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this is the story all about how the Schrodinger equation applies to the free particle what we mean by a free particle imagine an electron for instance floating in the vacuum of space it never encounters anything it never runs into anything how that enters the Schrodinger equation is that there is effectively no potential anywhere so the time independent Schrodinger equation were back to one dimension now so dont think about a particle floating around in the vacuum of three-dimensional space its floating around in the vacuum of one dimensional space the left-hand side of our time independent Schrodinger equation is the Hamiltonian operator applied to the wavefunction this is in some sense the total energy which breaks down into a kinetic energy component here with the momentum of the particle squared divided by twice the mass and the potential energy part here where V of X is the potential energy that the particle would have to have to be found at a particular location in the contex
