Transition amplitudes

Solving the TDSE

Consider a Hamiltonian with the following structure

\[H = H_0 + \eps V(t)\]

where we know the eigenvectors \(\ket{n}\) of \(H_0\) and the associated eigenvalues \(E_n\). At any time \(t\) we can expand our state in the basis \(\ket{n}\) which we take to be orthonormal (as such a basis of eigenvectors of the Hermitian operator \(H_0\) always exists).:

\[\ket{\psi(t)} = \sum_n c_n(t) \ket{n}\]

We can then write the time-dependent Schroedinger equation

\[i\hbar \del_t \ket{\psi(t)} = (H_0 + \eps V)\ket{\psi(t)}\]

as

\[\sum_n i\hbar{\dot c}_n \ket{n} = \sum_n c_n(t)(E_n + \eps V)\ket{n}\]

taking the inner product with a specific eigenket \(\bra{n}\) we have

\[i\hbar {\dot c}_n = E_n c_n + \eps \sum_m \bra{n} V(t) \ket{m} c_m(t)\]

which is a coupled set of first order differential equations.

The interaction picture

Now the first time on the right hand side is a nuisance and at any rate does not correspond to a change in teh amplitude of \(c_n\). To see this, we redefine

\[c_n(t) = {\tilde c}_n(t) e^{- i E_n t/\hbar}\]

Making this substitution, we find that the time derivative acting on \(c_n\) includes a term that cancels the \(E_n c_n\) term. The upshot is:

(335)\[\begin{split}\begin{align} i\hbar {\dot{\tilde c}}_n & = \sum_m \bra{n} \eps V\ket{m} e^{i (E_n - E_m) t/\hbar} {\tilde c}_m(t)\\ & = \sum_m \bra{n} e^{i H_0 t/\hbar} \eps V e^{-i H_0 t/\hbar} \ket{m} {\tilde c}_m\\ & \equiv \sum_m \bra{n} \eps V_I(t) \ket{m} {\tilde c}_m(t) \end{align}\end{split}\]

where we define interaction picture operators as

\[\cO_I(t) = e^{i H_0 t/\hbar} \cO_S(t) e^{-i H_0 t/\hbar}\]

Here \(\cO_S(t)\) is a Schrodinger picture operator, which we have allowed to have explicit time-dependence (for example, if we want to subject a charged particle to a time-dependent electromagnetic field). We can also define the state in the interaction picture as

\[\ket{\psi(t)}_I = e^{i H_0 t/\hbar} \ket{\psi(t)} = \sum_n {\tilde c}(t)\ket{n}\]

This is like the Heisenberg picture, but instead of evolving the actual state from \(t\) back to \(t = 0\) with the full Hamiltonian, we do so with the unperturbed Hamiltonian \(H_0\). The difference between these two protocols is a measure of the degree of interaction. Indeed, we can see from equation (335) that the Schroedinger equation can be written as

\[i\hbar \del_t \ket{\psi(t)}_I = \eps V_I(t) \ket{\psi(t)}_I\]

Since we are interested in transitions (changes of the state) induced by interactions, this formulation isolates the questions at hand.