Question #198134

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)

Expert's answer

(a) The best L value is

(L2 - x2)2 = 0

L2 - x2 = 0

L2 = x2

**L= x**

(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.

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