Welcome to Lecture 6. In today's preparation lecture, we will put together several concepts we have already covered to solve the particle-in-a-box problem and use it to quantify real physical systems. This lecture is broken down into two parts. The first part introduces the Schrodinger equation, while the second part applies it to the particle-in-a-box problem. By looking at this problem, we can practice calculating some important quantum mechanical properties and even apply it to conjugated double bonds to get a reasonable estimation of their emission spectra. In this course, we will focus on the time-independent Schrodinger equation. One way it can be derived is as such. Think back to the activity where we examine solutions to the wave equation. To solve the wave equation, we use the solution where u of x of t is equal to capital X of x times capital T of t, meaning that it was a function totally of position times a function totally of time. The separable nature of the wave equation allows us to examine the spatial part being the x part and ignore the temporal part being the t component. For instance, pretend that t of t is equal to cosine omega t. Now this isn't a big stretch because we know that it will be a combination of sine and cosine functions, the solution to that part of the differential equation. At this point, however, we don't know what x of x is, but instead of writing it as a capital X, let's just rewrite it as a psi of x. Therefore, the solution to the wave equation u of x of t is equal to psi of x times cosine omega t. If we plug this into the wave equation, then the time part can be multiplied out and we were left with d squared psi by dx squared plus omega squared over v squared psi of x equals to zero. Using the de Broglie relation and assuming the total energy of a particle is the sum of its kinetic and potential energy yields negative h bar squared divided by 2m times d squared psi by dx squared plus u of x psi of x is equal to e psi of x. This is called the time independent Schrodinger equation and it describes the steady state of a system. H bar is called the reduced Planck's constant and is just h divided by 2 pi. E is the total energy of the system and use the potential that a particle of mass m is moving in. There is a time dependent version of this equation, however many problems can be solved by looking solely at the steady state solutions. So what is psi of x? What does it represent? According to De Broglie, particles connect as waves and one way that this was demonstrated was doing this double slit experiment where when a stream of particles was shot through two narrow slits, we had an interference pattern that appeared on the screen in front of it. So psi of x represents the amplitude of the wave that represents a particle. Classically if one were to take the square of the amplitude of a wave, then one gets the intensity of the wave. And this means that this is the power that is delivered by the wave over an area per cycle. However, for quantum mechanics, this interpretation is different. Basically, this intensity is written as i as proportional to psi star psi, meaning psi times its complex conjugate. Max Boren, who is pictured on the right, interpreted this intensity when multiplied by a spatial zone as the probability of finding a particle between x and x plus dx. This is the widely accepted view, and it means that the intensity of a particle, described by the Schrodinger equation, is actually a probability distribution function. This means that for a given scenario, we can solve the Schrodinger equation and get a wave function which describes our particle and then apply probabilistic theories to extract measurable quantities regarding our particle.