Potential/Kinetic Energy of Particles in Harmonic Oscillator

In summary, the average potential and kinetic energies in a harmonic oscillator are 2/3 and 4/3 of the ground state energy, respectively.
  • #1
messier992
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Homework Statement


I'm trying to reconcile the answers to two questions regarding the average potential and kinetic energies in simple harmonic oscillator

Question 1:
The average potential energy of the vibrational motion in the ground state of a
diatomic molecule is 12 meV. The average kinetic energy of the vibrational motion
in that state is:
Answer: 12meV?

Question 2:
The ground state energy of a particle in a harmonic oscillator potential is E. The average kinetic energy of that particle in that state is

Homework Equations


Potential Energy = 1/2*k*x^2
Total Energy = Kinetic + Potential Energy

The Attempt at a Solution


To find the ratio of potential to kinetic energy:
the total potential energy is given by the integral of x^2 from -1 to 1 => 2/3
the total kinetic energy is given by the integral of 1-x^2 from -1 to 1 => 4/3

Therefore, the average kinetic energy should be twice the average potential energy
 
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  • #2
messier992 said:

The Attempt at a Solution


To find the ratio of potential to kinetic energy:
the total potential energy is given by the integral of x^2 from -1 to 1 => 2/3
the total kinetic energy is given by the integral of 1-x^2 from -1 to 1 => 4/3

Therefore, the average kinetic energy should be twice the average potential energy

Where are you getting this from?
 
  • #3
messier992 said:
the total potential energy is given by the integral of x^2 from -1 to 1 => 2/3
the total kinetic energy is given by the integral of 1-x^2 from -1 to 1 => 4/3
If you are integrating for displacements ranging from -1 to +1 and expecting to compute an average over time you should first have reason to believe that the oscillator spends an equal amount of time near each possible displacement.

But it is clear that this is not the case. An oscillator spends more time (proportionately) at its extremes and less near the center. You need to integrate over something that changes smoothly with time. A phase angle, perhaps?
 
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Related to Potential/Kinetic Energy of Particles in Harmonic Oscillator

1. What is a Harmonic Oscillator?

A Harmonic Oscillator is a system that experiences a restoring force proportional to its displacement from its equilibrium position. This means that when the system is displaced from its equilibrium position, it will experience a force that will bring it back to its original position.

2. What is Potential Energy in a Harmonic Oscillator?

Potential Energy in a Harmonic Oscillator is the energy that is stored in the system due to its position relative to its equilibrium position. It is directly proportional to the displacement of the system from its equilibrium position and is at its maximum when the displacement is at its maximum.

3. What is Kinetic Energy in a Harmonic Oscillator?

Kinetic Energy in a Harmonic Oscillator is the energy that is possessed by the particles in the system due to their motion. It is directly proportional to the velocity of the particles and is at its maximum when the particles are at their maximum displacement from the equilibrium position.

4. How are Potential and Kinetic Energy related in a Harmonic Oscillator?

In a Harmonic Oscillator, potential and kinetic energy are interchangeable. As the particles move towards the equilibrium position, their kinetic energy decreases and their potential energy increases. When the particles reach the equilibrium position, their kinetic energy is at its minimum and their potential energy is at its maximum. As the particles move away from the equilibrium position, the opposite occurs, with kinetic energy increasing and potential energy decreasing.

5. How is the Potential/Kinetic Energy of particles in a Harmonic Oscillator calculated?

The potential and kinetic energy of particles in a Harmonic Oscillator can be calculated using the equations:
Potential Energy = 1/2 * spring constant * (displacement)^2
Kinetic Energy = 1/2 * mass * (velocity)^2
These equations take into account the properties of the system, such as the stiffness of the spring and the mass of the particles, as well as their displacement and velocity from the equilibrium position.

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