Chapter 5

# Minimum energy of an harmonic oscillator

Take 10 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

A harmonic oscillator in one dimension has a mass and a proper pulsation .

The potential energy is expressed by : The average position and the average quantity of movement of the oscillator are equal to zero.

Let be : The maximum indetermination on the position of the oscillator.

## Question

Prove that the minimum energy of the oscillator can be written as : ### Hint

• Use the uncertainty principle to determine .

• Derivate the total energy in function of .

### Solution

The energy of the harmonic oscillator is : The uncertainty principle gives : Hence, the minimum energy for the considered value of is : In order to determine the minimum of this energy, let us compute its derivative : It is equal to zero when : Which corresponds to the minimum energy : Heisenberg's inequalities determine the minimum energy of a quantum harmonic oscillator.

The harmonic oscillator has an important value in physics :

Any system evolving in a potential near a stable equilibrium position (hence a minimum of potential) can be represented by a harmonic oscillator for small oscillations near this position of equilibrium.

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Heisenberg's indeterminacy equations
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Schrödinger equation