Quantum Chemistry/Example 29

Quantum Mechanical Harmonic Oscillator Wavefunction 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: With: The first four Hermite polynomials are: Normalization of a Wavefunction 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 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 Prove that is normalized (using notation from class) , where