Quantum mechanics should really be called Wave Function Analysis because
everything we can ever hope to know about a quantum system is found out
through the calculation and analysis of its wave function.
The science of quantum mechanics is really the science of wave function
analysis. Everything that can be known about a quantum mechanical system
can be calculated from the Schrodinger equation. The Schrodinger equation is
a differential equation which instead of being a direct equation in itself
it is a framework for setting up other equations.
The Schrodinger equation is really only a simple eigenvalue
eigenfunction equation. This equation tells us how to calculate the energy
levels for a system and the probabilities for a system to be in a particular
quantum state. The first equation below simply means that the differential
operator simply multiplies the function it is acting on by a constant.
The function e^5x is a solution to this equation
E simply stands for the energy levels and H is the quantum Hamiltonian operator
where all the appropriate quantum operator assignments have been made. Once one has
figured out what the quantum Hamiltonian is and solved the Schrodinger equation for
the eigenvalue energy levels and eigenfunction probablity amplitudes one should then
know everything about a system that can actually be known. The time evolution of the
probablity amplitudes for the system can then be calculated also. The time evolution
of a quantum system is calculated by solving the Schrodinger equation for the time
dependence separately and then combining the eigenfunction solutions together.
