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Molecular Vibrations

Friday, March 18, 2022

Vibrations in Quantum Mechanics

In a classical system, a vibrating system as an object of mass mm on a spring of force constant kk has an oscillation frequency ω=k/m\omega=\sqrt{k/m}. It has an associated energy U=12kx2U=\frac{1}{2}kx^2, and a maximum amplitude of xmx_m, meaning it has a (constant) total energy of E=12kxm2E=\frac{1}{2}kx_m^2.

Quantum oscillators have only certain allowed energies:

EN=(N+12)ω=(N+12)hf    N=0,1,2,3,...E_N=\left(N+\frac{1}{2}\right)\hbar\omega=\left(N+\frac{1}{2}\right)hf~~~~N=0,1,2,3,...

The ground-state energy (N=0N=0) is therefore 12ω=12hf\frac{1}{2}\hbar\omega=\frac{1}{2}hf, and each energy level is spaced by ω=hf\hbar\omega=hf. Note: there is no energy state where E=0E=0 (since this would have a displacement of exactly zero and momentum of exactly zero, violating the uncertainty principle). Instead, the minimum energy 12hf\frac{1}{2}hf is often called the zero-point energy.

Allowed transitions

The oscillator can absorb or emit radiation to jump to different states. However, these states must differ in NN by exactly one unit. This is called a selection rule.

Vibrational selection rule:    ΔN=±1\text{Vibrational selection rule:}~~~~\Delta N=\pm 1

Vibrating Diatomic Molecules

Imagine a diatomic molecule oscillating.

Diatomic molecule vibrating

The two atoms have masses m1m_1 and m2m_2 and effective force constant kk. At the equilibrium, the only energy is kinetic energy so


By using a frame of reference where the center of mass of the molecule is constant, the total momentum must be zero. Therefore p1=p2=pp_1=p_2=p, meaning




This represents the system as if it were a single mass mm moving with momentum pp. The mm can be thought of as an effective mass, known as the reduced mass, and it is used when dealing with the vibration of the molecule.

Note: when the masses are the same, m=m1/2m=m_1/2, which makes sense because the atoms both move equally. In a situation where one atom is much greater mass than the other, the reduced mass is about equal to the smaller mass (since the larger atom has significant inertia).