# Quantum Chemistry/Example 14

Show using calculus the most probable position of a quantum harmonic oscillator in the ground state (n=0)

**Question:**

What is the most probable position of a quantum harmonic oscillator at the ground state? Calculate this using the probability density equation to find the most probable position at n=0.

Probability distribution

**Solution:**

The Hermite polynomial at n=0 is:

The normalization factor at n=0 is:

α is a constant and is equal to:

The probability distribution at n=0:

The most probable position is when the maximum probability distribution is:

Applying this partial derivative to the probability distribution gives:

The constants can be taken out of the derivative:

The derivative gives:

Since it is equal to zero the constants can be divided out leaving:

Since all of the parts are multiplied they can be divided out leaving:

The point where the probability distribution is at a maximum for the ground state of n=0 for the quantum harmonic oscillator is 0.