Quantum mechanics and the Schrodinger equation

The Schrodinger equation can be derived in different ways, you could for example follow Landau s derivation [1], However you do it, you will get the same 

When one solves for the possible energies for the particles in a potential U, the result is that you will have a continuous energy spectrum of free particles above the potential, and a discrete set of energy levels in the potential, see Fig.(1.1). The discrete set is called bound states and represent the energies of the particles, usually electrons, that are bound by the potential. This does not mean that the particles do not move, of course, only that they have a fix energy. 

Another big discovery of the early 20th century was the theory of relativity. One of the most novel discoveries was that particles moving with a speed near the speed of light behaved in different ways than more mundane objects like cars or apples. Notions such as time dilation , the twin paradox , and space-time continuum became well known. Many times, you do not have to bother with using relativistic equations for the description of particle movements, but in some cases you do, e.g. when trying to describe particles in big accelerators, and then one has to use the relativistic version of the Schrodinger equation, known as the Dirac equation. In fact, this is what is implemented in the computer codes I will describe later, but notations become very complicated when dealing with the 

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Brief introductions to quantum mechanics and the Schrodinger equation were given. The solutions of the equation were shown to contain information regarding the energies of the permitted levels and the mathematical forms of the associated wave functions. The wave functions accurately describe atomic orbitals. 

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