πβ h ep mp ) β h ( i+1 i ) Collecting the pieces together, and introducing the P limit, we have for the density matri ( ) P/ mp ( mp e βh = lim P πβ h d d P ep β h ( i+1 i ) + β P (U( i) + U( i+1 )) 1=, P+1= The partition function is obtained by setting =, which is equivalent to setting 1 = P+1 and integrating over, or equivalently 1. Thus, the epression for Q(β) becomes ( ) P/ mp ( 1 Q(β) = lim P πβ h d 1 d P ep β mω P ( i+1 i ) + 1 P U( i) P+1= 1
where we have introduced a frequency ω P = P β h When epressed in this way, the partition function, for a finite value of P, is isomorphic to a classical configuration integral for a P-particle system, that is a cyclic chain of particles, with harmonic nearest neighbor interactions and interacting with an eternal potential U()/P. That is, the partition function becomes Q(β) d 1 d P e βu eff( 1,..., P) where U eff ( 1,..., P ) = 1 mω P( i+1 i ) + 1 ] P U( i) Thus, for finite (if large) P the partition function in the discretized path integral representation can be treated as any ordinary classical configuration integral. Consider the integrand of Q(β) in the limit that all P points on the cyclic chain are at the same location. Then the harmonic nearest neighbor coupling (which is due to the quantum kinetic energy) vanishes and (1/P) P U( i) U(), and the integrand becomes e βu() which is just the true classical canonical position space distribution function. Therefore, the greater the spatial spread in the cyclic chain, the more quantum the system is, since this indicates a greater contribution from the quantum kinetic energy. The spatially localized it is, the more the system behaves like a classical system. It remains formally to take the limit that P. There we will see an elegant formulation for the density matri and partition function emerges. II. THE FUNCTIONAL INTEGRAL REPRESENTATION OF THE PATH INTEGRAL A. The continuous limit In taking the limit P, it will prove useful to define a parameter ε = β h P so that P implies ε. In terms of ε, the partition function becomes ( m ) P/ ( Q(β) = lim d 1 d P ep ε h ( ) )] m i+1 i + U( i ) P,ε πε h ε We can think of the points 1,..., P as specific points of a continuous functions (τ), where such that () = (τ = Pε) = (τ = β h): k = (τ = (k 1)ε) P+1= 1 3
4 1 P+1 ε 4ε Pε FIG. 1. τ Note that and that the limit lim ε ( ) k+1 k ε ε lim P,ε h = lim ε m ( ) (kε) ((k 1)ε) ε ( ) i+1 i + U( i )] is just a Riemann sum representation of the continuous integral 1 β h ( ) h m d + U((τ))] Finally, the measure ( ) P/ m lim P,ε πε h d 1 d P represents an integral overa all values that the function (τ) can take on between τ = and τ = β h such that () = (β h). We write this symbolically as D( ). Therefore, the P limit of the partition function can be written as (β h)= β h ( m Q(β) = d D( )ep ẋ + U((τ)) ()= = D( )ep 1 h β h ε 1 h ( m ẋ + U((τ)) = d 4
The above epression in known as a functional integral. It says that we must integrate over all functions (i.e., all values that an arbitrary function (τ) may take on) between the values τ = and τ = β h. It must really be viewed as the limit of the discretized integral introduced in the last lecture. The integral is also referred to as a path integral because it implies an integration over all paths that a particle might take between τ = and τ = β h such that () = (β h, where the paths are paramterized by the variable τ (which is not time!). The second line in the above epression, which is equivalent to the first, indicates that the integration is taken over all paths that begin and end at the same point, plus a final integration over that point. The above epression makes it clear how to represent a general density matri element ep( βh) : (β h)= e βh = D( )ep 1 h β h ( m ẋ + U((τ)) ()= which indicates that we must integrate over all functions (τ) that begin at at τ = and end at at τ = β h: ρ(, ; ) β β h FIG.. u Similarly, diagonal elements of the density matri, used to compute the partition function, are calculated by integrating over all periodic paths that satisfy () = (β h) = : 5
β h FIG. 3. τ Note that if we let β = it/ h, then the density matri becomes ρ(, ; it/ h) = e iht/ h = U(, ; t) which are the coordinate space matri elements of the quantum time evolution operator. If we make a change of variables τ = is in the path integral epression for the density matri, we find that the quantum propagator can also be epressed as a path integral: U(, ; t) = e iht/ h = (t)= ()= i D( )ep h t ( m ds ẋ(s) U((s)) Such a variable transformation is known as a Wick rotation. This nomenclature comes about by viewing time as a comple quantity. The propagator involves real time, while the density matri involves a transformation t = iβ h to the imaginary time ais. It is because of this that the density matri is sometimes referred to as an imaginary time path integral. B. Dominant paths in the propagator and density matri Let us first consider the real time quantum propagator. The quantity appearing in the eponential is an integral of 1 mẋ U() L(, ẋ) which is known as the Lagrangian in classical mechanics. We can ask, which paths will contribute most to the integral t m ] ds ẋ (s) U((s)) = 6 t dsl((s), ẋ(s)) = S]
