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

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

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

The Schrödinger equation is

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

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

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

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,

$E_0={h\nu }/{2}$

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:

${\nu }_n=\nu \left(n+\frac{1}{2}\right)$

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

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

The energy levels are given by

$E_m=\frac{m^2h^2}{8\pi ^2I}$

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

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

The energy levels are given by

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

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.

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