# 18: Quantum Mechanics and Molecular Energy Levels

• 18.1: Energy Distributions and Energy Levels
The probability that the energy of a particular molecule is in a particular interval is intimately related to the energies that it is possible for a molecule to have. Before we can make further progress in describing molecular energy distributions, we must discuss atomic and molecular energies. For our development of the Boltzmann equation, we need to introduce the idea of quantized energy states.
• 18.2: Quantized Energy - De Broglie's Hypothesis and the Schroedinger Equation
Subsequent to Planck’s proposal that energy is quantized, the introduction of two further concepts led to the theory of quantum mechanics. The first was Einstein’s relativity theory, and his deduction from it of the equivalence of matter and energy. The second was de Broglie’s hypothesis that any particle of mass m moving at velocity v , behaves like a wave. De Broglie’s hypothesis is an independent postulate about the structure of nature.
• 18.3: The Schrödinger Equation for A Particle in A Box
The particle in a box provides a convenient illustration of the principles involved in setting up and solving the Schrödinger equation. Besides being a good illustration, the problem also proves to be a useful approximation to many physical systems. The statement of the problem is simple. We have a particle of mass m that is constrained to move only in one dimension. For locations in the box, the particle has zero potential energy. Outside the box, the particle has infinite potential energy.
• 18.4: The Schrödinger Equation for a Molecule
• 18.5: Solutions to Schroedinger Equations for Harmonic Oscillators and Rigid Rotors
• 18.6: Wave Functions, Quantum States, Energy Levels, and Degeneracies
We approximate the wave function for a molecule by using a product of approximate wave functions, each of which models some subset of the motions that the molecule undergoes. In general, the wave functions that satisfy the molecule’s Schrödinger equation are degenerate; that is, two or more of these wave functions have the same energy.
• 18.7: Particle Spins and Statistics- Bose-Einstein and Fermi-Dirac Statistics
The spin of a particle is an important quantum mechanical property. It turns out that quantum mechanical solutions depend on the spin of the particle being described. Particles with integral spins behave differently from particles with half-integral spins. When we treat the statistical distribution of these particles, we need to treat particles with integral spins differently from particles with half-integral spins.