HW #7

1. Free particle path integral

a) Propagator

To simplify the notation, we write t = t'' - t', x = x'' - x' and work in 1D.  Since \(TraditionalForm\`\(\([\)\(x\_i, p\_j\)\(]\)\) = 
        i\ \ δ\_\(i\ j\)\), 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),

	|x', 
            t'⟩ = 
          (e^(+i H t'/ℏ)) | x', 
        0⟩, \(TraditionalForm\`⟨\(\(\(x \*""", 
              t\) \*"""\)\(|\)\) = ⟨x \*""", 
            0 | \(\(e\^\(\(-i\)\ H\ \(t \*"""/ℏ\)\)\)\(.\)\)\) 

Therefore,

	\(TraditionalForm\`⟨x'', t'' | x', 
          t'⟩ = \(⟨x'' | 
              e\^\(\(-i\)\ H\ \((t'' - t')\)/ℏ\) | 
              x'⟩ = ∫d\ p ⟨x'' | p⟩ ⟨p | e\^\(\(-i\)\ H\ t/ℏ\) | 
                  x'⟩\)\)
\(TraditionalForm\`\(\(=\)\(∫d\ p\ \(e\^\(i\ p\ x''/ℏ\)\/\((2  π\ ℏ)\)\^\(1/2\)\) ⟨p | 
                e\^\(\(-i\)\ \((p\^2/2  m)\)\ t/ℏ\) | 
                x'⟩\)\)\)
\(TraditionalForm\`\(\(=\)\(∫d\ p\ \(e\^\(i\ p\ x''/ℏ\)\/\((2  π\ ℏ)\)\^\(1/2\)\) \(e\^\(\(-i\)\ \((p\^2/2  
                      m)\)\ t/ℏ\)\) 
            e\^\(\(-i\)\ p\ x'/ℏ\)\/\((2  π\ ℏ)\)\^\(1/2\)\)\)\)
\(TraditionalForm\`\(\(=\)\(∫d\ p\ \(e\^\(i\ p\ x/ℏ\)\/\(2  π\ ℏ\)\) e\^\(\(-i\)\ p\^2\ t/2\ m\ ℏ\)\)\)\)
\(TraditionalForm\`\(\(=\)\(∫d\ p\ \(1\/\(2  π\ ℏ\)\) \(exp(\(-i\)\ \(t\/\(2\ m\ ℏ\)\) \((p - \(m\ x\)\/t)\)\^2 + 
                i\ \(m\ x\^2\)\/\(2  ℏ\ t\))\)\)\)\)
\(TraditionalForm\`\(\(=\)\(\(1\/\(2  π\ ℏ\)\) \(\@\(\(2  π\ m\ ℏ\)\/\(i\ t\)\)\) e\^\(i\ m\ x\^2/2  ℏ\ t\)\)\)\)
\(TraditionalForm\`\(\(=\)\(\(\@\(\(\(m\)\(\ \)\)\/\(2\ π\ i\ ℏ\ t\)\)\) e\^\(i\ m\ x\^2/2  ℏ\ t\)\)\)\)

The analogous expression in three dimensions is simply

\(TraditionalForm\`⟨\(\(x\ ''\)\& -> \), 
          t'' | \(x\& -> \)', 
          t'⟩ = \(\((\(\(m\)\(\ \)\)\/\(2\ π\ i\ ℏ\ t\))\)\^\(3/2\)\) 
          e\^\(i\ m \( x\& -> \)\^2/2  ℏ\ t\)\) .

b) Action in exponent

For the classical trajectory, the velocity is simply (Overscript[v,  ->])  = (((Overscript[x,  ->]) )/t), and hence the action is (Subscript[S, c]) = ∫(1/2) 
              m (((d  (Overscript[x,  ->]) )/(d t))) d t = (1/2) 
              m ((((((Overscript[x,  ->]) )/t)))^2) 
              t = ((m (((Overscript[x,  ->]) )^2))/(2 t)), and so the exponent of the propagator is indeed (i (Subscript[S, c])/ℏ).

c) Partition function

The partition function from statistical mechanics is

	\(TraditionalForm\`Z = ∑\+\(n = □\)\%∞⟨n | e\^\(\(-β\)\ H\) | n⟩\),
	
where \(TraditionalForm\`\(\(\(|\)\(n\)\)\(\ \)\(⟩\)\)\) can denote elements of any basis.  Obviously, the Hamiltonian eigenstates themselves are generally most useful for calculating the sum directly; however, we can use the basis elements \(TraditionalForm\`\(\(\(|\)\(x\& -> \)\)\(\ \)\(⟩\)\)\) as well:
	
	\(TraditionalForm\`Z = ∫\ \(d\^3\) 
            x\ ⟨\ \(x\& -> \) | 
                e\^\(\(-β\)\ H\) | \(x\& -> \)\ ⟩\).
	
