Summary of perturbation theory equations

Feb 2, 2002

Summary of perturbation theory equations Summary of perturbation theory equations

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Time independent perturbation expansion

Suppose we have a reference Hamiltonian H₀ for which we know all of the eigenvalues and eigenfunctions:

H₀ F⁰_n = E⁰_n F⁰_n.

(1)

Now we want to approximate the eigenvalues E_n and eigenfunctions F_n of total Hamiltonian H º H₀ + H₁, where the second term is small compared to the reference Hamiltonian term. If the n^th zero-order eigenstate (E⁰_n) is not degenerate, then we can make the following expansion. We will use the shorthand notation áF⁰_k|H₁|F⁰_mñ º V_km.

E_n » E⁰_n + V_nn +

å
m ¹ n

|V_mn|²

E⁰_n-E⁰_m

+ O(V³).

(2)

F_n » F⁰_n +

å
m ¹ n

F⁰_m

V_mn

E⁰_n-E⁰_m

+ O(V²).

(3)

If, on the other hand, the zero-order eigenstate (E⁰_n) is degenerate with one or more other eigenstates, another method must be used. Suppose there are N such degenerate states which we will label {F⁰_{n_i}}, where i = 1,2,¼N. We suppose that we can find N new zero-order states {F^0a} from linear combinations of the original states, by diagonalizing the following N ×N matrix:

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E⁰_n₁+ V_{n₁ n₁}

V_{n₁ n₂}

V_{n₁ n₃}

V_{n₁ n_N}

V_{n₂ n₁}

E⁰_n₂+ V_{n₂ n₂}

V_{n₂ n₃}

V_{n₃ n_N}

V_{n_N n₁}

V_{n_N n₃}

E⁰_{n_N}+ V_{n_N n_N}

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C^a_n₁

C^a_n₂

C^a_{n_N}

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= E^a

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C^a_n₁

C^a_n₂

C^a_{n_N}

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(4)

The energy eigenvalues {E^a} correspond to corrections up to first order in the perturbation for this system. Each eigenvalue E^a corresponds to a linear combination of the zero order eigenfunctions in terms of the coefficients {C^a_{n_i}}:

F^0a =

N
å
i = 1

C^a_{n_i} F⁰_{n_i}.

(5)

Time dependent perturbation expansion

Now suppose that the perturbation depends on time. We will focus on the case in which there is a harmonic time dependence which is ``turned on'' at time t = 0:

H₁(t) = V(r) ( e^{i wt} + e^{-i wt} ) Q(t),

(6)

where Q(t) denotes the Heaviside step function. If the system is initially (t < 0) in the zero order state F⁰_n, the effects of the perturbation to first order in V is given by

F_n(r,t) » F⁰_n(r)e^{-i E⁰_n t/(^h/_2p)} +

å
m

c⁽¹⁾_m(t)F⁰_m(r)e^{-i E⁰_m t/(^h/_2p)},

(7)

where

c⁽¹⁾_m(t) = -

V_mn

(^h/_2p)

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e^{i(w_mn+ w) t} - 1

w_mn+ w

e^{i(w_mn- w) t} - 1

w_mn- w

ù
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û

(8)

In this expression, w_mn º [(E⁰_m - E⁰_n)/( (^h/_2p) )]. For large times t, it can be shown that the squared modulus of the exitation coefficient c⁽¹⁾_m(t) determines the transition rate:

R_{n® m} =

|c⁽1)_m(t)|²

2 p

(^h/_2p)²

|V_mn|² ( d(w_mn+ w) + d(w_mn- w)),

(9)

R_{n® m} »

2 p

(^h/_2p)

|V_mn|² ( d(E⁰_m - E⁰_n + (^h/_2p)w) + d(E⁰_m-E⁰_n- (^h/_2p)w)),

(10)

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On 2 Feb 2002, 16:22.