Numerical methods. Note 14

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Feynman's path integral method in quantum mechanics.

Numerical Methods. Note « 14 »

Feynmans path integral.' Consider a one-dimentional motion of a particle with mass m, coordinate x and Lagrangian L(x,x´ )=^m/₂x´²-V(x). Suppose |x_a⟩ and |x_b⟩ are the eigenfunctions of the coordinate operator with the eigenvalues x_a and x_b. Suppose U(t_b,t_a)=e^-iH(t_b-t_a) is the time-evolution operator: Ψ(x,t_b)=U(t_b,t_a)Ψ(x,t_a). Then the propagator K_{x_bx_a}(t_b,t_a) ≡ ⟨x_b|U(t_b,t_a)|x_a⟩ is intuitively defined defined as the Feynman's integral over all classical paths x(t) connecting (x_a,t_a) and (x_b,t_b),

K_{x_bx_a}(t_b,t_a) = ∫ _x(t) D[x(t)] e^{ⁱ/_h S[x(t)]}

where D[x(t)] is certain functional meausure, and S[p]=∫ dt L(x,x´ ) is the action calculated along the path x(t).

Connection to classical mechanics. The stationary phase approach says that when h→0 only the stationary path, that is where ^δS/_δx=0, gives nonvanishing contribution to the integral. This is apparently the classical variational principle which defines the classical path.

Lattice discretization (and regularization). Let us discretize time in N equidistant slices t₀=t_a,t₁,...,t_N=t_b, with Δt=^(t_b-t_a)/_N

K_{x_bx_a}(t_b,t_a) ≈ (^m/_{2π i h Δt})^N/2 ∫ dx(t₁)...dx(t_N-1) exp(ⁱ/_h ∑_n S_n)

where the short time action is defined using the midpoint rule,

S_n = Δt L[¹/₂(x_n+x_n-1),¹/_Δt(x_n-x_n-1)]

The (broken-line) path is now defined by the set of values {x(t₁),x(t₂),...,x(t_N-1)}

Monte-Carlo evaluation of the path integral: generate random paths {x(t₁),x(t₂),...,x(t_N-1)} using some algorithm and add up their contributions to the path integral.

Space discretization With the space discretized also, x={x₀,x₁,...,x_D}, the short-time propagator becomes a complex matrix

K_ij(Δt)=(^m/_{2π i h Δt})^1/2 exp(^iΔt/_h L[¹/₂(x_j+x_i),¹/_Δt(x_j-x_i)])

and consequently the finite time propagator K_ij(t)=⟨x_i|U(t)|x_j⟩ becomes a product of short-time propagators, K(t)=Δx^N-1K(Δt)^N.

Imaginary time propagation. Partition function e^-βH.

Real time propagation. The energy levels can be extracted by a Fourier transform of the trace of the propagator:

Tr U(t) = ∑_n ⟨ n | exp(-ⁱ/_hHt) | n ⟩ = ∑_n exp(-ⁱ/_hE_n t)

Problems

Consider (the lowest states of) a one-dimentional oscillator with time and space discretized path integral method with real- and complex-time propagation.