The mean internal energy of a one-dimensional harmonic oscillator in equilibrium with a heat bath of temperature T is in the units of kBT.Correct answer is '1'. Can you explain this answer?

Question

Answer

The mean internal energy of a one-dimensional harmonic oscillator in equilibrium with a heat bath of temperature T is indeed equal to 1 in the units of kBT. Let's understand this concept in detail:

Harmonic Oscillator:
A harmonic oscillator is a system that exhibits simple harmonic motion, which is characterized by a restoring force that is directly proportional to the displacement from the equilibrium position. In the case of a one-dimensional harmonic oscillator, the potential energy is given by the equation U(x) = 1/2 kx^2, where k is the spring constant and x is the displacement from the equilibrium position.

Internal Energy:
The internal energy of a system refers to the total energy associated with the microscopic motion and interactions of particles within the system. In the case of a harmonic oscillator, the internal energy is directly related to the potential energy of the system.

Equilibrium with a Heat Bath:
When a harmonic oscillator is in equilibrium with a heat bath, it means that the system is in thermal equilibrium with its surroundings and exchanging energy in the form of heat. The heat bath provides a constant temperature T to the harmonic oscillator.

Mean Internal Energy:
The mean internal energy, denoted by , represents the average value of the internal energy over a large number of oscillations. It is given by the formula = (1/2) kBT, where kB is the Boltzmann constant.

Explanation:
The mean internal energy of the harmonic oscillator in equilibrium with a heat bath of temperature T is 1 in the units of kBT. This means that on average, the internal energy of the system is equal to kBT/2.

The factor of 1/2 arises from the equipartition theorem, which states that each degree of freedom of a system in thermal equilibrium contributes an average energy of (1/2) kBT. In the case of a one-dimensional harmonic oscillator, there is only one degree of freedom associated with its motion along the x-axis.

Since the potential energy of a one-dimensional harmonic oscillator is given by U(x) = 1/2 kx^2, the average potential energy is (1/2) k, where represents the mean square displacement. This mean square displacement is directly related to the temperature T and is equal to kB T / k.

Substituting this value into the average potential energy equation, we get (1/2) k = (1/2) kBT.

Thus, the mean internal energy of a one-dimensional harmonic oscillator in equilibrium with a heat bath of temperature T is 1 in the units of kBT.

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