known as the action integral. Since we are integrating over a comple eponential ep(is/ h), which is oscillatory, those paths away from which small deviations cause no change in S (at least to first order) will give rise to the dominant contribution. Other paths that cause ep(is/ h) to oscillate rapidly as we change from one path to another will give rise to phase decoherence and will ultimately cancel when integrated over. Thus, we consider two paths (s) and a nearby one constructed from it (s) + δ(s) and demand that the change in S between these paths be S + δ] S] = Note that, since () = and (t) =, δ() = δ(t) =, since all paths must begin at and end at. The change in S is δs = S + δ] S] = Epanding the first term to first order in δ, we obtain t δs = ds L(, ẋ) + L ] L δẋ + ẋ δ t dsl( + δ, ẋ + δẋ) t L(, ẋ) = The term proportional to δẋ can be handled by an integration by parts: t ds L t ẋ δẋ = ds L t d L δ = ẋ dt ẋ δ because δ vanishes at and t, the surface term is, leaving us with t δs = ds d L dt ẋ + L ] δ = t t t dsl(, ẋ) ] L L ds δẋ + ẋ δ ds d L dt ẋ δ Since the variation itself is arbitrary, the only way the integral can vanish, in general, is if the term in brackets vanishes: d L dt ẋ L = This is known as the Euler-Lagrange equation in classical mechanics. For the case that L = mẋ/ U(), they give d U (mẋ) + dt = mẍ = U which is just Newton s equation of motion, subject to the conditions that () =, (t) =. Thus, the classical path and those near it contribute the most to the path integral. The classical path condition was derived by requiring that δs = to first order. This is known as an action stationarity principle. However, it turns out that there is also a principle of least action, which states that the classical path minimizes the action as well. This is an important consideration when deriving the dominant paths for the density matri, which takes the form (β h= ρ(, ; β) = D( )ep 1 h β h ( m ẋ(τ) + U((τ)) ()= The action appearing in this epression is S E ] = β h m ẋ + U((τ))] = β h H(, ẋ) which is known as the Euclidean action and is just the integral over a path of the total energy or Euclidean Lagrangian H(, ẋ). Here, we see that a minimum action principle is needed, since the smallest values of S E will contribute most to the integral. Again, we require that to first order S E + δ] S E ] =. Applying the same logic as before, we obtain the condition 7
d H ẋ H = mẍ = U() which is just Newton s equation of motion on the inverted potential surface U(), subject to the conditions () =, (β h) =. For the partition function Q(β), the same equation of motion must be solved, but subject to the conditions that () = (β h), i.e., periodic paths. The density matri for the free particle III. DOING THE PATH INTEGRAL: THE FREE PARTICLE H = P m will be calculated by doing the discrete path integral eplicitly and taking the limit P at the end. The density matri epression is ( ) P/ mp ρ(, ; β) = lim P πβ h Let us make a change of variables to d d P ep u 1 = 1 u k = k k mp β h ( i+1 i 1=, P+1= k = (k 1) k+1 + 1 k The inverse of this transformation can be worked out eplicitly, giving 1 = u 1 k = P+1 l=1 k 1 l 1 u l + P k + 1 P The Jacobian of the transformation is simply 1 1/ 1 /3 J = det 1 3/4 1 = 1 Let us see what the effect of this transformation is for the case P = 3. For P = 3, one must evaluate According to the inverse formula, Thus, the sum of squares becomes ( 1 ) + ( 3 ) + ( 3 4 ) = ( ) + ( 3 ) + ( 3 ) 1 = u 1 = u + 1 u 3 + 1 3 + 3 3 = u 3 + 3 + 1 3 u 1 8
( ) + ( 3 ) + ( 3 ) = u + 3 u 3 + 1 3 ( ) From this simple emple, the general formula can be deduced: ( i+1 i ) = k= Thus, substituting this transformation into the integral gives ρ(, ; β) = ( ) 1/ m P ( ) 1/ mk P πβ h πβ h k= = 1 u + 3 3 1 u 3 + 1 3 ( ) k k 1 u k + 1 P ( ) du du P ep ] m k P β h u k ep m ] β h ( ) k= where and the overall prefactor has been written as m k = k k 1 m ( ) P/ ( ) 1/ mp m P ( ) 1/ mk P πβ h = πβ h πβ h k= Now each of the integrals over the u variables can be integrated over independently, yielding the final result ( ) 1/ m ρ(, ; β) = πβ h ep m ] β h ( ) In order to make connection with classical statistical mechanics, we note that the prefactor is just 1/λ, where λ ( πβ h λ = m )1/ ( βh = πm is the kinetic prefactor that showed up also in the classical free particle case. In terms of λ, the free particle density matri can be written as ρ(, ; β) = 1 λ e π( ) /λ Thus, we see that λ represents the spatial width of a free particle at finite temperature, and is called the thermal de Broglie wavelength. ) 1/ 9