The next step is to solve the second order differential equation (13) above for u(y) Putting the values from (8) and (10) in equation (7): We must now calculate the derivatives of that will be substitued into the With this fact we can guess that ( y) will be as e ±. Substituting this new variable into Equation (5) above yeilds:įor very large values of y, the term is negligible in comparison to the y 2 term.
#FIRST HARMONIC SERIES#
The first step in the power series method is to perform aĬhange of variables by introducing the dimensionless variable, Series method is used to derive the wave function and the eigenenergies for the
The Equation for the Quantum Harmonic Oscillator is a second order differentialĮquation that can be solved using a power series. Placing this potential in the one dimensional, time-independent Schr ödinger Where is the natural frequency, k is the spring constant, and m is the mass of theįor convenience in this calculation, the potential for the harmonic oscillator is The classical potential for a harmonic oscillator is derivable from Hooke’s law. A firm understanding of the principles governing the harmonic oscillator is prerequisite to any substantial study of quantum mechanics.ġ.1 The Schrodinger Equation for the Harmonic Oscillatorģ Solved Harmonic Oscillator Problems 1 Solution of the Schrodinger Equation The solution to this simple system can then be used on them. Systems with nearly unsolvable equations are often broken down into smaller systems. It is one of the first applications of quantum mechanics taught at an introductory quantum level. An exact solution to the harmonic oscillator problem is not only possible, but also relatively easy to compute given the proper tools. The equation for these states is derived in section 1.2. The harmonic oscillator has only discrete energy states as is true of the one-dimensional particle in a box problem. This equation is presented in section 1.1 of this manual.
The Harmonic Oscillator is characterized by the its Schr ö dinger Equation. Here, harmonic motion plays a fundamental role as a stepping stone in more rigorous applications. Almost all potentials in nature have small oscillations at the minimum, including many systems studied in quantum mechanics. The potential for the harmonic ocillator is the natural solution every potential with small oscillations at the minimum. Any vibration with a restoring force equal to Hooke’s law is generally caused by a simple harmonic oscillator. Harmonic motion is one of the most important examples of motion in all of physics.