Quantum Mechanics

Perturbation Theory

Time-Independent Perturbation

A Toy Model of Qubit

General Ideas of Perturbation Theory

Exact solutions are rare. => We have to rely on perturbation theory to go beyond them and make the most of these exact solutions.

Separation of scales: physics often takes place at different energy scales, e.g. … → quarks (GeV) → nucleus (MeV) → atoms (eV) → molecules (100meV) → … => We can refine our descriptions by adding perturbative corrections progressively.

Time-independent perturbation: given the spectrum (eigenstates and eigenenergies) of (Subscript[H, 0]), find the spectrum of H=(Subscript[H, 0])+λ V, in power series of λ (given that λ is small).

Time-dependent perturbation: given the (bare) propagator (Subscript[U, 0])(t)=(e^(-i (Subscript[H, 0])t)), find the propagator

U(t)=𝒯 exp(-i (∫_0^t)dt' H(t')),
H(t)=(Subscript[H, 0])+λ V(t),

in power series of λ (given that λ is small). [We will explain the notations in Eq. () later.]

Conceptually: to establish effective Hamiltonians, to analyze renormalization group flows …

Practically: to calculate scattering amplitudes, response functions, spectral weights …

Qubit Model and its Exact Solution

H(λ)=(Subscript[H, 0])+λ V,

H(λ)=(σ^z)+λ (σ^x)
≏(({{1, λ}, {λ, -1}})).

H(λ)(Ket[(Subscript[ψ, ±])(λ)])=(Subscript[E, ±])(λ)(Ket[(Subscript[ψ, ±])(λ)]).

Eigenenergies

(Subscript[E, ±])(λ)=±(Sqrt[1+(λ^2)]).

Eigenstates

(Ket[(SubPlus[ψ])(λ)])=(((1+(Sqrt[1+(λ^2)]))(Ket[↑])+λ(Ket[↓]))/(Sqrt[2(1+(λ^2)+(Sqrt[1+(λ^2)]))])),
(Ket[(SubMinus[ψ])(λ)])=(((1+(Sqrt[1+(λ^2)]))(Ket[↓])-λ(Ket[↑]))/(Sqrt[2(1+(λ^2)+(Sqrt[1+(λ^2)]))])).

Taylor Expansion

As a reminder, the Taylor expansion of a function f(λ) is given by

f(λ)=(∑_(k=0)^∞)(((∂_λ^k)f(0))/(k!))(λ^k)
=f(0)+f'(0)λ+((f''(0))/2)(λ^2)+(((f^((3)))(0))/6)(λ^3)+….

Applying Eq. () to Eq. () and Eq. (), we get

(Subscript[E, ±])=±(1+((λ^2)/2)-((λ^4)/8)+…),

(Ket[(SubPlus[ψ])(λ)])=(Ket[↑])+(λ/2)(Ket[↓])-((λ^2)/8)(Ket[↑])-((3 (λ^3))/16)(Ket[↓])+((11 (λ^4))/128)(Ket[↑])+…,
(Ket[(SubMinus[ψ])(λ)])=(Ket[↓])-(λ/2)(Ket[↑])-((λ^2)/8)(Ket[↓])+((3 (λ^3))/16)(Ket[↑])+((11 (λ^4))/128)(Ket[↓])+….

Non-Degenerate Perturbation Theory

Problem Setup

This implies

H(0)=(Subscript[H, 0]),
(∂_λ)H(0)=V,
(∂_λ^2)H(0)=(∂_λ^3)H(0)=…=0.

To simplify the notation, we will suppress the argument λ if it is evaluated at λ=0.

H=(Subscript[H, 0]),
(∂_λ)H=V,
(∂_λ^2)H=(∂_λ^3)H=…=0.

Rule: Everything is treated as a function of λ, like f(λ). But if the dependence on λ is not explicitly spelt out, we assume it to be the function evaluated at λ=0, i.e. f≡f(Subscript[|, λ=0])=f(0).

H(λ)(Ket[n(λ)])=(Subscript[E, n])(λ)(Ket[n(λ)]).

(Subscript[E, n])(λ) is the nth energy level. It is a real number depending on λ.

(Ket[n(λ)]) is the nth eigenstate (in correspondence to (Subscript[E, n])(λ)). It is a state vector in the Hilbert space that can change with λ. Note: The notation (Ket[n(λ)]) does not imply that the index n is λ dependent, it should be understood as

(Ket[n(λ)])=(∑_m)(Subscript[ψ, n m])(λ)(Ket[m]).

Hellmann-Feynman Theorems

Applying (∂_λ) to both sides of H(Ket[n])=(Subscript[E, n])(Ket[n]),

(∂_λ)H(Ket[n])+H(Ket[(∂_λ)n])=(∂_λ)(Subscript[E, n])(Ket[n])+(Subscript[E, n])(Ket[(∂_λ)n]).

Note: (Ket[(∂_λ)n]) stands for the derivative of the state (Ket[n]) (not the index n)

(Ket[(∂_λ)n])=(Subscript[((∑_m)(∂_λ)(Subscript[ψ, n m])(λ)(Ket[m])), λ=0]).

It does not imply that the integer index n can be differentiated.

Overlap with (Bra[m]) from the left,

(Bra[m])(∂_λ)H(Ket[n])+(Bra[m])H(Ket[(∂_λ)n])=(∂_λ)(Subscript[E, n])(BraKet[m, n])+(Subscript[E, n])(BraKet[m, (∂_λ)n]).

Using (Bra[m])H=(Bra[m])(Subscript[E, m]),

(Bra[m])(∂_λ)H(Ket[n])+(Subscript[E, m])(BraKet[m, (∂_λ)n])=(∂_λ)(Subscript[E, n])(BraKet[m, n])+(Subscript[E, n])(BraKet[m, (∂_λ)n])
=>(Bra[m])(∂_λ)H(Ket[n])=(∂_λ)(Subscript[E, n])(BraKet[m, n])+((Subscript[E, n])-(Subscript[E, m]))(BraKet[m, (∂_λ)n]).

Note that (∂_λ)H=V and (BraKet[m, n])=(Subscript[δ, m n]),

When m=n, Eq. () implies

This is the first Hellmann-Feynman theorem.

When m≠n, Eq. () implies

This is the second Hellmann-Feynman theorem. Tip: the energy denominator is always given by the energy of the state that is being differentiated minus the energy of the other state.

Energy Corrections

(Subscript[E, n])(λ)=(∑_(k=0)^∞)(((∂_λ^k)(Subscript[E, n]))/(k!))(λ^k)=(Subscript[E, n])+(∂_λ)(Subscript[E, n])λ+(1/2)(∂_λ^2)(Subscript[E, n])(λ^2)+….

We already know from Eq. () that

(∂_λ)(Subscript[E, n])=(Subscript[V, n n])=(Bra[n])(∂_λ)H(Ket[n]).

We continue to evaluate

(∂_λ^2)(Subscript[E, n])=(∂_λ)(Bra[n])(∂_λ)H(Ket[n])
=(Bra[(∂_λ)n])(∂_λ)H(Ket[n])+(Bra[n])(∂_λ^2)H(Ket[n])+(Bra[n])(∂_λ)H(Ket[(∂_λ)n]).

