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Higher Order Perturbation Theory

(This is also discussed in Merzbacher [2] though with a slightly different notation.

We start from the time-dependent Schrödinger equation

$\displaystyle i\partial_t \vert\Psi(t)\rangle = H(t) \vert\Psi(t)\rangle.$     (3.16)

The state \bgroup\color{col1}$ \vert\Psi(t)\rangle$\egroup at time \bgroup\color{col1}$ t$\egroup is obtained from the state \bgroup\color{col1}$ \vert\Psi(t_0)\rangle$\egroup at time \bgroup\color{col1}$ t_0$\egroup by application of the time evolution operator \bgroup\color{col1}$ U(t,t_0)$\egroup via
$\displaystyle \vert\Psi(t)\rangle = U(t,t_0)\vert\Psi(t_0)\rangle.$     (3.17)

If \bgroup\color{col1}$ H(t)=H$\egroup is time-independent, we have
$\displaystyle U(t,t_0) = e^{-iH(t-t_0)},$   time-independent Hamiltonian$\displaystyle .$     (3.18)

For arbitrary \bgroup\color{col1}$ H(t)$\egroup, we have
$\displaystyle i\partial_t U(t,t_0) = H(t) U(t,t_0),\quad U(t_0,t_0)=1.$     (3.19)

We now assume a form
$\displaystyle H(t) = H_0 + V(t).$     (3.20)

We solve this differential equation by introducing the interaction picture time-evolution operator \bgroup\color{col1}$ \tilde{U}(t,t_0)$\egroup,
$\displaystyle \tilde{U}(t,t_0)$ $\displaystyle =$ $\displaystyle e^{i{H_0}t} U(t,t_0) e^{-i{H_0}t_0}$ (3.21)
$\displaystyle i\partial_t \tilde{U}(t,t_0)$ $\displaystyle =$ $\displaystyle - H_0 \tilde{U}(t,t_0) + i(-i) e^{i{H_0}t} H(t) U(t,t_0)e^{-i{H_0}t_0}$  
  $\displaystyle =$ $\displaystyle - H_0 \tilde{U}(t,t_0) + e^{i{H_0}t} [H_0 + V(t)] e^{-i{H_0}t}e^{i{H_0}t}U(t,t_0)e^{-i{H_0}t_0}$  
  $\displaystyle =$ $\displaystyle - H_0 \tilde{U}(t,t_0) + e^{i{H_0}t} [H_0 + V(t)] e^{-i{H_0}t} \tilde{U}(t,t_0)$  
  $\displaystyle =$ $\displaystyle \tilde{V}(t) \tilde{U}(t,t_0)$ (3.22)
$\displaystyle \tilde{V}(t)$ $\displaystyle =$ $\displaystyle e^{i{H_0}t} V(t) e^{-i{H_0}t}.$ (3.23)


next up previous contents index
Next: States Up: Time-Dependent Perturbation Theory Previous: Time-Independent Hamiltonian   Contents   Index
Tobias Brandes 2005-04-26