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Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy, and the uncertainty principle and is generally used in single-body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.
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Lippmann-Schwinger solution
What's wrong with this general solution of the Lippmann-Schwinger equation:
$$
|\psi_k \rangle=|\phi_k \rangle+G_k V|\psi_k \rangle
$$
Taking the inner product with $\langle\phi_{k'}|$
\begin{align}
…