We observe that β H looks an awful lot like \(TraditionalForm\`i\ H\ t/\), except that the latter is purely imaginary whereas the former is purely real.  Therefore, we define the "Euclidean" time τ by the analytic continuation t ->  -i τ and get

	\(TraditionalForm\`Z = ∫\ \(\(\(d\)\(\ \)\)\^3\) 
            x\ ⟨\ \(x\& -> \) | 
                e\^\(\(-H\)\ τ\ /\ \) | \(x\& -> \)\ ⟩\),
	
in which we set \(TraditionalForm\`τ\  = \ \(\(\)\(\ \)\(β\)\(\ \)\)\), the thermal quantum timescale.  Noting that \(TraditionalForm\`⟨\ \(x\& -> \), 
        0 | 
            
            e\^\(\(-i\)\ \ H\ t\ /\ \)\  = \ ⟨\ \(x\& -> \), \(\(t\)\(|\)\)\) with "Minkowski" time in the Heisenberg picture (see part (a)), we get

	\(TraditionalForm\`Z = ∫\ \(\(\(d\)\(\ \)\)\^3\) 
            x\ ⟨\ \(x\& -> \), \(-i\)\ \ β | \(\(\(x\)\(\ \)\)\& -> \), 0⟩\).
	
The conversion to Euclidean time is already complete, since \(TraditionalForm\`\(x\& -> \)\_f = \(\(x\& -> \)\_i = \(x\& -> \)\)\).  This is because if the topology of Minkowski time is an open line from -∞ to ∞, the topology of Euclidean time must be a circle.  Periodic functions of time become hyperbolic, and hyperbolic functions become 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 computing the path integral, we can just convert our result for part (a) to get

	\(TraditionalForm\`Z = ∫\(d\^3\) 
            x\ \ \(\((\(\(m\)\(\ \)\)\/\(2\ π\ \(\^2\) β\))\)\^\(3/2\)\) e\^\(\(-m\)\ 0\ /\ 2 \( \^2\) β\)\)

\(TraditionalForm\`Z = \(V(\(\(m\)\(\ \)\)\/\(2\ π\ \(\^2\) β\))\)\^\(3/2\)\)

where V is the volume of the system.  This is nothing but the single-particle partition function for the classical ideal gas in three dimensions, as expected.

It is interesting that changing from Minkowski time to Euclidean time would effect a change from a propagator that obeys the Schrödinger equation to a diffusion kernel that obeys the heat equation, and moreso that substituting \(TraditionalForm\`\ β\) for the Euclidean time yields the thermodynamic partition function per unit volume.

d) Superfluid transition temperature in He-4

In Euclidean time, the action integral is

	\(TraditionalForm\`S\_E = \(∮\_\(\(\ \ \)\(0\)\)\^\(\ β\)\ 
            d\ τ\ L\  = ∮\_\(\(\ \ \)\(0\)\)\^\(\ β\)\ 
            d\ τ\ \ \(m\/2\) \((\(d\ \(x\& -> \)\)\/\(d\ τ\))\)\^2\)\)

where all paths are periodic, including particle exchange operations.

If one imagines Euclidean 1+1 spacetime as a cylinder, the trajectories of two umolested particles are just single loops around.  However, if we switch the particles, the trajectories cross -- the trajectory starting at particle 1 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 make one more 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 might simply make the thermal quantum substitutions λ (the thermal de Broglie wavelength) and \(TraditionalForm\`\ β\) for |d  (Overscript[x,  ->]) | = d x and d τ respectively.  While this makes sense for d τ, one must be careful with d x.  Trying it, one would find the result \(TraditionalForm\`S\_E\ ~\ N\ \), where we define N to be the number of loops around the Euclidean spacetime cylinder, with all the other constants cancelling.  That is, it would quantize "too far" -- we need to retain some length scale that's relevant to the inter-particle dynamics that changes the normal fluid to a superfluid. 