Note that (∂_λ^2)H=0 according to the setup in Eq. ().

(∂_λ^2)(Subscript[E, n])=(Bra[(∂_λ)n])(∂_λ)H(Ket[n])+(Bra[n])(∂_λ)H(Ket[(∂_λ)n])
=(∑_m)((BraKet[(∂_λ)n, m])(Bra[m])(∂_λ)H(Ket[n])+(Bra[n])(∂_λ)H(Ket[m])(BraKet[m, (∂_λ)n]))
=(∑_m)(((Subscript[V, n m])/((Subscript[E, n])-(Subscript[E, m])))(Subscript[V, m n])+(Subscript[V, n m])((Subscript[V, m n])/((Subscript[E, n])-(Subscript[E, m]))))
=2(∑_m)(((Subscript[V, n m])(Subscript[V, m n]))/((Subscript[E, n])-(Subscript[E, m]))).

(∂_λ^2)(Subscript[E, n])=(∑_m)((BraKet[(∂_λ)n, m])(Bra[m])(∂_λ)H(Ket[n])+(Bra[n])(∂_λ)H(Ket[m])(BraKet[m, (∂_λ)n]))
=(∑_(m≠n))((BraKet[(∂_λ)n, m])(Bra[m])(∂_λ)H(Ket[n])+(Bra[n])(∂_λ)H(Ket[m])(BraKet[m, (∂_λ)n]))+((BraKet[(∂_λ)n, n])(Bra[n])(∂_λ)H(Ket[n])+(Bra[n])(∂_λ)H(Ket[n])(BraKet[n, (∂_λ)n]))
=2(∑_(m≠n))(((Subscript[V, n m])(Subscript[V, m n]))/((Subscript[E, n])-(Subscript[E, m])))+(Subscript[V, n n])((BraKet[(∂_λ)n, n])+(BraKet[n, (∂_λ)n]))
=2(∑_(m≠n))(((Subscript[V, n m])(Subscript[V, m n]))/((Subscript[E, n])-(Subscript[E, m])))+(Subscript[V, n n])(∂_λ)(BraKet[n, n]).

(∂_λ^2)(Subscript[E, n])=2(∑_(m≠n))(((Subscript[V, n m])(Subscript[V, m n]))/((Subscript[E, n])-(Subscript[E, m]))).

Comment on Gauge Fixing

(BraKet[n, n])=1
=>(BraKet[(∂_λ)n, n])+(BraKet[n, (∂_λ)n])=(∂_λ)(BraKet[n, n])=(∂_λ)1=0
=>Re(BraKet[n, (∂_λ)n])=0.

(Ket[n(λ)])->(e^(i ϕ(λ)))(Ket[n(λ)]),

(BraKet[n, (∂_λ)n])→(BraKet[n, (∂_λ)n])+i (∂_λ)ϕ.

State Corrections

(Ket[n(λ)])=(∑_(k=0)^∞)((Ket[(∂_λ^k)n])/(k!))(λ^k)=(Ket[n])+(Ket[(∂_λ)n])λ+(1/2)(Ket[(∂_λ^2)n])(λ^2)+….

(Ket[(∂_λ)n])=(∑_(m≠n))(Ket[m])(BraKet[m, (∂_λ)n])=(∑_(m≠n))(Ket[m])((Subscript[V, m n])/((Subscript[E, n])-(Subscript[E, m]))).

(Ket[(∂_λ^2)n])=(∂_λ)(∑_(m≠n))(Ket[m])(((Bra[m])(∂_λ)H(Ket[n]))/((Subscript[E, n])-(Subscript[E, m])))
=(∑_(m≠n))((Ket[(∂_λ)m])(((Bra[m])(∂_λ)H(Ket[n]))/((Subscript[E, n])-(Subscript[E, m])))+(Ket[m])(((Bra[(∂_λ)m])(∂_λ)H(Ket[n]))/((Subscript[E, n])-(Subscript[E, m])))+(Ket[m])(((Bra[m])(∂_λ)H(Ket[(∂_λ)n]))/((Subscript[E, n])-(Subscript[E, m])))-(Ket[m])(((Bra[m])(∂_λ)H(Ket[n]))/((((Subscript[E, n])-(Subscript[E, m])))^2))((∂_λ)(Subscript[E, n])-(∂_λ)(Subscript[E, m])))
=(∑_(m≠n))((∑_(l≠m))(Ket[l])((Subscript[V, l m])/((Subscript[E, m])-(Subscript[E, l])))((Subscript[V, m n])/((Subscript[E, n])-(Subscript[E, m])))+(∑_(l≠m))(Ket[m])((Subscript[V, m l])/((Subscript[E, m])-(Subscript[E, l])))((Subscript[V, l n])/((Subscript[E, n])-(Subscript[E, m])))+(∑_(l≠n))(Ket[m])((Subscript[V, m l])/((Subscript[E, n])-(Subscript[E, m])))((Subscript[V, l n])/((Subscript[E, n])-(Subscript[E, l])))-(Ket[m])((Subscript[V, m n])/((((Subscript[E, n])-(Subscript[E, m])))^2))((Subscript[V, n n])-(Subscript[V, m m])))

(Ket[(∂_λ^2)n])=(∑_(m≠n))((∑_(l≠m,n))((Ket[l])((Subscript[V, l m])/((Subscript[E, m])-(Subscript[E, l])))((Subscript[V, m n])/((Subscript[E, n])-(Subscript[E, m])))+(Ket[m])((Subscript[V, m l])/((Subscript[E, m])-(Subscript[E, l])))((Subscript[V, l n])/((Subscript[E, n])-(Subscript[E, m])))+(Ket[m])((Subscript[V, m l])/((Subscript[E, n])-(Subscript[E, m])))((Subscript[V, l n])/((Subscript[E, n])-(Subscript[E, l]))))+(Ket[n])((Subscript[V, n m])/((Subscript[E, m])-(Subscript[E, n])))((Subscript[V, m n])/((Subscript[E, n])-(Subscript[E, m])))+(Ket[m])((Subscript[V, m n])/((Subscript[E, m])-(Subscript[E, n])))((Subscript[V, n n])/((Subscript[E, n])-(Subscript[E, m])))+(Ket[m])((Subscript[V, m m])/((Subscript[E, n])-(Subscript[E, m])))((Subscript[V, m n])/((Subscript[E, n])-(Subscript[E, m])))-(Ket[m])(((Subscript[V, m n])(Subscript[V, n n]))/((((Subscript[E, n])-(Subscript[E, m])))^2))+(Ket[m])(((Subscript[V, m n])(Subscript[V, m m]))/((((Subscript[E, n])-(Subscript[E, m])))^2)))
=(∑_(m≠n))((∑_(l≠m,n))((Ket[l])((Subscript[V, l m])/((Subscript[E, m])-(Subscript[E, l])))((Subscript[V, m n])/((Subscript[E, n])-(Subscript[E, m])))+(Ket[m])((Subscript[V, m l])/((Subscript[E, m])-(Subscript[E, l])))((Subscript[V, l n])/((Subscript[E, n])-(Subscript[E, m])))+(Ket[m])((Subscript[V, m l])/((Subscript[E, n])-(Subscript[E, m])))((Subscript[V, l n])/((Subscript[E, n])-(Subscript[E, l]))))-(Ket[n])(((Subscript[V, n m])(Subscript[V, m n]))/((((Subscript[E, n])-(Subscript[E, m])))^2))-2(Ket[m])(((Subscript[V, m n])(Subscript[V, n n]))/((((Subscript[E, n])-(Subscript[E, m])))^2))+2(Ket[m])(((Subscript[V, m n])(Subscript[V, m m]))/((((Subscript[E, n])-(Subscript[E, m])))^2)))

