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Perturbations on the fine structure energy levels of the hydrogen atom; electronic configurations and spectral terms; fine structure; Stark and Zeeman effects; intercombination lines.
5.1 In the space H2 of the polarization states of photons, consider the basis |e1适 | e2适 consisting of the vectors that represent two rectilinear states of polarization along two orthogonal directions. Determine the matrix that, in the above basis, represents the following observables (see Problem 4.3).
1.1 According to the model proposed by J.J. Thomson at the beginning of the 20th century, the atom consists of a positive charge Ze (Z is the atomic number) uniformly distributed inside a sphere of radius R, within which Z pointlike electrons can move.
Note. The states of photon rectilinear polarization denoted by ê1,2 in the previous chapter will from now on be denoted by |e1,2 适 in the sequel we shall use the notation | e1,2适 for vectors in the complex Hilbert space, whereas ê1,2 will stand for vectors in the ‘physical’ real configuration space.
Coherent and incoherent radiation; photoelectric effect; transitions in dipole approximation; angular distribution and polarization of the emitted radiation; lifetimes.
Rotational energy levels of polyatomic molecules; entangled states and density matrices; singlet and triplet states; composition of angular momenta; quantum fluctuations; EPR paradox; quantum teleportation.
Perturbations in one-dimensional systems; Bender–Wu method for the anharmonic oscillator; Feynman–Hellmann and virial theorems; “nocrossing theorem”; external and internal perturbations in hydrogen-like ions.
Orbital angular momentum: states with l = 1 and representations; rotation operators; spherical harmonics; tensors and states with definite angular momentum ( l = 1, l = 2).
Note. The problems in this chapter are based on what is known as Old Quantum Theory: Bohr and de Broglie quantization rules. Those situations are treated in which the results will substantially be confirmed by quantum mechanics and some problems of statistical mechanics are proposed where the effects of quantization are emphasized.
Note. The exercises in this chapter regard the fundamental concepts at the basis of quantum mechanics: Problem 3.3 exposes all the “interpretative drama” of quantum mechanics, which is why its somewhat paradoxical aspects are discussed in detail in the solution.