Generally, the relevant length scale is the "mean free path" (Subscript[l, f]), which is the average distance a particle travels between collisions.  In the low temperature regime where the particles are evenly distributed in Boltzmann fashion as in part (c), multiple bosons would pile up as a condensate.  That is, many particles would share the same ground state wavefunction; moreover, the classical interactions between particles would be sphere-like, with no "screening" effects (and ignoring mean field effects). In this case, 

	\(TraditionalForm\`l\_f\  ≈ \ \((\(\@2\) n\ ×\ σ)\)\^\(-1\) ≈ \((\(\@2\) n\ ×\ \(\(π\ n\^\(\(-2\)/3\)\)\/4\))\)\^\(-1\) ≈ \(2\^\(3/2\)\/π\) 
          n\^\(\(-1\)/3\)\) 

where n is the number density, and the factor of (Sqrt[2]) comes from the Maxwell-like distribution of particle velocities.  (If a particle of interest were much faster than all the other particles, we would just use (((n × σ))^(-1)), which is easy to see geometrically.)  Note we just substituted (n^(-1/3)) for the cross-sectional diameter.

Let us try d x  ≈  (Subscript[l, f]):
	
	\(TraditionalForm\`S\_E ≈ ∮\_\(\(\ \ \)\(0\)\)\^\(\ β\)\ 
            d\ τ\ \ \(m\/2\) \((l\_f\/\(\ β\))\)\^2 = \(m\/2\) \((l\_f\/\(\ β\))\)\^2\ \(∮\_\(\(\ \ \)\(0\)\)\^\(\ β\)\ d\ τ\)\)
\(TraditionalForm\`\(\(=\)\(\(m\/2\) \((l\_f\/\(\ β\))\)\^2\ N\ \ β\  = \ \(\(m\/2\) \(8\/\(\(π\^2\) 
                    n\^\(2/3\)\)\) \(1\/\(\ β\)\) 
              N = \(\(4\)  \(m\)\(\ \)\)\/\(\(π\^2\) \(n\^\(2/3\)\) \ β\)\ \(\(N\)\(\ \ \)\(.\)\)\)\)\)\)

Now, we want the temperature at which the change in (Subscript[S, E]) is \(TraditionalForm\`\) with each additional loop:

	\(TraditionalForm\`Δ\ S\_E = \(\(\(4\)  \(m\)\(\ \)\)\/\(\(π\^2\) \(n\^\(2/3\)\) \ β\) ≃ \ ⟹\ 
              T\_λ ≃ \(\(π\^2\) \(\^2\) n\^\(2/3\)\)\/\(\(4\) \( k\_B\) \(m\)\(\ \)\) = \(\(π\^2\) \(\^2\) ρ\^\(2/3\)\)\/\(4 \( k\_B\) m\^\(5/3\)\)\)\)

where ρ is the mass density at the superfluid transition (Subscript[T, λ]).  Let us compute it, with a figure of 7.798 lb/ft^3 for the mass density of liquid He-4 @ 4 K (from the liquid helium safety data sheet; 4.22 K is the boiling point according to Wikipedia):

\(\(\(π\^2\) \(\^2\) ρ\^\(2/3\)\)\/\(4 \( k\_B\) m\^\(5/3\)\) /. {k\_B -> 1.38*10\^\(-23\),  -> 1.055*10\^\(-34\), 
        m -> 4*1.66*10\^\(-27\), ρ -> 7.798*16}\)

This result is embarrassingly close to the measured value of 2.1768 K (Wikipedia), for having used such hand-wavy arguments!

