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Time-Dependent Schrodinger Equation
In quantum mechanics, the Schrodinger equation is the analog to the second law of motion of classical mechanics. That is, the Schrodinger equation plays the role of helping us describe the motion and behavior of the system (such as an electron) and it does this by using a quantity known as the wave function. In the same way that position and momentum are used to completely describe the motion of an object in classical mechanics, we use the wave function to completely describe the motion of the system in quantum mechanics. As stated in the lecture, the time-dependent Schrodinger equation is not something that was derived but rather was invited by Erwin Schrodinger to explain how systems move on the atomic and subatomic level. It was then tested extensively through experiments and confirmed to be true. From the equation, we see that knowing what the wave function looks like at some initial condition,say at time of zero seconds, allows us to completely determine what the future motion of the system will look like to the extent that quantum mechanics allows us.
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