The quantum-mechanical solutions show that the amplitude of the oscillating particle goes from −∞ to +∞. However, the probability of finding a particle at the large distances from the equilibrium is extremely small.
Therefore, one could use an approximate form of a harmonic oscillator wavefunction shown below:
𝜓(𝑥)=(𝐿 −𝑥 )
where the amplitude of an oscillation, x, is limited to values smaller than L (from -L to +L), and L is a variational parameter.
Use the variational method and the proposed form of a trial function to find the following:
(a) the best L value
(b) the best energy for the lowest vibrational state (zero point vibration)
(a) The best L value is
(L2 - x2)2 = 0
L2 - x2 = 0
L2 = x2
(b) Each electronic energy level is itself composed of smaller vibrational energy level,
so, the best energy level for the lowest vibrational state (zero point vibration) is zero.