This is a short note concerning the thermal and quantum fluctuations of a harmonic oscillator. This mathematical analysis may be useful for considering the fluctuations of bond lengths involved in molecular dynamics simulations for e.g. proteins, in which the associated bonding interactions are usually described in terms of (classical) force fields based on the harmonic potentials. We are mainly concerned with the degree of quantum effects and the validity of classical-mechanical approximations.

Let us consider the Schrödinger equation for a particle with the mass m and the coordinate x,

confined in the one-dimensional harmonic potential,

where k is the spring constant and  is the Planck constant. The eigenfunction  and the eigenvalue of energy,

are explicitly obtained [1], where the frequency  is introduced. The quantum-mechanical probability density at the position x is then given by

where  and refers to the Hermite polynomials [2].

We here consider the statistical average of the probability density over the canonical ensemble at temperature T in the thermal equilibrium. Recalling the population density of the eigenstate n in the canonical ensemble:

with  and  being the Boltzmann constant, we can with  and  being the Boltzmann constant, we can calculate the statistical probability density at the position x as

Employing the integral representation for the Hermite polynomials as [2,3]

we can express Eq. (6) as

Then, carrying out the summation over n with

we find

Finally, performing the Gaussian integrals over the variables s and t, we obtain

which shows a Gaussian distribution around x = 0. Though this expression itself is known in the literature [4,5], the present derivation is very simple and straightforward.

In the limit of zero temperature (T → 0), we see

which is the distribution represented by the ground state (n = 0). On the other hand, in the high-temperature (classical) limit (β → 0), we find

which is the Boltzmann distribition with the potential U(x).

It is interesting to compare the above result with the fully classical-mechanical derivation. Starting with the Newtonian equation of motion,

we find a solution,

with an initial condition of  and  where A represents the amplitude. Then, introducing the period  we can calculate the probability density at the position x as

where the amplitude is related to the total energy E of harmonic oscillator as  It is here remarked that the probability distribution diverges at  in the microcanonical distribution with a given energy E. Then, transforming from the microcanonical ensemble to the canonical ensemble with a given temperature T in the thermal equilibrium, we calculate the statistical probability density at the position  as

The final integration in Eq. (17) can be carried out through a change of variables as  thus leading to the Boltzmann distribution, Eq. (13).

By using the quantum-mechanical probability density obtained above, we can evaluate the fluctuation (variance) of particle position around the stable point x = 0 as

Due to  we find

in the zero temperature limit. Recalling  in the classical (high-temperature) limit, on the other hand, we find

It is noted that we see  because of  We also see  owing to  Thus, the evaluation of  in Eq. (18) is given as a combination of quantum and thermal contributions. Since we observe

the contributions from the thermal (classical) and quantum fluctuations are dominant in the high-temperature (or low-frequency) and low-temperature (or high-frequency) regions, respectively.

Acknowledgement

S.T. would like to acknowledge the Grants-in-Aid for Scientific Research (No. 21K06098) from the Ministry of Education, Cultute, Sports, Science and Technology (MEXT).

1. Schiff LI. Quantum Mechanics. McGraw-Hill, Singapore. 1968.
2. Gradshteyn IS, Ryzhik IM, Jeffrey A. Table of Integrals, Series, and Products. Academic Press, San Diego. 1994.
3. Tanaka S. Information geometrical characterization of the Onsager-Machlup process. Chem Phys Lett. 2017; 689:152-155.
4. Messiah A. Quantum Mechanics Vol. 1. North-Holland, Amsterdam. 1967.
5. Schönhammer K. Quantum versus thermal fluctuations in the harmonic chain and experimental implications. Am J Phys. 2014; 82: 887-895.