(∑_(m≠n))(∑_(l≠m,n))(Ket[l])((Subscript[V, l m])/((Subscript[E, m])-(Subscript[E, l])))((Subscript[V, m n])/((Subscript[E, n])-(Subscript[E, m])))=(∑_(m≠n))(∑_(l≠m,n))(Ket[m])((Subscript[V, m l])/((Subscript[E, l])-(Subscript[E, m])))((Subscript[V, l n])/((Subscript[E, n])-(Subscript[E, l]))),

therefore

(Ket[(∂_λ^2)n])=(∑_(m≠n))((∑_(l≠m,n))(Ket[m])(Subscript[V, m l])(Subscript[V, l n])((1/(((Subscript[E, l])-(Subscript[E, m]))((Subscript[E, n])-(Subscript[E, l]))))+(1/(((Subscript[E, m])-(Subscript[E, l]))((Subscript[E, n])-(Subscript[E, m]))))+(1/(((Subscript[E, n])-(Subscript[E, m]))((Subscript[E, n])-(Subscript[E, l])))))-(Ket[n])(((Subscript[V, n m])(Subscript[V, m n]))/((((Subscript[E, n])-(Subscript[E, m])))^2))-2(Ket[m])(((Subscript[V, m n])(Subscript[V, n n]))/((((Subscript[E, n])-(Subscript[E, m])))^2))+2(Ket[m])(((Subscript[V, m n])(Subscript[V, m m]))/((((Subscript[E, n])-(Subscript[E, m])))^2)))
=(∑_(m≠n))((∑_(l≠m,n))2(Ket[m])(((Subscript[V, m l])(Subscript[V, l n]))/(((Subscript[E, n])-(Subscript[E, m]))((Subscript[E, n])-(Subscript[E, l]))))-(Ket[n])(((Subscript[V, n m])(Subscript[V, m n]))/((((Subscript[E, n])-(Subscript[E, m])))^2))-2(Ket[m])(((Subscript[V, m n])(Subscript[V, n n]))/((((Subscript[E, n])-(Subscript[E, m])))^2))+2(Ket[m])(((Subscript[V, m n])(Subscript[V, m m]))/((((Subscript[E, n])-(Subscript[E, m])))^2)))

(Ket[(∂_λ^2)n])=(∑_(m≠n))((∑_(l≠n))2(Ket[m])(((Subscript[V, m l])(Subscript[V, l n]))/(((Subscript[E, n])-(Subscript[E, m]))((Subscript[E, n])-(Subscript[E, l]))))-2(Ket[m])(((Subscript[V, m n])(Subscript[V, n n]))/((((Subscript[E, n])-(Subscript[E, m])))^2))-(Ket[n])(((Subscript[V, n m])(Subscript[V, m n]))/((((Subscript[E, n])-(Subscript[E, m])))^2)))

Summary of Results

H(λ)=(Subscript[H, 0])+λ V→(Subscript[H, 0])+V.

(Subscript[H, 0])=(∑_n)(Ket[n])(Subscript[E, n])(Bra[n]), V=(∑_(m,n))(Ket[m])(Subscript[V, m n])(Bra[n]),

(Subscript[H, 0])+V=(∑_n)(Ket[n(V)])(Subscript[E, n])(V)(Bra[n(V)]).

Physical Intuitions

(Subscript[H, 0])+V≏(({{ , ⋮,  , ⋮,  }, {⋯, (Subscript[E, n])+(Subscript[V, n n]), ⋯, (Subscript[V, n m]), ⋯}, { , ⋮,  , ⋮,  }, {⋯, (Subscript[V, m n]), ⋯, (Subscript[E, m])+(Subscript[V, m m]), ⋯}, { , ⋮,  , ⋮,  }})),

To the 1st order, (Subscript[E, n])(V) simply takes out the diagonal matrix element of (Subscript[H, 0])+V, which amounts to re-evaluating the energy expectation value on the old eigenstate (Ket[n]):

(Subscript[E, n])+(Subscript[V, n n])=(Bra[n])(Subscript[H, 0])+V(Ket[n]).

State hybridization: the 1st order correction of (Ket[n(V)]) hybridizes (mixes) the original state (Ket[n]) with all the other (Ket[m]) states that are connected to (Ket[n]) by non-vanishing (Subscript[V, m n]).

(Ket[n(V)])=(Ket[n])+(∑_(m≠n))(Ket[m])((Subscript[V, m n])/((Subscript[E, n])-(Subscript[E, m])))

The hybridization coefficient is

proportional to the transition rate (Subscript[V, m n]),

inversely proportional to the energy level spacing ((Subscript[E, n])-(Subscript[E, m])).

Row[Block[{a=0.5,En=0,Em=#,lev},
lev[id_]:={Line[{#,0}&/@{-a,a}],Text["\!\(\[LeftBracketingBar]"<>id<>"⟩\)",{0,0},{0,-1}],Text["\!\(E\_"<>id<>"\)",{a,0},{-1.5,0}]};
Graphics[{Translate[lev["n"],{-1,En}],Translate[lev["m"],{1,Em}],{Arrowheads[{{0.1,0.5}}],Dashed,AbsoluteThickness[1],Arrow[{{-1+a,En},{1-a,Em}}]},Text["\!\(V\_\(m n\)\)",{0,(En+Em)/2},{-1.2Sign[#],0}]},ImageSize->100]]&/@{2,-2},Spacer[20]]

When (Subscript[E, n])-(Subscript[E, m])->0 (energy levels become degenerated) => the hybridization coefficient diverges => signifies the breakdown of the non-degenerate perturbation theory. This scenario is called a level resonance.

Level repulsion: the 2nd order energy correction always repel the levels apart (by making the higher level higher and the lower level lower).

