HW7.nb HW #7. Free particle path integral a) Propagator To siplify the notation, we write t t t, x x x and work in D. Since x i, p j i i j, we can just construct the 3D solution. First of all, because the base kets evolve according to the "wrong sign" Schrödinger equation (see pp. 87 89), Therefore, x, t e i H t ħ x, 0, x", t" x", 0 e i H t"ħ. x, t x, t x e i H t t ħ x d px pp e i H tħ x d p d p d p 2Π ħ 2 ei p x ħ 2Π ħ 2 ei p x ħ p e i p2 2 tħ x ei p x ħ e i p2 2 tħ ei p xħ 2Π ħ ei p2 t2 ħ d p 2Π ħ expi t 2 ħ 2Π ħ 2Π ħ e i x2 2ħ t i t e i x2 2ħ t 2 Π i ħ t 2Π ħ 2 x p t 2 i x2 2ħ t The analogous expression in three diensions is siply x, t x, t 2 Π i ħ t 32 e i x 2 2ħ t. b) Action in exponent For the classical trajectory, the velocity is siply v x, and hence the action is t S c dx 2 d t d t 2 x 2 t t 2 x, and so the exponent of the propagator is indeed i S 2 t c ħ.
HW7.nb 2 c) Partition function The partition function fro statistical echanics is Z n n e Β H n, where n can denote eleents of any basis. Obviously, the Hailtonian eigenstates theselves are generally ost useful for calculating the su directly; however, we can use the basis eleents x as well: Z d 3 x x e Β H x. We observe that Β H looks an awful lot like i H t, except that the latter is purely iaginary whereas the forer is purely real. Therefore, we define the "Euclidean" tie Τ by the analytic continuation t i Τ and get Z d 3 x x e H Τ x, in which we set Τ Β, the theral quantu tiescale. Noting that i H t x, 0 e the Heisenberg picture (see part (a)), we get Z d 3 x x, i Β x, 0. x, t with "Minkowski" tie in The conversion to Euclidean tie is already coplete, since x f x i x. This is because if the topology of Minkowski tie is an open line fro to, the topology of Euclidean tie ust be a circle. Periodic functions of tie becoe hyperbolic, and hyperbolic functions becoe periodic. Accordingly, in the exponent of the path integral, the action integral is now on a loop, and all trajectories return to their origin. Instead of coputing the path integral, we can just convert our result for part (a) to get Z d 3 x 2 Π 2 Β 32 e 0 22 Β 2 Π 2 Β 32 Z V where V is the volue of the syste. This is nothing but the single particle partition function for the classical ideal gas in three diensions, as expected. It is interesting that changing fro Minkowski tie to Euclidean tie would effect a change fro a propagator that obeys the Schrödinger equation to a diffusion kernel that obeys the heat equation, and oreso that substituting Β for the Euclidean tie yields the therodynaic partition function per unit volue.
HW7.nb 3 d) Superfluid transition teperature in He 4 In Euclidean tie, the action integral is S E 0 Β d Τ L 0 Β d Τ 2 d x 2 d Τ where all paths are periodic, including particle exchange operations. If one iagines Euclidean + spacetie as a cylinder, the trajectories of two uolested particles are just single loops around. However, if we switch the particles, the trajectories cross the trajectory starting at particle attaches to the start of the trajectory of particle 2 after wrapping around the cylinder, and vice versa. In order for this switching operation to be undone (i.e., for the trajectories to be closed), the trajectories have to ake one ore trip around, to connect to where they started originally. So, with a switching operation, each particle has an average of one extra loop in calculating the action. Naively, one ight siply ake the theral quantu substitutions Λ (the theral de Broglie wavelength) and Β for dx d x and d Τ respectively. While this akes sense for d Τ, one ust be careful with d x. Trying it, one would find the result S E N, where we define N to be the nuber of loops around the Euclidean spacetie cylinder, with all the other