With both light and electrons, one was faced with the so-called "wave-particle duality"; both could be regarded as waves for some purposes and as par… - Rudolf Peierls

Born approximation is a familiar and convenient approximation in handling scattering problems. It is adequate, or at least informative, in so many cases that we tend to develop the habit of using its first-order term without always checking the conditions for its applicability.

The atoms which constitute a solid consist of nuclei and electrons. For a description of the state of the solid it is not, however, necessary to specify the state of all the Z electrons of each atom, since we can eliminate most or all of them by a principle that is familiar from the theory of molecules. ... Since the atomic nuclei are much heavier than the electrons, they move much more slowly, and it is therefore reasonable to start from the approximation in which they are taken to be taken to be at rest, though not necessarily in the regular positions.

Any theoretical physicist has met, in his introduction to the subject, the simplest examples of Schrödinger's equation, including the harmonic oscillator. In demonstrating its solution, it is usually shown that for energies satisfying the usual quantum condition, E = (n + ½)ħω (1.1.1)
where n is a non-negative integer and ω the frequency, the equation has a solution satisfying the correct boundary conditions. It is equally important to know that these are the only solutions, i.e., that for an energy not equal to (1.1.1) no admissible solution exists. This negative statement is not usually proved in elementary treatments, or else it is deduced from quite elaborate discussions of the convergence and behavior of a certain infinite series. It is therefore surprising to find that the result can be seen without any complicated algebra.