One of the most important topics in time-dependent quantum mechanics is the description of spectroscopy, which refers to the study of matter through its interaction with electromagnetic radiation. Classically, light–matter interactions are a result of an oscillating electromagnetic field resonantly interacting with charged particles in the matter, most often bound electrons. We observe these processes either through changes to the light induced by the matter, such as absorption or emission of new light fields, or by light-induced changes to the matter, such as ionization and photochemistry. By studying such processes as a function of the control variables for the light field (amplitude, frequency, polarization, phase, etc.), we can deduce properties of the samples. 7.1: Introduction to Light-Matter Interactions Different types of spectroscopy give you different perspectives. This indirect contact with the microscopic targets means that the interpretation of spectroscopy requires a model, whether it is stated or not. Modeling and laboratory practice of spectroscopy are dependent on one another, and spectroscopy is only as useful as its ability to distinguish different models. This makes an accurate description of the underlying physical process governing the interaction of light and matter important. 7.2: Classical Light–Matter Interactions As a starting point, it is helpful to first summarize the classical description of electromagnetic fields. 7.3: Quantum Mechanical Electric Dipole Hamiltonian Now we are in a position to substitute the quantum mechanical momentum for the classical. 7.4: Relaxation and Line-Broadening Let’s describe absorption to a state that is coupled to a continuum. What happens to the probability of absorption if the excited state decays exponentially? 7.5: Absorption Cross-Sections The rate of absorption induced by a monochromatic electromagnetic field can be extended to obtain the magnitude of the transition dipole matrix element from absorption spectra . 7.6: Appendix - Review of Free Electromagnetic Field Here we review the derivation of the vector potential for the plane wave in free space. 7.7: Appendix - Magnetic Dipole and Electric Quadrupole Transitions