The Schrodinger wave equation generalizes the fitting-in-of-waves procedure.
The waves that "fit" into the region to which the particle is contained can be recognized "by inspection" only for a few simple systems. For other problem a mathematical procedure must be used. The Schrodinger wave equation, suggested by Erwin Schrodinger in 1926, provides one method for doing this. You will see, when we again do the particle-on-a-line problem, that this equation extends the pictorial fitting-in-of-waves procedure.
Think of the method in which the Schrodinger equation is used as the counterpart of the more familiar classical parts in which Newton's laws are used. Recall that equations, such as ƒ = ma, based on Newton's law are presented without derivation. These laws let us calculate the dynamic behavior of ordinary objects. We accept Newton's laws and the equation derived from because the results are agree from experiment. Schrodinger's equation is also presented without derivation. We accept the results that we obtain by using it because in all cases where the results have been tested, they have been in agreement with experiment. Just as one uses and trusts ƒ = ma, so one must use and, to the extent that seems justified, trust the Schrodinger equation.
The Schrodinger equation, as with the direct use of the de Broglie waves, leads to waves from which all other information follows. From these waves, we obtain immediately the allowed energies of any confined particle and the probability of the particle being at various positions.
We begin by writing the form of the Schrodinger equation that lets us deduce the waves, and then the energies and position probabilities, for a particle that moves along one dimension. Let x be the variable that locates positions along this dimension. The behavior of the particle depends on the potential energy that it would have at various positions. Let U (x) be the mathematical function that describes the potential energy. The Schrodinger equation requires us to supply this function and to indicate the mass m of the particle being treated.
Solutions of the Schrodinger equation are in the form of mathematical functions that shows the amplitude of the wave at various x places. The square of this function gives the relative probability of the particles being at various positions. The energies for which these probabilities of the particles exist are the energies "allowed" to the particle.
The Schrodinger equation can be viewed as a method in which wave properties yield the total energy of a particle as the sum of its potential and kinetic energies. The potential energy contribution is given by the Schrodinger equation as a "weighting" of the potential energy at each position according to the value of the wave function at that position. The kinetic energy contribution of the first term can be appreciated by reference to the particle on a line results. The particle-on-a-line example produced the quite general result that waves for the highest energy of the wave function, the greater the kinetic energy, the greater the curvature of the wave function.
The general energy relation:
KE + PE = total energy
Becomes the one-dimensional Schrodinger equation;
-h^{2}/8∏2m Χ d^{2}?/dx^{2} + U(x)v = ε?
The potential energy contribution is given by the Schrodinger equation as a "weighting" of the potential energy at each position according to the value of the wave function amplitude at that position.
The kinetic-energy contribution fo the first term can be appreciated by reference to the particle-on-a-line results. The particle-on-a-line example produced the quite general result that the waves for the higher energy states had more nodes than the waves for the greater the curvature of the wave function, the greater the kinetic energy. This shows up in the Schrodinger equation as a relation between the second derivate of the wave function and the kinetic energy.
The behavior of a particle is deduced by finding a function and the kinetic energy will solve the differential equation after an appropriate expression for U (x) has been substituted. Solution functions generally exist for certain values for the allowed energies of the particle. The probability function also obtained from the solution function. In general may be either a real or a complex function. To allow for the second possibility, we should write not a sign but where implies the product of the wave function and its complex conjugate. Here we do not deal with problems that lead to complex wave functions. The probability is given by the simple squared term.