(Subscript[E, n])+(∑_m)(((|(Subscript[V, n m])|)^2)/((Subscript[E, n])-(Subscript[E, m])))({{{, ({{<(Subscript[E, n]), if (Subscript[E, n])<(Subscript[E, m]),}, {>(Subscript[E, n]), if (Subscript[E, n])>(Subscript[E, m]).}})}})

Block[{a=0.5,En=0,Em=2,lev},
lev[id_]:={Line[{#,0}&/@{-a,a}],Text["\!\(\[LeftBracketingBar]"<>id<>"⟩\)",{0,0},{0,-1}],Text["\!\(E\_"<>id<>"\)",{a,0},{-1.5,0}]};
Graphics[{Translate[lev["n"],{-1,En}],Translate[lev["m"],{1,Em}],{Arrowheads[{{0.08,0.5}}],Dashed,AbsoluteThickness[1],Arrow@BSplineCurve[{{-1+a,En},{0,Em},{1-a,Em}}],Arrow@BSplineCurve[{{1-a,Em},{0,En},{-1+a,En}}]},Text["\!\(V\_\(m n\)\)",{-1+a,(En+Em)/2},{0,-1.2}],Text["\!\(V\_\(n m\)\)",{1-a,(En+Em)/2},{0,1.2}]},ImageSize->120]]

Application: the Qubit Model

(Subscript[H, 0])+V≏(({{1, λ}, {λ, -1}})).

Unperturbed spectrum. (Ket[↑]): (Subscript[E, ↑])=+1, (Ket[↓]): (Subscript[E, ↓])=-1.

State hybridization:

(Ket[↑])'=(Ket[↑])+((Subscript[V, ↓↑])/((Subscript[E, ↑])-(Subscript[E, ↓])))(Ket[↓])+…=(Ket[↑])+(λ/2)(Ket[↓])+…,
(Ket[↓])'=(Ket[↓])+((Subscript[V, ↑↓])/((Subscript[E, ↓])-(Subscript[E, ↑])))(Ket[↑])+…=(Ket[↓])-(λ/2)(Ket[↑])+….

Level repulsion:

(Subscript[E, ↑]^′)=(Subscript[E, ↑])+(((Subscript[V, ↑↓])(Subscript[V, ↓↑]))/((Subscript[E, ↑])-(Subscript[E, ↓])))+…=+1+((λ^2)/2)+…,
(Subscript[E, ↓]^′)=(Subscript[E, ↓])+(((Subscript[V, ↓↑])(Subscript[V, ↑↓]))/((Subscript[E, ↓])-(Subscript[E, ↑])))+…=-1-((λ^2)/2)+….

A Numerical Interlude: Jacobi Algorithm*

Algorithm

Given a Hamiltonian H (as a matrix).

Pick an off-diagonal element (Subscript[H, m n]) (labeled by two indices m and n), called a pivot. In practice, the largest (Subscript[H, m n]) is taken as the pivot, but that is not necessary in general.

(({{ , ⋮,  , ⋮,  }, {⋯, (Subscript[H, n n]), ⋯, (Subscript[H, n m]), ⋯}, { , ⋮,  , ⋮,  }, {⋯, (Subscript[H, m n]), ⋯, (Subscript[H, m m]), ⋯}, { , ⋮,  , ⋮,  }})).

The goal is to suppress (Subscript[H, m n]).

First introduce two angles θ and ϕ to parameterize the following ratio

((Subscript[H, m n])/((Subscript[H, n n])-(Subscript[H, m m])))=(1/2)(e^(i ϕ))tan 2θ.

Then construct a unitary matrix G (called the Givens matrix)

G=(({{𝟙,  ,  ,  ,  }, { , cos θ,  , -(e^(-i ϕ))sin θ,  }, { ,  , 𝟙,  ,  }, { , (e^(i ϕ))sin θ,  , cos θ,  }, { ,  ,  ,  , 𝟙}})).

The following transformation will eliminate (Subscript[H, m n]) and brings H closer to diagonal

H->H'=(SuperDagger[G]) H G.

Take the new Hamiltonian H' and start over again (until it is sufficiently diagonalized).

Demonstration

Graphics[{{EdgeForm[Directive[Black,AbsoluteThickness[1]]],ComplexColor[#],Disk[ReIm[#],0.14]}&/@Exp[I 2π Range[16]/16],Text[#1,0.6ReIm[#2]]&@@@{{"+1",1},{"-1",-1},{"+I",I},{"-I",-I}}},ImageSize->80]

Block[{L=5,H0,Jacobi,Hs},
SeedRandom[0];
H0=Chop[ Hermitianize[RandomVariate[NormalDistribution[],{L,L,2}].{1,I}]];
Jacobi[H_]:=Block[{m,n,r,ϕ,θ,G},
{{m,n}}=Position[#,Max@#]&@Abs@LowerTriangularize[H,-1];
r=H[[m,n]]/(H[[n,n]]-H[[m,m]]);
ϕ=Arg[r];
θ=ArcTan[2 Abs[r]]/2;
G=IdentityMatrix@Dimensions@H;
G[[n,n]]=G[[m,m]]=Cos[θ];
G[[m,n]]=-Sin[θ]Exp[I ϕ];
G[[n,m]]=Sin[θ]Exp[-I ϕ];
Chop[G.H.ConjugateTranspose[G]]
];
Hs=NestList[Jacobi,H0,20];
Column@{Row[ComplexMatrixPlot[#,FrameTicks->None,Mesh->All,MeshStyle->AbsoluteThickness[0.7],ImageSize->50]&/@Hs,"→"],
Grid[{Prepend[Sort@Diagonal@First@Hs,"Initial:"],Prepend[Sort@Diagonal@Last@Hs,"Final:"],Prepend[Sort@Eigenvalues@H0,"Exact:"]}]}
]

Block[{L=5,H0,mask,Jacobi,Hs},
SeedRandom[2];
H0=Chop[ Hermitianize[RandomVariate[NormalDistribution[],{L,L,2}].{1,I}]];
mask=Normal@SparseArray[{i_,j_}/;i>2&&j≤2:>1,Dimensions[H0]];
Jacobi[H_]:=Block[{m,n,r,ϕ,θ,G},
{{m,n}}=Position[#,Max@#]&@Abs[mask H];
r=H[[m,n]]/(H[[n,n]]-H[[m,m]]);
ϕ=Arg[r];
θ=ArcTan[2 Abs[r]]/2;
G=IdentityMatrix@Dimensions@H;
G[[n,n]]=G[[m,m]]=Cos[θ];
G[[m,n]]=-Sin[θ]Exp[I ϕ];
G[[n,m]]=Sin[θ]Exp[-I ϕ];
Chop[G.H.ConjugateTranspose[G]]];
Hs=NestList[Jacobi,H0,20];
Row[ComplexMatrixPlot[#,MeshStyle->AbsoluteThickness[0.7],FrameTicks->None,Mesh->All,ImageSize->50]&/@Hs,"→"]
]

Perturbative Limit

(Subscript[E, n])=(Subscript[H, n n]),
(Subscript[V, m n])=(Subscript[H, m n]) (for m≠n).

(e^(i ϕ))θ≃((Subscript[V, m n])/((Subscript[E, n])-(Subscript[E, m]))).

G≃(({{𝟙,  ,  ,  ,  }, { , 1,  , ((Subscript[V, n m])/((Subscript[E, m])-(Subscript[E, n]))),  }, { ,  , 𝟙,  ,  }, { , ((Subscript[V, m n])/((Subscript[E, n])-(Subscript[E, m]))),  , 1,  }, { ,  ,  ,  , 𝟙}})).

