# 4.6: Selection Rules for the Particle-in-a-Box

- Page ID
- 4493

This section explores the use of symmetry to determine selection rules. Here we derive an analytical expression for the transition dipole moment integral for the particle-in-a-box model. The result that the magnitude of this integral increases as the length of the box increases explains why the absorption coefficients of the longer cyanine dye molecules are larger. We use the transition moment integral and the trigonometric forms of the particle-in-a-box wavefunctions to get Equation \(\ref{4-27}\) for an electron making a transition from orbital \(i\) to orbital \(f\).

\[ \begin{align} \mu _T &= \dfrac {-2e}{L} \int \limits _0^L \sin \left (\dfrac {f \pi x}{L} \right ) x \sin \left ( \dfrac {i \pi x }{L} \right ) dx \\[4pt] &= \dfrac {-2e}{L} \int \limits _0^L x \sin \left (\dfrac {f \pi x}{L} \right ) \sin \left ( \dfrac {i \pi x }{L} \right ) dx \label {4-27} \end{align}\]

Simplify the integral in Equation \(\ref{4-27}\) by substituting the product-to-sum trigonometric identity

\[\sin \psi \sin \theta = \dfrac {1}{2} \left[ \cos (\psi - \theta ) - \cos (\psi + \theta) \right] \label {4-28}\]

and also redefine the sum and difference terms:

\[ \Delta n = f - i \nonumber\]

and

\[n_T = f + i \nonumber \]

So Equation \(\ref{4-27}\)

\[ \begin{align} \mu _T &= \dfrac {-e}{L} \int \limits _0^L x \left[ \cos \left (\dfrac {\Delta n \pi x}{L} \right ) - \cos \left (\dfrac {n_T \pi x}{L} \right ) \right ] dx \\[4pt] &= \dfrac {-e}{L} \left[ \int \limits _0^L x \cos \left (\dfrac {\Delta n \pi x}{L} \right ) dx - \int \limits _0^L x \cos \left (\dfrac { n_T \pi x}{L} \right ) dx \right ] \label{step 3} \end{align}\]

These two definite integrals can be directly evaluated using this relationship

\[ \int \limits _0^L x \cos (ax) dx = \left[ \dfrac {1}{a^2} \cos (ax) + \dfrac {x}{a} \sin (ax) \right]^L_0 \label {4-29}\]

where \(a\) is any nonzero constant. Using Equation \ref{4-29} in Equation \ref{step 3} produces

\[T = \dfrac {-e}{L} {\left(\dfrac {L}{\pi}\right)}^2 \left[ \dfrac {1}{\Delta n^2} (\cos (\Delta n \pi) - 1) - \dfrac {1}{n^2_T} (\cos (n_T \pi) - 1) + \dfrac {1}{\Delta n} \sin (\Delta n \pi ) - \dfrac {1}{n_T} \sin (n_T \pi) \right] \label {4-31}\]

From Examples \(\PageIndex{1}\) and \(\PageIndex{2}\), we can formulate the selection rules for the particle-in-a-box model: Transitions are forbidden if \(Δn = f - i\) is an even integer. Transitions are allowed if \(Δn = f - i \) is an odd integer. In the next section we will see that these selection rules can be understood in terms of the symmetry of the wavefunctions.

Through the evaluation of the transition moment integral, we can understand why the spectra of cyanine dyes are very simple. The spectrum for each dye consists only of a single peak because other transitions have very much smaller transition dipole moments. We also see that the longer molecules have the larger absorption coefficients because the transition dipole moment increases with the length of the molecule.

## Contributors

David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski ("Quantum States of Atoms and Molecules")