By Kaoru Yamanouchi

At a degree available to complicated undergraduates, this textbook explains the elemental position of quantum mechanics in settling on the constitution, dynamics, and different houses of molecules. Readers will come to appreciate the quantum-mechanical foundation for harmonic oscillators, angular momenta and scattering procedures. workouts are supplied to assist readers deepen their clutch of the basic phenomena.

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**Sample text**

73) From Eqs. 73), εn = εn∗ . This indicates that εn is a real number. By the way, the probability with which the system is detected at some point in the whole space should be 1. 3 The Harmonic Oscillator and Its Applications 43 which corresponds to taking the complex conjugate of ψm , multiplying it with ψn , and performing the integration. Putting this in a short form, we describe them as ψm |ψn , or m|n when it is sufficient to specify only the m and n. Namely, m|n = ψm |ψn = ∗ ψm ψn dτ. 75) Additionally, when the operation of an operator H on ψn results in ψn , that is, when ψn = H ψn , the scalar product of ψm and ψn is given by ψm |ψn = ψm |H ψn , which is rewritten as ψm |H |ψn .

4) = μ|v A − v B |2 + k(rAB − re )2 . 2 2 The H here is called the Hamiltonian of the molecular vibration. Molecular vibration is described as a motion on the straight line that connects the atoms A and B, so it is convenient if we align this line with the x axis. When we consider a system where the value of B’s x coordinate xB is larger than the value of A’s x coordinate xA (xB > xA ), the internuclear distance xAB is xAB = xB − xA , and the x coordinate is represented as the deviation from the equilibrium internuclear distance, which is given by x = xAB − re .

When the system is in the n-th state, the diagonal element hnn ˆ of the matrix is the expectation value of the operator h. In this section, we will calculate matrix elements using eigenfunctions of harmonic oscillators, and apply it to determining expectation values of energy and selection rules in vibrational spectra. 1 Hermitian Operators and the Bracket Notation All operators representing physical quantities such as positions, momenta, and energies are classified as Hermitian operators. 66) holds with arbitrary wave functions ψm and ψn .