PATH INTEGRAL for HARMONIC OSCILLATOR

Transcription

1 PATH ITEGRAL for HAROIC OSCILLATOR I class, I have showed how to use path itegral formalism to calculate the partitio fuctio of a quatum system. Formally, ] ZT = Tr e it Ĥ = ÛT, 0 = dx 0 Ux 0, T ; x 0, 0 = dx 0 = xt =x 0 x0=x 0 D xt] e isxt] xt =x0 Dxt] e isxt] where Sxt] = T 0 dt ẋ V x is the Lagragia actio fuctioal. ote the boudary coditios i the last path itegral i eq. : xt is required to be periodic i time, xt = x0 but there are o separate iitial or fial coditios. I class, I evaluated the path itegral for the harmoic oscillator, but I was deliberately igorig all issues of covergece ad hopig that all the pre-expoetial factors would somehow tae care of themselves. I this ote I remedy the problem ad do the calculatio right. Actually, there are two separate covergece problems. defied via time discretizatio accordig to Formally, the path itegral is xt =x0 / Dxt] e isxt] = lim dx πit dx exp is discr x, x,...,, x, 3 so there is a obvious covergece problem of the cotiuum-time limit of. But eve for fiite there is a separate covergece problem of a dimesioal itegral of a rapidlyoscillatig but uiular fuctio e is. I fact, for this itegral does ot coverge, ot eve coditioally, so it must be re-defied via aalytic cotiuatio.

2 The usual aalytic cotiuatio eeps the x = xt real but maes the time itself imagiary, t = it E which rus from 0 to T = iβ. I field theory, t E is called the Euclidea time because the 4D spacetime spaed by x, x, x 3, x 4 = t E is Euclidea rather ta iowsi. Goig from the real iowsi time t to the real Euclidea time t E = it turs the oscillatig phase fuctio e isdiscr = exp it = x x V x ] T/ of the discretized path itegral ito a real arrowly peaed fuctio exp S discr where the discretized Euclidea actio S discr E = β = x x + V x β/ E 4 receives positive cotributio from both the Kietic ad the potetial eergy terms. I the cotiuous Euclidea time limit S E xt E ] = β ad the Euclidea path itegral becomes 0 dx + V x] xβ=x0 Z E β = Tr e βĥ] = Dxt E ] e SExtE]. 7 ote that this Euclidea partitio fuctio is precisely the partitio fuctio of Statistical echaics, so it is well worth calculatig i its ow right. Ulie the iowsi-time path Itegral, the Euclidea path itegral is well defied because for each fiite we have a absolutely coverget itegral / dx πβ dx exp SE discr x, x,...,, x, 8 ad the cotiuous-euclidea-time limit usually behaves well. 5 6 Cosequetly, the techical defiitio of the iowsi-time path itegral is othig but the aalytic cotiuatio of the Euclidea-time PI bac to iowsi time t = it E.

3 So, after all these prelimiaries, let is calculate the Euclidea path itegral for the harmoic oscillator. The Euclidea actio of the oscillator S E = β 0 ] dx + ω x 9 discretizes to SE discr x,..., x = β x x + ω β = x ]. 0 which is a quadratic fuctio of the itegratio variables x,..., x. Cosequetly, the discretized path itegral Zβ, = / πβ d x exp S discr E x,..., x is Gaussia ad may be evaluated exactly. Ufortuately, the determiat of the quadratic form 0 is rather formidable, so the best way to evaluate the itegral is to diagoalize the actio as a quadratic form. The cotiuum-time Euclidea actio is diagoalized via Fourier trasform xt E = + = S E x] = β / e πite/β y, ω + π y : β ote that the frequecies here are discrete because the Euclidea time is periodic; also, y = y. For the discretized actio 0 however, we eed the discrete Fourier trasform x = e πi/ y 3 = where the discrete frequecies are defied ulo, i.e. y 0 y, y y, etc., etc.; agai, the frequecy es y are complex, but the complete set of y,... y is self-cojugate 3

4 as y = y. The ey formula of the discrete Fourier trasform is Cosequetly, e πi l/ = δ l. 4 x = x x = y y 5 ad liewise x x = e πi/ y y 6 where the latter follows from x x = / e πi/ e πi/ y. Thus ad therefore SE discr y ] = 4 si π β + ω β y, 7 Zbeta, ω, = / J πβ = J d y e Sdiscr E y 4 si π + ω β / 8 where J is the Jacobia of the discrete Fourier trasform 3. To evaluate this Jacobia, we perform the Fourier trasform twice: y = m / e πim/ z m, x = / e πi/ y = z, 9 4

5 which immediately tells us that det x ] = det x = ± y z m ad hece J = det x / y = ad therefore Zβ, ω, = 4 si π + ω β /. 0 At this poit, let me use without proof a somewhat obscure mathematical formula = si π which allows me to re-write the discretized partitio fuctio as Zβ, ω, = ωβ = 4 si π = ωβ + = =, + ω β ω β 4 si π / / To evaluate the large limit of this partitio fuctio physically, the cotiuous time limit, we approximate 4 si π/ π for, ad liewise 4 si π/ π for, while for the remaiig es si π/ = O ad hece Cosequetly, Zβ, ω, ωβ ωβ = + + ω β + ω β ω β 4 si π π π /.. It remais to evaluate the ifiite product i the last formula. + ω β / π 3 Cosider Zωβ as a aalytic fuctio of a complex argumet. Wheever ay factor of o the right had side has 5

6 a zero i the complex ωβ plae, Zωβ has a zero ad ditto for the poles. Also, the product coverges, so these are the oly poles ad zeroes of the Zωβ The idividual factors at had are /ωβ ad = + ω β π π ωβ + πi ωβ πi for =,, 3,.... Thus, the Zωβ fuctio has o zeroes ad it has poles at ωβ = πi for all itegers positive, egative ad zero. I other words, it has the same poles ad zeroes as the / sihωβ/ fuctio ad ideed, there is a well ow formula sihz = z + = + z π. Thus, at the ed of the log path-itegral calculatio, we arrive at a rather simple formula Z E β = sihωβ/ 4 i Euclidea time, ad by aalytic cotiuatio to iowsi time Z T = i siωt/. 5 Expadig the latter partitio fuctio ito a sum of e iet phases, we have i siωt/ = e iωt/ e iωt = =0 exp it + ω, 6Spectrum which immediately tells us that the harmoic oscillator has o-degeerate eergy spectrum with eigevalues E = + ω. Of course, we ew that log before this calculatio, but it cofirms that properly applied path-itegral formalism does yield the correct spectrum. 6