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 ground state? Calculate this by deriving an equation to find the most probable position.


Solution:

Using the above formula, apply the definite integral:

From the above equation we can find the normalized eigenfunction. Which results in the formula for the ground state of the harmonic oscillator,

Substitute α with the following,

This gives the following,

From this equation we can determine the formula to find the position which is equal to the 'x' variable

Thus the probable position of the quantum harmonic oscillator is determined to be 0.