Scattering States, Wave Packets, Transition Amplitudes, and Tunneling in the Energy Domain: A Time-Free Formulation of Quantum Processes

This paper develops an energy-domain formulation of time-independent non-relativistic quantum processes, including scattering, wave packets, transition amplitudes, and tunneling. A fixed-energy spectral component is written as ${\Psi }_E\left(x,z\right)={\psi }_E\left(x\right){\chi }_z\left(E\right)$ , while the complete state $\Psi \left(x,z\right)$ is obtained by summing or integrating over the spectral components. The energy derivative is defined on the spectral phase ${\chi }_z\left(E\right)$ , with the energy-dependent spatial eigenfunction ${\psi }_E\left(x\right)$ retained as an element of the spectral basis. For a specified reference energy $E_{\text{ref}}$ , the spectral phase is ${\chi }_z\left(E\right)=\text{exp}\left[？\text{i}{\left(E？E_{\text{ref}}\right)}^2\left(z？z_0\right)/？\right]$ . Fixed-energy reflection and transmission amplitudes, tunneling probabilities, probability currents, transition matrix elements, and selection rules retain their standard