(({{(Ket[n])', (Ket[m])'}}))=(({{(Ket[n]), (Ket[m])}}))G≃(({{(Ket[n]), (Ket[m])}}))(({{1, ((Subscript[V, n m])/((Subscript[E, m])-(Subscript[E, n])))}, {((Subscript[V, m n])/((Subscript[E, n])-(Subscript[E, m]))), 1}})),

(Ket[n])'≃(Ket[n])+(Ket[m])((Subscript[V, m n])/((Subscript[E, n])-(Subscript[E, m]))),
(Ket[m])'≃(Ket[m])+(Ket[n])((Subscript[V, n m])/((Subscript[E, m])-(Subscript[E, n]))).

Degenerate Perturbation Theory

General Ideas

Block[{L=5,Jacobi,mask,K,V,H0,H1,H2,cmp},
SeedRandom[2];
mask=Normal@SparseArray[{i_,j_}/;i>2&&j≤2:>1,{L,L}];
Jacobi[H_]:=Block[{m,n,r,ϕ,θ,G},{m,n}=First@Position[#,Max@#]&@Abs[mask H];
r=H[[m,n]]/(H[[n,n]]-H[[m,m]]);
ϕ=Arg[r];
θ=ArcTan[2 Abs[r]]/2;
G=IdentityMatrix@Dimensions@H;
G[[n,n]]=G[[m,m]]=Cos[θ];
G[[m,n]]=-Sin[θ]Exp[I ϕ];
G[[n,m]]=Sin[θ]Exp[-I ϕ];
Chop[G.H.ConjugateTranspose[G]]];
K=DiagonalMatrix[{-1,-1,1,1,1}];V=Chop[Hermitianize[RandomVariate[NormalDistribution[],{L,L,2}].{1,I}]];
V=V-DiagonalMatrix@Diagonal@V;
H0=K+0.3V;
H1=Nest[Jacobi,H0,25];
mask=Normal@SparseArray[{i_,j_}/;i>j&&!(i>2&&j≤2):>1,Dimensions[H1]];
H2=Nest[Jacobi,H1,7];
cmp=ComplexMatrixPlot[#,Mesh->Block[{mesh={{0,AbsoluteThickness[1]},{1,AbsoluteThickness[0.3]},{2,AbsoluteThickness[1]},{3,AbsoluteThickness[0.3]},{4,AbsoluteThickness[0.3]},{5,AbsoluteThickness[1]}}},{mesh,mesh}],FrameTicks->None,ImageSize->60]&;
Style[Row[{Column[{cmp[K],"+       ",Row[{"λ ",cmp[V]}]},Alignment->Right]," = ",Row[cmp/@{H0,H1,H2},"→"]}],FontFamily->"CMU"]]

First apply non-degenerate perturbation theory to bring the Hamiltonian to diagonal blocks in the degenerate subspaces.

Each diagonal block represents an effective Hamiltonian within the degenerated subspace.

previously: perturbative correction to each energy level → now: perturbative correction to each effective Hamiltonian.

Then go on with each effective Hamiltonian:

If the degeneracy has been lifted (typically), we proceed with non-degenerate perturbation in each block.

If the diagonal elements are still fully degenerated, we proceed with exact diagonalization (no perturbative approach available in this case).

Generalized Hellmann-Feynman Theorems

(Ket[n])(Overscript[⟶, generalize])(Ket[n α])

n: principal quantum number, labels degenerate subspaces.

α: secondary quantum number, labels orthogonal degenerate state within each subspace.

(Ket[n α]) form a set of orthonormal basis: (BraKet[m α, n β])=(Subscript[δ, m n])(Subscript[δ, α β]).

The perturbation operator V can be represented in this basis:

H(λ)=(Subscript[H, 0])+λ V,

(Subscript[E, n,α β])(λ) is the matrix element of the effective Hamiltonian in the nth degenerate subspace.

(Subscript[E, n,α β])(λ) should take the form of

(Subscript[E, n,α β])≡(Subscript[E, n,α β])(0)=(Subscript[E, n])(Subscript[δ, α β]),

to restore Eq. () in the λ=0 unperturbed limit.

The Hermitian conjugate version of Eq. () reads

(Bra[m γ(λ)])H(λ)=(∑_δ)(Bra[m δ(λ)])(Subscript[E, m,δ γ]^( * ))(λ)=(∑_δ)(Subscript[E, m,γ δ])(λ)(Bra[m δ(λ)]),

where we have assumed that the effective Hamiltonian is  Hermitian: (Subscript[E, m,δ γ]^( * ))(λ)=(Subscript[E, m,γ δ])(λ).

(Bra[m γ])(∂_λ)H(Ket[n β])=(∑_α)(∂_λ)(Subscript[E, n,α β])(BraKet[m γ, n α])+(∑_α)(BraKet[m γ, (∂_λ)n α])(Subscript[E, n,α β])-(∑_δ)(Subscript[E, m,γ δ])(BraKet[m δ, (∂_λ)n β])
=(∂_λ)(Subscript[E, n,γ β])(Subscript[δ, m n])+((Subscript[E, n])-(Subscript[E, m]))(BraKet[m γ, (∂_λ)n β])

When m=n, the first Hellmann-Feynman theorem:

When m≠n, the second Hellmann-Feynman theorem:

Effective Hamiltonian

(∂_λ)(Subscript[E, n,α β])=(Subscript[V, n α,n β]),
(∂_λ^2)(Subscript[E, n,α β])=2(∑_(m≠n))(∑_γ)(((Subscript[V, n α,m γ])(Subscript[V, m γ,n β]))/((Subscript[E, n])-(Subscript[E, m]))),
(Ket[(∂_λ)n α])=(∑_(m≠n))(∑_β)(Ket[m β])((Subscript[V, m β,n α])/((Subscript[E, n])-(Subscript[E, m]))).

(the matrix element of) the effective Hamiltonian to the 2nd order in λ

the corrected basis state to the 1st order in λ

Application: A Spin-1 Model

Basis: (Ket[+1]),(Ket[0]),(Ket[-1]).

The matrix representations for spin operators (S^x) and (S^z)

(S^x)≏(1/(Sqrt[2]))(({{0, 1, 0}, {1, 0, 1}, {0, 1, 0}})),(S^z)≏(({{1, 0, 0}, {0, 0, 0}, {0, 0, -1}})).

H(λ)=(Subscript[H, 0])+λ V,
(Subscript[H, 0])=((((S^z)))^2)≏(({{1, 0, 0}, {0, 0, 0}, {0, 0, 1}})),
V=(S^x)+(S^z)≏(({{1, (1/(Sqrt[2])), 0}, {(1/(Sqrt[2])), 0, (1/(Sqrt[2]))}, {0, (1/(Sqrt[2])), -1}})).

({{n, states, energy}, {1, (Ket[+1]),(Ket[-1]), (Subscript[E, 1])=1}, {0, (Ket[0]), (Subscript[E, 0])=0}})

Corrected basis (use ′ to denote the perturbed result)

(Ket[±1])'=(Ket[±1])+(Ket[0])((Subscript[V, 0,±1])/((Subscript[E, 1])-(Subscript[E, 0])))λ+…=(Ket[±1])+(λ/(Sqrt[2]))(Ket[0])+…,
(Ket[0])'=(Ket[0])+(Ket[+1])((Subscript[V, +1,0])/((Subscript[E, 0])-(Subscript[E, 1])))λ+(Ket[-1])((Subscript[V, -1,0])/((Subscript[E, 0])-(Subscript[E, 1])))λ+…
=(Ket[0])-(λ/(Sqrt[2]))((Ket[+1])+(Ket[-1]))+….

