|CH301H - Principles
of Chemistry I: Honors
Fall 2011, Unique 51040
Lecture Summary, 29 September 2011
The Schrodinger equation is the foundation of the wave-particle
formulation of quantum mechanics. The Schrodinger equation cannot
be derived; it was the product of remarkable insight by Erwin
Schrodinger, and has been successful at explaining observed quantum
measurements. The Schrodinger equation takes the form:
H(psi) = E(psi)
psi is called the wavefunction - it is a function that describes the amplitude of a wave at any position x.
H is a mathematical operator, or set of instructions about what to do with the wavefunction. These instructions could be "multiply by 2," "take the derivative," "take the second derivative," etc.
E is a constant that describes the energy of the wavefunction psi at any position x.
The Schrodinger equation is therefore a very special equation. It says that the system is described by a function which, if you do something to it like take the second derivative, you get your exact function back, times a constant which represents the energy of the wavefunction.
The wavefunction psi has some interesting properties.
1) It must solve the Schrodinger equation.
2) It must be continuous.
3) The probability of the wavefunction, psi2, must be normalized. This is to make sure that the system described by the wavefunction allows for finding the system somewhere in allowed space.
This last condition is particularly important and provides a useful boundary condition for solving problems. It is important to remember that this condition applies to the probability of the wavefunction, not the wavefunction itself.
Particle-in-a-box: The Schrodinger equation can be solved exactly for a handful of problems. The simplest of these problems is called particle-in-a-box, and is a classic example of the utility of the Schrodinger equation.
Let's define a particle confined to a 1-dimensional box of length L in which the potential energy of the particle inside the box is zero, and outside the box is infinite. Based on these conditions, we can find a wavefunction psi and the energy of the particle E. By finding the correct wavefunction and solving for E, we find that it is quantized automatically - no assumptions required. Our solution to the particle-in-the-box is attached here: