Question

Physical Chemistry..

Explain what is the difference in terms of energy between the ground states of rigid rotor and of harmonic oscillators?

Answer 1

It's very simple.

Ground state energy of a rigid rotor is zero, while that of a harmonic ossilator is not zero but equal to $h^2 \omega / 8\pi ^2$ , which is called it's ZERO-POINT ENERGY.

Explanation:

Energy of a rigid rotor is expressed as

Bell+1) with B 杞 2I

B is the rotational constant; 'l' is the rotational quantum number;

so, at the ground state, i.e., when l=0, E_l=0 too.

but , enegy of a harmonic ossilator is expressed as

win

where n is the quantum number. So here we see, even at the ground state, i.e., when n =0, E_n is not zero but $h^2 \omega / 8\pi ^2$ .

This '$h^2 \omega / 8\pi ^2$' is called the zero-point energy of the harmonic ossilator. Because of the zero-point energy, the position and momentum of the oscillator in the ground state are not fixed (as they would be in a classical oscillator), but have a small range of variance, in accordance with the Heisenberg uncertainty principle. This zero-point energy further has important implications in quantum field theory and quantum gravity.

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