Effective Hamiltonian

n=1 subspace

(Subscript[H, 1]^eff)=(Ket[+1])'((Subscript[E, 1])+(Subscript[V, +1,+1])λ+(((Subscript[V, +1,0])(Subscript[V, 0,+1]))/((Subscript[E, 1])-(Subscript[E, 0])))(λ^2))(Bra[+1])'+(Ket[-1])'((Subscript[E, 1])+(Subscript[V, -1,-1])λ+(((Subscript[V, -1,0])(Subscript[V, 0,-1]))/((Subscript[E, 1])-(Subscript[E, 0])))(λ^2))(Bra[-1])'+(Ket[+1])'((Subscript[V, +1,-1])λ+(((Subscript[V, +1,0])(Subscript[V, 0,-1]))/((Subscript[E, 1])-(Subscript[E, 0])))(λ^2))(Bra[-1])'+(Ket[-1])'((Subscript[V, -1,+1])λ+(((Subscript[V, -1,0])(Subscript[V, 0,+1]))/((Subscript[E, 1])-(Subscript[E, 0])))(λ^2))(Bra[+1])'+…
=(Ket[+1])'(1+λ+((λ^2)/2))(Bra[+1])'+(Ket[-1])'(1-λ+((λ^2)/2))(Bra[-1])'+(Ket[+1])'((λ^2)/2)(Bra[-1])'+(Ket[-1])'((λ^2)/2)(Bra[+1])'+…

On the corrected basis (Ket[+1])',(Ket[-1])', (Subscript[H, 1]^eff) can be represented as a 2×2 matrix

(Subscript[H, 1]^eff)≏(({{1+λ+((λ^2)/2), ((λ^2)/2)}, {((λ^2)/2), 1-λ+((λ^2)/2)}})).

The degeneracy is lifted => we can proceed with non-degenerate perturbation in the next iteration.

n=0 subspace

(Subscript[H, 0]^eff)=(Ket[0])'((Subscript[E, 0])+(Subscript[V, 0,0])λ+(((Subscript[V, 0,+1])(Subscript[V, +1,0])+(Subscript[V, 0,-1])(Subscript[V, -1,0]))/((Subscript[E, 0])-(Subscript[E, 1])))(λ^2))(Bra[0])'+…
=(Ket[0])'(-(λ^2))(Bra[0])'+…

(Subscript[H, 1]^eff)=(Ket[+1])'(Subscript[E, +1]^′)(Bra[+1])'+(Ket[-1])'(Subscript[E, -1]^′)(Bra[-1])'+(Ket[+1])'(Subscript[V, +1,-1]^′)(Bra[-1])'+(Ket[-1])'(Subscript[V, -1,+1]^′)(Bra[+1])',
(Subscript[E, ±1]^′)=1±λ+((λ^2)/2),
(Subscript[V, +1,-1]^′)=(Subscript[V, -1,+1]^′)=((λ^2)/2), (Subscript[V, +1,+1]^′)=(Subscript[V, -1,-1]^′)=0.

Corrected states

(Ket[+1])''=(Ket[+1])'+(Ket[-1])'((Subscript[V, -1,+1]^′)/((Subscript[E, +1]^′)-(Subscript[E, -1]^′)))+…=(Ket[+1])'+(λ/4)(Ket[-1])'+…
(Ket[-1])''=(Ket[-1])'+(Ket[+1])'((Subscript[V, +1,-1]^′)/((Subscript[E, -1]^′)-(Subscript[E, +1]^′)))+…=(Ket[-1])'-(λ/4)(Ket[+1])'+…

Plugging in Eq. (),

(Ket[+1])''=(Ket[+1])+(λ/(Sqrt[2]))(Ket[0])+(λ/4)(Ket[-1])+((λ^2)/(4(Sqrt[2])))(Ket[0])+…
(Ket[-1])''=(Ket[-1])+(λ/(Sqrt[2]))(Ket[0])-(λ/4)(Ket[+1])-((λ^2)/(4(Sqrt[2])))(Ket[0])+…

The (λ^2) terms should not be included, because the expansion is only reliable to the 1st order in λ.

Corrected energies

(Subscript[E, +1]^″)=(Subscript[E, +1]^′)+(Subscript[V, +1,+1]^′)+(((Subscript[V, +1,-1]^′)(Subscript[V, -1,+1]^′))/((Subscript[E, +1])-(Subscript[E, -1])))=1+λ+((λ^2)/2)+((λ^3)/8)+…
(Subscript[E, -1]^″)=(Subscript[E, -1]^′)+(Subscript[V, -1,-1]^′)+(((Subscript[V, -1,+1]^′)(Subscript[V, +1,-1]^′))/((Subscript[E, -1])-(Subscript[E, +1])))=1-λ+((λ^2)/2)-((λ^3)/8)+…

The (λ^3) terms should not be included, because the expansion is only reliable to the 2nd order in λ.

({{eigenenergies, eigenstates}, {(Subscript[E, +1]^″)=1+λ+((λ^2)/2)+𝒪((λ^3)), (Ket[+1])''=(Ket[+1])+(λ/(Sqrt[2]))(Ket[0])+(λ/4)(Ket[-1])+𝒪((λ^2))}, {(Subscript[E, 0]^′)=-(λ^2)+𝒪((λ^3)), (Ket[0])'=(Ket[0])-(λ/(Sqrt[2]))((Ket[+1])+(Ket[-1]))+𝒪((λ^2))}, {(Subscript[E, -1]^″)=1-λ+((λ^2)/2)+𝒪((λ^3)), (Ket[-1])''=(Ket[-1])+(λ/(Sqrt[2]))(Ket[0])-(λ/4)(Ket[+1])+𝒪((λ^2))}})

Grid[{Series[#1,{λ,0,2}],PowerExpand@Series[#2/Sqrt[#2.#2],{λ,0,1}].(Ket/@{"+1","0","-1"})}&@@@Thread@Eigensystem[{{1,0,0},{0,0,0},{0,0,1}}+λ{{1,1/Sqrt[2],0},{1/Sqrt[2],0,1/Sqrt[2]},{0,1/Sqrt[2],-1}}]]

Time-Dependent Perturbation

Time-Dependent Perturbation Theory

Problem Setup

Time-independent perturbation: corrections to energy levels, states, effective Hamiltonians.

Time-dependent perturbation: corrections to time-evolution operators (propagators, Green’s functions).

The spectrum of (Subscript[H, 0]) is known

(Subscript[H, 0])(Ket[n])=(Subscript[E, n])(Ket[n]).

The basis states (Ket[n]) are fixed (time-independent), because (Subscript[H, 0]) is time-independent.

