Quantum Chemistry/Example 29

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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)

, where