Quantum numbers zero-point energy

FIGURE 1.14 The arrangement known as a particle in a box, in which a particle of mass m is confined between two impenetrable walls a distance L apart. The first six wavefunctions and their energies are shown. The numbers on the left are values of the quantum number n. Note that there is a zero-point energy because n cannot be 0. 

Note from (9) that a molecule has some vibrational energy (a half quantum ) even if its vibrational quantum number, v, is zero. This residual vibrational energy (which presumably persists even at 0° K) is called zero-point energy. It is connected with the uncertainty in the relative positions of the atoms in low energy molecules as dictated by the Heisenberg principle (p. 7). 

These energy levels are numbered ty, the vibrational quantum number. At , = 0, the potential energy has its lowest value, which is not the energy of the potential minimum. The former exceeds the latter by Ml h- Pi. This is the so-called zero point energy. This energy cannot be removed from the molecule, even at temperatures approaching absolute zero. 

---