All the time dependence is ascribed to the operator V(t), which can be represented in the eigenbasis of (Subscript[H, 0])

V(t)=(∑_(m,n))(Ket[m])(Subscript[V, m n])(t)(Bra[n]).

(Subscript[V, m n])(t) is expected to be small (compared to the energy scale of (Subscript[H, 0])) through out the time t.

This defines the unitary operator U(t), called the time-evolution operator. Once we know U(t), we can apply it to any initial state (Subscript[(Ket[ψ(0)]), 𝒮]) to obtain the final state (Subscript[(Ket[ψ(t)]), 𝒮]) (we don’t need to solve the Schrödinger equation over and over again).

i(∂_t)(Subscript[U, 0])(t)=(Subscript[H, 0])(Subscript[U, 0])(t).

({{(Subscript[H, 0]), (Overscript[⟶, +V(t)]), H(t)}, {↓,  , ↓}, {(Subscript[U, 0])(t), (Overscript[⟶, +???]), U(t)}})

Interaction Picture

State-based formalism.

One can show that

i(∂_t)(Subscript[(Ket[ψ(t)]), ℐ])=i(∂_t)((e^(i (Subscript[H, 0])t))(Subscript[(Ket[ψ(t)]), 𝒮]))
=i(∂_t)((e^(i (Subscript[H, 0])t)))(Subscript[(Ket[ψ(t)]), 𝒮])+(e^(i (Subscript[H, 0])t))i(∂_t)(Subscript[(Ket[ψ(t)]), 𝒮])
=(e^(i (Subscript[H, 0])t))(-(Subscript[H, 0]))(Subscript[(Ket[ψ(t)]), 𝒮])+(e^(i (Subscript[H, 0])t))((Subscript[H, 0])+V(t))(Subscript[(Ket[ψ(t)]), 𝒮])
=(Subscript[U, 0]^†)(t)V(t)(Subscript[(Ket[ψ(t)]), 𝒮])
=(Subscript[U, 0]^†)(t)V(t)(Subscript[U, 0])(t)(Subscript[U, 0]^†)(t)(Subscript[(Ket[ψ(t)]), 𝒮]).

Define the perturbation in the interaction picture:

The time-evolution of (Subscript[(Ket[ψ(t)]), ℐ]) is described by

Operator-based formalism.

Block[{r=0.05},Graphics[{{Arrowheads[0.06],AbsoluteThickness[1],Arrow[{{-0.2,0},{1.2,0}}],Line[{0,# r/3}&/@{-1,1}]},{Text["\!\( t\)",{1.2,0},{-1,0}],Text["\!\(0\)",{0,0},{1,1}]},{Arrowheads[{{0.05,0.5}}],{Red,Arrow@BSplineCurve[{{0,r},{1,r},{1+r/2,r},{1+r,r/2},{1+r,0}}],Text["\!\(U(t)\)",{0.5,r},{0,-1}]},{Blue,Arrow@BSplineCurve[{{1+r,0},{1+r,-r/2},{1+r/2,-r},{1,-r},{0,-r}}],Text["\!\(U\_0\%†(t)\)",{0.5,-r},{0,1}]}}},ImageSize->180]]

(Subscript[U, ℐ])(t) captures the “additional” unitary evolution implemented by U(t) compared to the reference (Subscript[U, 0])(t). Following the similar derivation in Eq. (), we can show that (Subscript[U, ℐ])(t) is governed by

subject to the initial condition: (Subscript[U, ℐ])(0)=𝟙. The solution of (Subscript[U, ℐ])(t) can be used

to provide the universal solution for (Subscript[(Ket[ψ(t)]), ℐ])=(Subscript[U, ℐ])(t)(Subscript[(Ket[ψ(0)]), ℐ]),

and to construct U(t)=(Subscript[U, 0])(t)(Subscript[U, ℐ])(t).

Dyson Series

i(Subscript[U, ℐ])(t)-i(Subscript[U, ℐ])(0)=i(∫_0^t)dt'(∂_(t'))(Subscript[U, ℐ])(t')=(∫_0^t)dt'(Subscript[V, ℐ])(t')(Subscript[U, ℐ])(t'),

(Subscript[U, ℐ])(t)=𝟙-i(∫_0^t)dt'(Subscript[V, ℐ])(t')+(((-i))^2)(∫_0^t)dt'(Subscript[V, ℐ])(t')(∫_0^(t'))dt''(Subscript[V, ℐ])(t'')(Subscript[U, ℐ])(t'').

Green’s Function

(Subscript[V, ℐ])((Subscript[t, k]))(Subscript[V, ℐ])((Subscript[t, k-1]))…(Subscript[V, ℐ])((Subscript[t, 1]))
=(Subscript[U, 0]^†)((Subscript[t, k]))V((Subscript[t, k]))(Subscript[U, 0])((Subscript[t, k]))(Subscript[U, 0]^†)((Subscript[t, k-1]))V((Subscript[t, k-1]))(Subscript[U, 0])((Subscript[t, k-1]))…(Subscript[U, 0]^†)((Subscript[t, 1]))V((Subscript[t, 1]))(Subscript[U, 0])((Subscript[t, 1])).

(Subscript[V, ℐ])((Subscript[t, k]))(Subscript[V, ℐ])((Subscript[t, k-1]))…(Subscript[V, ℐ])((Subscript[t, 1]))
=(Subscript[U, 0]^†)(t)(Subscript[G, 0])(t,(Subscript[t, k]))V((Subscript[t, k]))(Subscript[G, 0])((Subscript[t, k]),(Subscript[t, k-1]))V((Subscript[t, k-1]))… (Subscript[G, 0])((Subscript[t, 2]),(Subscript[t, 1]))V((Subscript[t, 1]))(Subscript[G, 0])((Subscript[t, 1]),0).

U(t)=(∑_(k=0)^∞)(((-i))^k)(∫_0^t)d(Subscript[t, k])(∫_0^(Subscript[t, k]))d(Subscript[t, k-1])…(∫_0^(Subscript[t, 2]))d(Subscript[t, 1])(Subscript[G, 0])(t,(Subscript[t, k]))V((Subscript[t, k]))(Subscript[G, 0])((Subscript[t, k]),(Subscript[t, k-1]))V((Subscript[t, k-1]))… (Subscript[G, 0])((Subscript[t, 2]),(Subscript[t, 1]))V((Subscript[t, 1]))(Subscript[G, 0])((Subscript[t, 1]),0).

G(t,t') can be calculated in power series of V(t) given (Subscript[G, 0])(t,t').

Since U(t)=G(t,0), we also know how to calculate U(t) in power series of V(t).

Feynman Diagrams

A directed single-line link: the bare Green’s function from one time to another,

=(Subscript[G, 0])(t,t').

The arrow specifies the direction of time (from past to future).

A solid node: the perturbation operator at a particular time,

t=-i V(t).

Connecting links and nodes: identifying the time together

({{(Subscript[t, 0]),  , (Subscript[t, 1]),  , (Subscript[t, 2])}})=(Subscript[G, 0])((Subscript[t, 2]),(Subscript[t, 1]))(-i V((Subscript[t, 1])))(Subscript[G, 0])((Subscript[t, 1]),(Subscript[t, 0])).

