My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
tion, may be defined in the same way:
∫
Ψ(R, τ) = G(R, R ′ ; τ)Ψ(R ′ , 0) dR ′ . (3.43)
Comparing these two equations and using the definition of f gives a simple relationship
between the original and modified Green’s functions:
˜G(R, R ′ ; τ) = Ψ T (R)G(R, R ′ 1
; τ)
Ψ T (R ′ ) . (3.44)
This demonstrates that ˜G is not symmetric (since G is): this lack of symmetry is a
consequence of the fact that the operator
ˆF = 1 2 ∇2 − ∇ · v − v · ∇ + E T − E L , (3.45)
which appears on the right-hand side of equation (3.38), is not Hermitian.
Of course, the exact analytical form of ˜G is not known. However, as before, an
approximation which is valid for small τ can be used. A formal expression for ˜G is
the following:
˜G(R ′ , R; τ) = 〈 R ′∣ ∣e τ ˆF ∣ ∣R 〉 = 〈 R ′∣ ∣e −τ( ˆT + ˆV) ∣ ∣R 〉 (3.46)
where two new operators have been introduced for convenience:
ˆT = − 1 2 ∇2 + (∇ · v) + v · ∇ (3.47)
ˆV = E L − E T . (3.48)
The operators ˆT and ˆV represent modified kinetic and potential energies respectively.
The kinetic energy operator is associated with the drift-diffusion process, while the
potential energy operator comes from the rate equation. Using the Trotter-Suzuki
formula, 4 the Green’s function can be approximately factorised:
˜G(R ′ , R; τ) = 〈 R ′∣ ∣e − 1 2 τ ˆVe −τ ˆT e − 1 2 τ ˆV ∣ ∣R 〉 + O [ τ 3]
= e − 1 2 τV(R′ ) 〈 R ′∣ ∣e −τ ˆT ∣ ∣R 〉 e − 1 2 τV(R) + O [ τ 3]
= W (R ′ , R; τ)G D (R ′ , R; τ) + O [ τ 3] (3.49)
4 The formula gives an approximate form for the exponential of a sum of two operators:
e −τ(Â+ ˆB) = e −τ ˆB/2 e −τÂeτB/2
+ O [ τ 3] .
ˆ
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