18.5: Solutions to Schroedinger Equations for Harmonic Oscillators and Rigid Rotors

We can approximate the wavefunction for a molecule by partitioning it into wavefunctions for individual translational, rotational, vibrational, and electronic modes. The wavefunctions for each of these modes can be approximated by solutions to a Schrödinger equation that approximates that mode. Our objective in this chapter is to introduce the quantized energy levels that are found.

Translational modes are approximated by the particle in a box model that we discuss above.

Vibrational modes are approximated by the solutions of the Schrödinger equation for coupled harmonic oscillators. The vibrational motion of a diatomic molecule is approximated by the solutions of the Schrödinger equation for the vibration of two masses linked by a spring. Let the distance between the masses be \(r\) and the equilibrium distance be \(r_0\). Let the reduced mass of the molecule be \(\mu\), and let the force constant for the spring be \(\lambda\). From classical mechanics, the potential energy of the system is

\[V\left(r\right)=\frac{\lambda {\left(r-r_0\right)}^2}{2} \nonumber \]

and the vibrational frequency of the classical oscillator is \[\nu =\frac{1}{2\pi }\sqrt{\frac{\lambda }{\mu }} \nonumber \]

The Schrödinger equation is

\[-\left(\frac{h^2}{8{\pi }^2\mu }\right)\frac{d^2\psi }{dr^2}+\frac{\lambda {\left(r-r_0\right)}^2}{2}\psi =E\psi \nonumber \]

The solutions to this equation are wavefunctions and energy levels that constitute the quantum mechanical description of the classical harmonic oscillator. The energy levels are given by

\[E_n=h\nu \left(n+\frac{1}{2}\right) \nonumber \]

where the quantum numbers, \(n\), can have any of the values \(n=0,\ 1,\ 2,\ 3,\ \dots .\) The lowest energy level, that for which \(n=0\), has a non-zero energy; that is,

\[E_0={h\nu }/{2} \nonumber \]

The quantum mechanical oscillator can have infinitely many energies, each of which is a half-integral multiple of the classical frequency, \(\nu\). Each quantum mechanical energy corresponds to a quantum mechanical frequency:

\[{\nu }_n=\nu \left(n+\frac{1}{2}\right) \nonumber \]

A classical rigid rotor consists of two masses that are connected by a weightless rigid rod. The rigid rotor is a dumbbell. The masses rotate about their center of mass. Each two-dimensional rotational motion of a diatomic molecule is approximated by the solutions of the Schrödinger equation for the motion of a rigid rotor in a plane. The simplest model assumes that the potential term is zero for all angles of rotation. Letting \(I\) be the molecule’s moment of inertia and \(\varphi\) be the rotation angle, the Schrödinger equation is

\[-\left( \frac{h^2}{8\pi ^2I}\right) \frac{d^2\psi }{d \varphi ^2}=E\psi \nonumber \]

The energy levels are given by

\[E_m=\frac{m^2h^2}{8\pi ^2I} \nonumber \]

where the quantum numbers, \(m\), can have any of the values \(m=1,\ 2,\ 3,\ \dots .,\)(but not zero). Each of these energy levels is two-fold degenerate. That is, two quantum mechanical states of the molecule have the energy \(E_m\).

The three-dimensional rotational motion of a diatomic molecule is approximated by the solutions of the Schrödinger equation for the motion of a rigid rotor in three dimensions. Again, the simplest model assumes that the potential term is zero for all angles of rotation. Letting \(\theta\) and \(\varphi\) be the two rotation angles required to describe the orientation in three dimensions, the Schrödinger equation is

\[-\frac{h^2}{8{\pi }^2I}\left(\frac{1}{\mathrm{sin} \theta} \frac{\partial }{\partial \theta } \left(\mathrm{sin} \theta \frac{\partial \psi }{\partial \theta }\right)+\frac{1}{\mathrm{sin}^2 \theta }\frac{d^2\psi }{d{\varphi }^2}\right)=E\psi \nonumber \]

The energy levels are given by

\[E_J=\frac{h^2}{8{\pi }^2I}J\left(J+1\right) \nonumber \]

where the quantum numbers, \(J\), can have any of the values \(J=0,\ 1,\ 2\ ,3,\ \dots .\) \(E_J\) is \(\left(2J+1\right)\)-fold degenerate. That is, there are \(2J+1\) quantum mechanical states of the molecule all of which have the same energy, \(E_J\).

Equations for the rotational energy levels of larger molecules are more complex.