2. Propagator of harmonic oscillator

a) Propagator with energy eigenvalues

As in 1(a) above,

	\(TraditionalForm\`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 Hamiltonian eigenstates \(TraditionalForm\`\(\(\(|\)\(n\)\)\(\ \)\(⟩\)\)\):

	\(TraditionalForm\`\(\(\ \)\(⟨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 (Subscript[t, f]) - (Subscript[t, i]) = -i τ, we obtain the desired result

\(TraditionalForm\`K = ∑\+\(n = 0\)\%∞\(\(\( ψ\_n\)(
                  x\_f)\)\^*\) \(\(ψ\_n\)(x\_i)\) 
            e\^\(\(-E\_n\) τ\ /\ \)\).

b) Leading behavior

We implement the harmonic oscillator propagator and make the substitution for (Subscript[t, f]) - (Subscript[t, i]) = -i τ = 
          i ln(ϵ)/ω:

\(\(kho = \(\@\(\(m\ ω\)\/\(2  π\ I\ \ Sin[ω \((t - t0)\)]\)\)\) 
            Exp[\((\(I\ m\ ω\)\/\(2\ \ Sin[ω \((t - t0)\)]\))\) \((\((x\^2 + x0\^2)\) Cos[ω \((t - t0)\)] - 
                    2\ x\ x0)\)] /. \((t - t0)\) -> 
            I\ Log[ϵ]/ω;\)\)

As τ -> ∞, ϵ -> 0, so we can expand it around ϵ = 0:

Series[kho, {ϵ, 0, 1}]

We see that the leading order is  (ϵ^(1/2)) as expected.

c) Expansion to arbitrary order

Expand the propagator to order 10 + 1/2 = 21/2:

khos = Series[kho, {ϵ, 0, 11}];

Extract the wavefunctions:

\(khos0 = 
      Simplify[\((SeriesCoefficient[khos, 1] /. x0 -> x)\)\^\(1/2\), 
        Assumptions -> { > 0, m > 0, ω > 0, 
            x ∈ Reals}]\)
\(khos5 = 
      Simplify[\((SeriesCoefficient[khos, 11] /. x0 -> x)\)\^\(1/2\), 
        Assumptions -> { > 0, m > 0, ω > 0, 
            x ∈ Reals}]\)
\(khos10 = 
      Simplify[\((SeriesCoefficient[khos, 21] /. x0 -> x)\)\^\(1/2\), 
        Assumptions -> { > 0, m > 0, ω > 0, 
            x ∈ Reals}]\)

So we find the wavefunctions:

\(\(ψk0 = \(E\^\(-\(\(m\ x\^2\ ω\)\/\(2\ \)\)\)\ \((\(m\ ω\)\/)\)\^\(1/4\)\)\/π\^\(1/4\);\)\)
\(\(ψk5 = \(E\^\(-\(\(m\ x\^2\ ω\)\/\(2\ \)\)\)\ \((\(m\ ω\)\/)\)\^\(3/4\)\ \((15\ \^2\ x - 20\ \ m\ x\^3\ ω + 4\ m\^2\ x\^5\ ω\^2)\)\)\/\(2\ \@15\ \^2\ π\^\(1/4\)\);\)\)
\(\(ψk10 = \(1\/\(720\ \@7\ π\^\(1/4\)\)\) \((E\^\(-\(\(m\ x\^2\ ω\)\/\(2\ \)\)\)\ \((\(m\ ω\)\/\^21)\)\^\(1/4\)\ \((945\ \^5 - 
                  9450\ \^4\ m\ x\^2\ ω + 
                  12600\ \^3\ m\^2\ x\^4\ ω\^2 - 
                  5040\ \^2\ m\^3\ x\^6\ ω\^3 + 
                  720\ \ m\^4\ x\^8\ ω\^4 - 
                  32\ m\^5\ x\^10\ ω\^5)\))\);\)\)

d) Graph and check normalization

Let us plot our wavefunctions:

\(\(nums = {m -> 1,  -> 1, ω -> 
            1};\)\)
Plot[ψx0 /. nums, {x, -3, 3}];
Plot[ψx5 /. nums, {x, -6, 6}];
Plot[ψx10 /. nums, {x, -7, 7}];

And show that they're normalized to 1:

\(Integrate[ψk0\^2, {x, \(-∞\), ∞}, \ 
      Assumptions -> { > 0, 
          m > 0, ω > 0}]\)
\(Integrate[ψk5\^2, {x, \(-∞\), ∞}, \ 
      Assumptions -> { > 0, 
          m > 0, ω > 0}]\)
\(Integrate[ψk10\^2, {x, \(-∞\), ∞}, \ 
      Assumptions -> { > 0, m > 0, ω > 0}]\)

3. Discretized HO path integral [optional]

See path integral notes pp. 15-19.