constants cancelling. That is, it would quantize "too far" we need to retain soe length scale that s relevant to the inter particle dynaics that changes the noral fluid to a superfluid. Generally, the relevant length scale is the "ean free path" l f, which is the average distance a particle travels between collisions. In the low teperature regie where the particles are evenly distributed in Boltzann fashion as in part (c), ultiple bosons would pile up as a condensate. That is, any particles would share the sae ground state wavefunction; oreover, the classical interactions between particles would be sphere like, with no "screening" effects (and ignoring ean field effects). In this case, l f 2 n Σ 2 n Π n23 232 4 Π n3 where n is the nuber density, and the factor of 2 coes fro the Maxwell like distribution of particle velocities. (If a particle of interest were uch faster than all the other particles, we would just use n Σ, which is easy to see geoetrically.) Note we just substituted n 3 for the cross sectional diaeter. Let us try d x l f : S E 0 Β d Τ 2 l f 2 l f Β 2 N Β Β 2 2 8 Π 2 n 23 2 l f Β 2 Β d Τ 0 Β N 4 Π 2 n 23 Β N. Now, we want the teperature at which the change in S E is with each additional loop: S E 4 Π 2 n 23 Β T Λ Π2 2 n 23 4k B Π2 2 Ρ 23 4k B 53 where Ρ is the ass density at the superfluid transition T Λ. Let us copute it, with a figure of 7.798 lb/ft^3 for the ass density of liquid He 4 @ 4 K (fro the liquid heliu safety data sheet; 4.22 K is the boiling point according to Wikipedia):
HW7.nb 4 Π 2 2 Ρ 23. k B.38 0 23,.055 0 34, 4.66 0 27, Ρ 7.798 6 4k B 53 2.83 This result is ebarrassingly close to the easured value of 2.768 K (Wikipedia), for having used such hand wavy arguents! 2. Propagator of haronic oscillator a) Propagator with energy eigenvalues As in (a) above, K x f, t f x i, t i x f, t i e i H t f t i x i, t i. We can insert the unity operator, on the basis of Hailtonian eigenstates n : x f, t f x i, t i n 0 x f, t i e i H t f t i n n x i, t i n 0 x f, t i n n x i, t i e i E n t f t i n 0 Ψ n x f Ψ n x i e i E n t f t i. Making the usual substitution t f t i i Τ, we obtain the desired result K n 0 Ψ n x f Ψ n x i e E n Τ. b) Leading behavior We ipleent the haronic oscillator propagator and ake the substitution for t f t i i Τ i lnε Ω: Ω kho 2Π I SinΩt t0 Exp t t0 I LogΕ Ω; As Τ, Ε 0, so we can expand it around Ε 0: Serieskho, Ε, 0, x2 Ω 2 x02 Ω 2 Ω Ε OΕ 32 Π I Ω 2 SinΩt t0 x 2 x0 2 CosΩt t0 2 x x0. We see that the leading order is Ε 2 as expected.
HW7.nb 5 c) Expansion to arbitrary order Expand the propagator to order 0 2 2 2: khos Serieskho, Ε, 0, ; Extract the wavefunctions: khos0 SiplifySeriesCoefficientkhos,. x0 x 2, Assuptions 0, 0, Ω 0, x Reals khos5 SiplifySeriesCoefficientkhos,. x0 x 2, Assuptions 0, 0, Ω 0, x Reals khos0 SiplifySeriesCoefficientkhos, 2. x0 x 2, Assuptions 0, 0, Ω 0, x Reals x2 Ω 2 Ω 4 Π 4 x2 Ω 2 Ω 34 Abs5 2 x 20 x 3 Ω 4 2 x 5 Ω 2 2 5 2 Π 4 720 7 Π 4 x2 Ω 2 Ω 2 4 Abs945 5 9450 4 x 2 Ω 2600 3 2 x 4 Ω 2 5040 2 3 x 6 Ω 3 720 4 x 8 Ω 4 32 5 x 0 Ω 5 So we find the wavefunctions: x 2 Ω Ψk0 4 ; Π 4 2 Ω x 2 Ω Ψk5 34 5 2 x 20 x 3 Ω 4 2 x 5 Ω 2 2 ; 5 2 Π 4 Ψk0 2 Ω 720 7 Π 4 x2 Ω 2 Ω 4 945 5 9450 4 x 2 Ω 2 2600 3 2 x 4 Ω 2 5040 2 3 x 6 Ω 3 720 4 x 8 Ω 4 32 5 x 0 Ω 5 ; d) Graph and check noralization Let us plot our wavefunctions:
HW7.nb 6 nus,, Ω ; PlotΨx0. nus, x, 3, 3; PlotΨx5. nus, x, 6, 6; PlotΨx0. nus, x, 7, 7; 0.7 0.6 0.5 0.4 0.3 0.2 0. -3-2 - 2 3 0.4 0.2-6 -4-2 2 4 6-0.2-0.4 0.4 0.2-6 -4-2 2 4 6-0.2-0.4 And show that they re noralized to :
HW7.nb 7 IntegrateΨk0 2, x,,, Assuptions 0, 0, Ω 0 IntegrateΨk5 2, x,,, Assuptions 0, 0, Ω 0 IntegrateΨk0 2, x,,, Assuptions 0, 0, Ω 0 3. Discretized HO path integral [optional] See path integral notes pp. 5 9.