Note: in the diagram, the time flows along the arrow from left to right; but in the operator product, the operator acts in sequence from right to left. Things are mirror image (left-right reversed) to each other with respect to the “=” sign.

If the time is not labeled explicitly, then

the time of the outmost node (the initial and final nodes) is fixed, (convention: (Subscript[t, 0]) - the initial time, t - the final time),

the time of the internal node will be automatically integrated over, and the integration goes through all possible arrangements preserving the time-ordering.

=(Subscript[G, 0])(t,(Subscript[t, 0])),
=(-i )(∫_(Subscript[t, 0])^t)d(Subscript[t, 1])(Subscript[G, 0])(t,(Subscript[t, 1]))V((Subscript[t, 1]))(Subscript[G, 0])((Subscript[t, 1]),(Subscript[t, 0])),
=(((-i ))^2)(∫_(Subscript[t, 0])^t)d(Subscript[t, 2])(∫_(Subscript[t, 0])^(Subscript[t, 2]))d(Subscript[t, 1])(Subscript[G, 0])(t,(Subscript[t, 2]))V((Subscript[t, 2]))(Subscript[G, 0])((Subscript[t, 2]),(Subscript[t, 1]))V((Subscript[t, 1]))(Subscript[G, 0])((Subscript[t, 1]),(Subscript[t, 0])),
…

A directed double-line link: the dressed Green’s function from one time to another,

=G(t,(Subscript[t, 0])).

If we turn off the perturbation, i.e.V(t)=0 or =0, Eq. () reduces to

=,

as all the diagrams containing the node will vanish. This simply restores G(t,(Subscript[t, 0]))=(Subscript[G, 0])(t,(Subscript[t, 0])) in the absence of perturbation.

In the presence of V(t), the propagator is dressed order-by-order by scattering with the perturbation . For example,  describes that the system is first propagated to an intermediate time, acted by the perturbation operator, and then continued to propagate to the final time. Other diagrams describe higher order processes. The full propagation is the sum of all possible processes.

Energy Level Transitions

Transition Probability

G(t,(Subscript[t, 0]))≃(Subscript[G, 0])(t,(Subscript[t, 0]))-i (∫_(Subscript[t, 0])^t)d(Subscript[t, 1])(Subscript[G, 0])(t,(Subscript[t, 1]))V((Subscript[t, 1]))(Subscript[G, 0])((Subscript[t, 1]),(Subscript[t, 0]))+…,

(Bra[f])G(t,(Subscript[t, 0]))(Ket[i])≃(Bra[f])(Subscript[G, 0])(t,(Subscript[t, 0]))(Ket[i])-i (∫_(Subscript[t, 0])^t)d(Subscript[t, 1])(Bra[f])(Subscript[G, 0])(t,(Subscript[t, 1]))V((Subscript[t, 1]))(Subscript[G, 0])((Subscript[t, 1]),(Subscript[t, 0]))(Ket[i])
=(e^(-i((Subscript[E, f])t+(Subscript[E, i])(Subscript[t, 0]))))(Subscript[δ, f i])-i(∫_(Subscript[t, 0])^t)d(Subscript[t, 1])(e^(-i (Subscript[E, f])(t-(Subscript[t, 1]))))(Bra[f])V((Subscript[t, 1]))(Ket[i])(e^(-i (Subscript[E, i])((Subscript[t, 1])-(Subscript[t, 0]))))
=(e^(-i((Subscript[E, f])t+(Subscript[E, i])(Subscript[t, 0]))))((Subscript[δ, f i])-i(∫_(Subscript[t, 0])^t)d(Subscript[t, 1])(Bra[f])V((Subscript[t, 1]))(Ket[i])(e^(i((Subscript[E, f])-(Subscript[E, i]))(Subscript[t, 1]))))

Fermi’s Golden Rule

V(t)=({{{, ({{V (e^(-i ω t)), t>0}, {0, t≤0}})}}).

(Subscript[P, i→f])(t)=(1/(ℏ^2))((|(∫_0^t)d(Subscript[t, 1])(Bra[f])V(Ket[i])(e^(i((Subscript[ω, f i])-ω)(Subscript[t, 1])))|)^2)
=(1/(ℏ^2))((|(Bra[f])V(Ket[i])(((e^(i((Subscript[ω, f i])-ω)t))-1)/(i((Subscript[ω, f i])-ω)))|)^2)
=(1/(ℏ^2))((|(Bra[f])V(Ket[i])|)^2)(((((sin(((Subscript[ω, f i])-ω)t/2))/(((Subscript[ω, f i])-ω)/2))))^2)

Plot[Sinc[α]^2,{α,-15,15},Epilog->{Text["\!\(t=1\)",Scaled@{1,1},{1.5,1.5}]},FrameLabel->{"\!\(α=(ω\_\(f i\)-ω)/2\)","\!\((sin(α t)/α)\^2\)"},PlotStyle->Black,PlotRange->All,ImageSize->200]

(lim_(t→∞))(1/t)(((((sin α t)/α)))^2)=π δ(α)=2π δ(2α).

One might worry that in the t->∞ limit, the transition probability is in fact diverging. How can we justify the use of perturbation theory? For a transition with (Subscript[ω, f i])≠ω, the “long time” limit is reached when t≫1/((Subscript[ω, f i])-ω), a value that can still be very short compared with the mean transition time, which is set by the matrix element ℏ/|(Bra[f])V(Ket[i])|.

Energy conservation is enforced in the long-time limit

(Subscript[E, f])-(Subscript[E, i])=ℏ ω,

such that transition occurs between levels in resonant with the the frequency of the perturbation.

Adiabatic Process

V(t)=({{{, ({{V (e^(t/τ)), t<0}, {0, t≥0}})}}).

Plot[Exp[t]UnitStep[-t],{t,-5,2},PlotStyle->Black,PlotRange->All,FrameLabel->{"\!\( t\)","\!\(V(t)\)"},ImageSize->180]

(Subscript[P, i→f])=(1/(ℏ^2))((|(∫_(-∞)^0)d(Subscript[t, 1])(Bra[f])V(Ket[i])(e^((Subscript[t, 1])/τ))(e^(i (Subscript[ω, f i])(Subscript[t, 1])))|)^2)
=(((|(Bra[f])V(Ket[i])|)^2)/(((((Subscript[E, f])-(Subscript[E, i])))^2)+(ℏ^2)/(τ^2))).

Plot[1/(ω^2+1),{ω,-5,5},PlotStyle->Black,FrameLabel->{"\!\((E\_f-E\_i)τ/ℏ\)","\!\(P\_\(i→f\)/\[LeftBracketingBar]V\_\(f i\)\[RightBracketingBar]\^2\)"},ImageSize->200]

(Ket[i(V)])=(Ket[i])+(∑_(m≠i))(Ket[m])((Subscript[V, m i])/((Subscript[E, i])-(Subscript[E, m])))+…,

((|(BraKet[f, i(V)])|)^2)=(((|(Subscript[V, f i])|)^2)/((((Subscript[E, i])-(Subscript[E, f])))^2)),