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Quantum Chemistry/Example 14
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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.