The Translational States of Atoms

The energy eigenfunction of a hydrogen atom is given in Eq. (17.1 -17) as a product of the center-of-mass factor and the relative factor 

If an atom is confined in a rectangular box its center of mass cannot move completely up to the walls of the box because of the electrons in the atom. However, if the box is much larger than the size of an atom it will be an excellent approximation to apply the formulas that apply to a particle of zero size to the translation of an atom in a box. The translational energy eigenfunctions would be represented by the normalized version of Eq. (15.3-21)  

The electronic energy levels of any atom other than the hydrogen atom cannot be represented by any simple formula, and we will usually rely on experimental data for the energy eigenvalues. The electronic energy levels are very widely spaced compared with translational energy levels. 

The energy difference between the ground state and first excited electronic level of a hydrogen atom is 10.2 eV. Compare this energy difference with the spacing between the ground state and first excited translational level of a hydrogen atom in a cubical box 0.100 m on a side. 

The ground translational state corresponds to rix = riy = = 1, which we denote by (111). 

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