Harmonic Oscillator Wavefunctions, Eigenenergies, Quantum Transition and Hermite Recurrences скачать в хорошем качестве

Harmonic Oscillator Wavefunctions, Eigenenergies, Quantum Transition and Hermite Recurrences

Quantum Chemistry Problem [Q21-05-00] ----------------------------------------- Question: (a) Describe the nature of the eigenfunctions of the 1-dimensional harmonic oscillator. (b) Show that the Hermite polynomials obey the following recurrence relations: Hn(y)=2yHn−1(y)−2(n−1)Hn−2(y), and Hn′(y)=2nHn−1(y). (c) Show how these recurrence relations are related to the effects of applying the position operator x on the eigenfunctions of the harmonic oscillator. ----------------------------------------- The harmonic oscillator is a key problem in quantum chemistry, because it is a model for molecular vibrations. Very different from classical mechanics, vibrational modes in quantum mechanics are discrete in energy. When a molecule absorbs light to undergo a vibrational excitation, it does so by absorbing a photon with the correct energy to match the vibrational quanta, moving the harmonic oscillator wavefunction up to the next eigenenergy. Because of this, understanding the quntum mechanics of the harmonic oscillator will be important for its application to molecular spectruscopy. So, how do you solve the Schrodinger equation for the harmonic oscillator? What is the nature of the wavefunctions of the harmonic oscillator and their corresponding eigenenergies? What is a "vibrational quanta"? Why are the vibrational eigenstates equally spaced? In this video, we will show show you how to use a simple scaling to simplify the Schrodinger equation for the harmonic oscillator. For this, the eigenfunctions of the harmonic oscillator are seen to be Gaussian functions multiplied by Hermite polynomials. From these, we will show you how to easily deduce that the spectrum of the harmonics oscillator should consist of equally-spaced energy levels, separated by the same unit of energy: h-bar times omega, ℏω, called a"vibrational quanta". From the original Schrodinger equation, we will show you how to derive an auxiliary eigenequation just for the Hermite polynomials, and use it to find higher-energy wavefunctions of the harmonic oscillator. The same eigenequation is used to prove two recurrence relations for the Hermite polynomials. We show how you can use these recurrence relations to derive an equation for the position operator X. The video will show you how the position operator, when operated on an eigenstate of the harmonic oscillator, will generate a wavefunction one quatum above, and another wavefunction one quantum below the original wavefunction. Because a light field oscillating along the direction of the bond between two atoms is exciting the vibration, the action of the light field corresponds to applying a position measurement on the length of the bond. Since application of the position operator X on the harmonic oscillator generates adjacent eigenstate above and below the original state, absorption of a photon by the harmonic oscillator allows it to make transitions only to the states immediately above or below its original energy. These facts will be important in the application of quantum mechanics to vibrational spectroscopy for understanding quantum chemistry.