Quantum Chemistry/Example 29
Quantum Mechanical Harmonic Oscillator Wavefunction[edit | edit source]
A system undergoing harmonic motion around an equilibrium is known as a harmonic oscillator.
In quantum chemistry, the harmonic oscillator refers to a simplified model often used to describe how a diatomic molecule vibrates. This is because it behaves like two masses on a spring with a potential energy that depends on the displacement from the equilibrium, but the energy levels are quantized and equally spaced. The potential energy is non-zero and can theoretically range from .
The wavefunction for the quantum harmonic oscillator is given by the Hermite polynomials multiplied by the Gaussian function. The general form of the wavefunction is:
The first four Hermite polynomials are:
Normalization of a Wavefunction[edit | edit source]
The probability of finding the particle in any state is given by the square of the wavefunction. Therefore normalizing a wavefunction in quantum mechanics means ensuring that the total probability of finding a particle in all possible positions is equal to 1. A normalized wavefunction is one that satisfies the normalization condition:
Normalization is important because it ensures that the probability of finding the particle somewhere in space is 100%.
Example[edit | edit source]
Show the derivation of the normalization factor of the v=1 state of the harmonic oscillator beginning from the unnormalized wavefunction.
Sub in v=1
Check[edit | edit source]
Prove that is normalized (using notation from class)