Note that an that the liit li! k+? k li P!;! h (k)? ((k? )) li! i i+? i + U( i ) is just a Rieann su representation of the continuous integral h h j +

Transcription

1 G5.65: Statistical Mechanics Notes for Lecture 5 I. THE FUNCTIONAL INTEGRAL REPRESENTATION OF THE PATH INTEGRAL A. The continuous liit In taking the liit P!, it will prove useful to ene a paraeter h P so that P! iplies!. In ters of, the partition function becoes P Q() li P ep? i+? i + U( i ) P!;! h h i! P + We can think of the points ; :::; P as specic points of a continuous functions (), where k ( (k? )) such that () ( P ) ( h): 4 P+ ε 4ε Pε FIG.. τ

2 Note that an that the liit li! k+? k li P!;! h (k)? ((k? )) li! i i+? i + U( i ) is just a Rieann su representation of the continuous integral h h j + U(()) Finally, the easure P li P!;! h P represents an integral overa all values that the function () can take on between an h such that () (h). We write this sybolically as D(). Therefore, the P! liit of the partition function can be written as Q() I (h) () D() ep? h D() ep? h _ + U(()) _ + U(()) The above epression in known as a functional integral. It says that we ust integrate over all functions (i.e., all values that an arbitrary function () ay take on) between the values an h. It ust really be viewe as the liit of the iscretize integral introuce in the last lecture. The integral is also referre to as a path integral because it iplies an integration over all paths that a particle ight take between an h such that () (h, where the paths are paraterize by the variable (which is not tie!). The secon line in the above epression, which is equivalent to the rst, inicates that the integration is taken over all paths that begin an en at the sae point, plus a nal integration over that point. The above epression akes it clear how to represent a general ensity atri eleent hj ep(?h)j i: hje?h j i (h) () D() ep? h _ + U(()) which inicates that we ust integrate over all functions () that begin at at an en at at h:

3 β h FIG.. τ Siilarly, iagonal eleents of the ensity atri, use to copute the partition function, are calculate by integrating over all perioic paths that satisfy () (h) : 3

4 β h FIG. 3. τ Note that if we let ith, then the ensity atri becoes (; ; ith) hje?ihth j i U(; ; t) which are the coorinate space atri eleents of the quantu tie evolution operator. If we ake a change of variables is in the path integral epression for the ensity atri, we n that the quantu propagator can also be epresse as a path integral: U(; ; t) hje?ihth j i (t) () i D() ep h s _(s)? U((s)) Such a variable transforation is known as a Wick rotation. This noenclature coes about by viewing tie as a cople quantity. The propagator involves real tie, while the ensity atri involves a transforation t?ih to the iaginary tie ais. It is because of this that the ensity atri is soeties referre to as an iaginary tie path integral. B. Doinant paths in the propagator an ensity atri Let us rst consier the real tie quantu propagator. The quantity appearing in the eponential is an integral of _? U() L(; _) which is known as the Lagrangian in classical echanics. We can ask, which paths will contribute ost to the integral sh _ (s)? U((s)) i 4 sl((s); _(s)) S[]

5 known as the action integral. Since we are integrating over a cople eponential ep(ish), which is oscillatory, those paths away fro which sall eviations cause no change in S (at least to rst orer) will give rise to the oinant contribution. Other paths that cause ep(ish) to oscillate rapily as we change fro one path to another will give rise to phase ecoherence an will ultiately cancel when integrate over. Thus, we consier two paths (s) an a nearby one constructe fro it (s) + (s) an ean that the change in S between these paths be S[ + ]? S[] Note that, since () an (t), () (t), since all paths ust begin at an en at. The change in S is S S[ + ]? S[] Epaning the rst ter to rst orer in, we obtain S s L(; _ sl( + ; _ + _)? L(; _) The ter proportional to _ can be hanle by an integration by parts: @ _ t because vanishes at an t, the surface ter is, leaving us with S s _ _ Since the variation itself is arbitrary, the only way the integral can vanish, in general, is if the ter in brackets @ This is known as the Euler-Lagrange equation in classical echanics. For the case that L _? U(), they give t ( which is just Newton's equation of otion, subject to the conitions that (), (t). Thus, the classical path an those near it contribute the ost to the path integral. The classical path conition was erive by requiring that S to rst orer. 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 iniizes the action as well. This is an iportant consieration when eriving the oinant paths for the ensity atri, which takes the for (; ; ) (h () The action appearing in this epression is S E [] D() ep? h h _ + U(())i _() + U(()) H(; _) which is known as the Eucliean action an is just the integral over a path of the total energy or Eucliean Lagrangian H(; _). Here, we see that a iniu action principle is neee, since the sallest values of S E will contribute ost to the integral. Again, we require that to rst orer S E [ + ]? S E []. Applying the sae logic as before, we obtain the conition 5

6 @ U() which is just Newton's equation of otion on the inverte potential surface?u(), subject to the conitions (), (h). For the partition function Q(), the sae equation of otion ust be solve, but subject to the conitions that () (h), i.e., perioic paths. The ensity atri for the free particle II. DOING THE PATH INTEGRAL: THE FREE PARTICLE H P will be calculate by oing the iscrete path integral eplicitly an taking the liit P! at the en. The ensity atri epression is (; ; ) li P! Let us ake a change of variables to P h P P ep? P h i( i+? i ) ;P + u u k k? ~ k ~ k (k? ) k+ + k The inverse of this transforation can be worke out eplicitly, giving u k The Jacobian of the transforation is siply J et P X+ l k? l? u l + P? k + P u??3?34 C A Let us see what the eect of this transforation is for the case P 3. For P 3, one ust evaluate Accoring to the inverse forula, (? ) + (? 3 ) + ( 3? 4 ) (? ) + (? 3 ) + ( 3? ) u u + u u Thus, the su of squares becoes 6

7 (? ) + (? 3 ) + ( 3? ) u + 3 u (? ) Fro this siple eple, the general forula can be euce: i ( i+? i ) Thus, substituting this transforation into the integral gives (; ; ) u u P ep where h P Y k k P h k k? u + 3 3? u (? ) k k? u k + P (? ) k k?? k k P h u k an the overall prefactor has been written as P P Y P k P h h h k ep? h (? ) Now each of the integrals over the u variables can be integrate over inepenently, yieling the nal result (; ; ) ep? h h (? ) In orer to ake connection with classical statistical echanics, we note that the prefactor is just, where h h is the kinetic prefactor that showe up also in the classical free particle case. In ters of, the free particle ensity atri can be written as (; ; ) e?(? ) Thus, we see that represents the spatial with of a free particle at nite teperature, an is calle the \theral e Broglie wavelength. 7