The Bethe Ansatz

Schrödinger, Heisenberg and Interaction pictures d.b.p

For a time-independent Hamiltonian \(H\), the Schrödinger equation for states \(|\psi^S (t) \rangle\) and its solution can be written

\begin{equation*} i \hbar \partial_t | \psi^S (t) \rangle = H | \psi^S (t) \rangle, \hspace{10mm} |\psi^S (t) \rangle = e^{-\frac{i}{\hbar} H t} | \psi^S (t=0) \rangle \end{equation*}

A time-dependent matrix element of some operator \({\cal O}^S\) thus reads

\begin{equation*} \langle \psi^S_1 (t) | {\cal O}^S | \psi^S_2 (t) \rangle \end{equation*}

where states are time-dependent, and operators are time-independent. This is the Schrödinger picture.

In the Heisenberg picture, the time dependence is shifted from the states to the operators:

\begin{equation*} \langle \psi^S_1 (t) | {\cal O} | \psi^S_2 (t) \rangle = \langle \psi^S_1 (t=0)| e^{\frac{i}{\hbar} Ht} {\cal O}^S e^{-\frac{i}{\hbar} H t} | \psi^S_2 (t=0) \rangle \equiv \langle \psi^H_1 | {\cal O}^H (t) |\psi^H_2 \rangle \end{equation*}

in which states are time-independent,

\begin{equation*} |\psi^H \rangle \equiv |\psi^S (t=0) \rangle \end{equation*}

and operators \({\cal O}^H (t) = e^{\frac{i}{\hbar} Ht} {\cal O}^S e^{-\frac{i}{\hbar} H t}\) obey the equation of motion

\begin{equation*} \frac{d}{dt} {\cal O}^H (t) = \frac{i}{\hbar} \left[ H, {\cal O}^H (t) \right] + e^{\frac{i}{\hbar} Ht} \partial_t {\cal O}^S e^{-\frac{i}{\hbar} H t} \equiv \frac{i}{\hbar} \left[ H, {\cal O}^H (t) \right] + \left[ \partial_t {\cal O}\right]^H. \end{equation*}

Let us now consider a generic, time-dependent Hamiltonian

\begin{equation*} H(t) = H_0 + V^S(t) \end{equation*}

in which \(H_0\) is the time-independent Hamiltonian of some exactly-solvable theory for which we know all the eigenstates, in other words for which we can provide a complete set of states \(|\alpha^0 \rangle\) such that

\begin{equation*} H_0 |\alpha^0\rangle = E_{\alpha^0} |\alpha^0\rangle. \end{equation*}

The operator \(V^S(t)\) (in the Schrödinger representation) then represents some perturbation/additional interaction which we would like to take into account. The idea of the interaction representation is to "Heisenbergize" using only \(H_0\), meaning that we define states and operators as (here from their Schrödinger representation)

\begin{equation*} |\psi^I (t) \rangle = e^{\frac{i}{\hbar} H_0 t} |\psi^S (t) \rangle, \hspace{10mm} {\cal O}^I (t) = e^{\frac{i}{\hbar} H_0 t} {\cal O}^S e^{-\frac{i}{\hbar} H_0 t}. \end{equation*}

The time evolution of states in the interaction representation can be simply obtained from the Schrödinger equation as

\begin{equation*} i\hbar \partial_t |\psi^I(t) \rangle = e^{\frac{i}{\hbar} H_0 t} \left[ -H_0 + H (t) \right] |\psi^S(t) \rangle = e^{\frac{i}{\hbar} H_0 t} V^S(t) |\psi^S (t) \rangle. \end{equation*}

This can be simply rewritten as

\begin{equation*} i\hbar \partial_t |\psi^I (t) \rangle = V^I (t) |\psi^I (t) \rangle. \label{eq:SEIR} \end{equation*}

Thus, in the interaction representation, the change of the phase of a wavefunction is driven solely by the interaction term, and the time evolution of an operator is driven solely by the exactly-solvable part of the Hamiltonian.

Formally, one can write a solution to the interaction picture Schrödinger equation as

\begin{equation*} |\psi^I(t) \rangle = U^I (t, t_0) | \psi^I (t_0) \rangle \end{equation*}

in terms of the propagator \(U^I\) in the interaction representation. If \(V^S (t)\) is in fact time-independent, we immediately have

\begin{equation*} U^I (t, t_0) = e^{\frac{i}{\hbar} H_0 t} e^{-\frac{i}{\hbar} H (t - t_0)} e^{-\frac{i}{\hbar} H_0 t_0}. \end{equation*}

For a generic time-dependent \(V^S (t)\), we have

\begin{equation*} i\hbar \partial_t U^I (t, t_0) |\psi^I(t_0) \rangle = V^I (t) U^I (t, t_0) |\psi^I (t_0) \rangle \end{equation*}

so the propagator satisfies the equation (with obvious boundary condition)

\begin{equation*} i\hbar \partial_t U^I (t, t_0) = V^I (t) U^I (t, t_0), \hspace{10mm} U^I (t_0, t_0) = 1. \end{equation*}

We can write an iterative solution to this. Integrating from \(t_0\) to \(t\) gives

\begin{equation*} U^I (t, t_0) = 1 + \frac{-i}{\hbar} \int_{t_0}^t dt' V^I (t') U^I (t', t_0) \end{equation*}

so we can develop the perturbative series

\begin{equation*} U^I (t, t_0) = 1 + \frac{-i}{\hbar} \int_{t_0}^t dt' V^I (t') + \left(\frac{-i}{\hbar} \right)^2 \int_{t_0}^t dt_1 V^I (t_1) \int_{t_0}^{t_1} dt_2 V^I (t_2) + ... \end{equation*}

This series can be represented as

\begin{align*} U^I (t, t_0) &= \sum_{n=0}^\infty \left(\frac{-i}{\hbar}\right)^n \int_{t_0}^t dt_1 \int_{t_0}^{t_1} dt_2 ... \int_{t_0}^{t_{n-1}} dt_n V^I (t_1) ... V^I(t_n) \nonumber \\ &= \sum_{n=0}^\infty \frac{(-i/\hbar)^n}{n!} \int_{t_0}^t dt_1 ... dt_n T_t \left[ V^I(t_1) ... V^I (t_n) \right] \end{align*}

in which we have introduced the time-ordering operator \(T_t\) acting as (here, for \({\cal O}\) operators which are bosonic in character)

\begin{equation*} T_t \left[ {\cal O}_1^I (t_1) {\cal O}_2^I (t_2) \right] \equiv \left\{ \begin{array}{ll} {\cal O}_1^I (t_1) {\cal O}_2^I (t_2), & t_1 > t_2 \\ \\ {\cal O}_2^I (t_2) {\cal O}_1^I (t_1), & t_1 < t_2 \end{array} \right. \end{equation*}

(with straightforward generalization to an arbitrary product of operators at different times).

The propagator in the interaction representation is thus compactly represented as

\begin{equation*} U^I (t, t_0) = T_t \left[ e^{-\frac{i}{\hbar} \int_{t_0}^t dt' V^I (t')} \right]. \end{equation*}

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Author: Jean-Sébastien Caux

Created: 2023-06-07 Wed 16:02