How do you derive, the “naive form” of the Lippmann-Schwinger Equation?

Question

I was looking for a derivation of the ‘Lippmann-Schwinger Equation’.

The wikipedia material at http://en.wikipedia.org/wiki/Lippmann%E2%80%93Schwinger_equation#Derivation does not (23rd Jan 2024), as I understand matters, actually contain a derivation of the equation! As part of the "derivation", it gives what it calls a “naive solution”, without actually showing how to get to it. Let’s call this naive solution, the ‘Naive Form’ of the Lippmann-Schwinger equation, this is shown below

\begin{equation*} |\psi\rangle= |\phi \rangle+\frac{ 1 }{ E-H_0 }V|\psi\rangle \end{equation*}

The only version of Sakurai’s$^1$ Modern Quantum Mechanics that I currently have, does not contain a derivation of this naive solution either.

My question is: How do you derive, the “naive form” of the Lippmann-Schwinger Equation?

Reference:

1, J.J.Sakurai, Ed San Fu Tuan, Modern Quantum Mechanics Revised Edition, Addison-Wesley Publishing Company (1994).

Answer 1

Sakurai$^1$ gives, Section 7.1, 'The Lippmann-Schwinger Equation'

\begin{equation*} H_0|\phi\rangle=E|\phi\rangle \tag{7.1.3} \end{equation*}

\begin{equation*} ( H_0+V)|\psi\rangle=E|\psi\rangle \tag{7.1.4} ~~~~~~~~~~~ \end{equation*}

Sakurai$^1$ say's, about the solution to ($\mathbf{7.1.4}$)

It may be argued that the desired solution is \begin{equation*} |\psi\rangle= |\phi \rangle+\frac{ 1 }{ E-H_0 }V|\psi\rangle \tag{7.1.5} \end{equation*}

but does not give any proof.

A proof.

Re-write ($\mathbf{7.1.3}$), and ($\mathbf{7.1.4}$) as

\begin{equation*} (E-H_0)|\phi \rangle =0~~~~~~ \tag{1} \end{equation*} \begin{equation*} (E-H_0)|\psi \rangle =V|\psi \rangle \tag{2} \end{equation*}

subtract ($\mathbf{1}$) from ($\mathbf{2}$), note that $(E-H_0)$ is a linear operator, giving \begin{equation*} (E-H_0) \left[ |\psi \rangle - |\phi \rangle \right] =V|\psi \rangle \end{equation*} Hence, \begin{align*} |\psi \rangle - |\phi \rangle&= \frac{ 1 }{ E-H_0 }V|\psi\rangle\\ |\psi \rangle &=|\phi \rangle+\frac{ 1 }{ E-H_0 }V|\psi\rangle \tag{3} \end{align*}

Other Information

Sakurai$^1$ discusses ($\mathbf{7.1.5}$) (which is the same as ($\mathbf{3}$) ), and say's that on it's own it has no meaning, but that this may be addressed by making $E$ slightly complex. He then gives the Lippmann-Schwinger equation in the form

\begin{equation*} |\psi^{(\pm )} \rangle =|\phi \rangle+\frac{ 1 }{ E-H_0\pm i\epsilon }V|\psi^{(\pm )}\rangle \tag{7.1.6} \end{equation*}

Reference:

1, J.J.Sakurai, Ed San Fu Tuan, Modern Quantum Mechanics Revised Edition, Addison-Wesley Publishing Company (1994).