Varieties of Wave EquationsEarly in the 20th century, electrons were shown to have wave properties, and the waveparticle duality became a part of our understanding of nature. The mathematics for describing the behavior of such electron waves might be expected to be similar to that for describing classical waves, such as the wave on a stretched string or a plane electromagnetic wave The wave equation developed by Erwin Schrodinger in 1926 shows some similarities in its onedimensional form: 
Index Schrodinger equation concepts  

Go Back 
The Postulates of Quantum Mechanics 
Index Schrodinger equation concepts  

Go Back 
The Wavefunction PostulateIt is one of the postulates of quantum mechanics that for a physical system consisting of a particle there is an associated wavefunction. This wavefunction determines everything that can be known about the system. The wavefunction is assumed here to be a singlevalued function of position and time, since that is sufficient to gaurantee an unambiguous value of probability of finding the particle at a particular position and time. The wavefunction may be a complex function, since it is its product with its complex conjugate which specifies the real physical probability of finding the particle in a particular state. Constraints on the wavefunction 
Index Schrodinger equation concepts Postulates of quantum mechanics  

Go Back 
Constraints on WavefunctionIn order to represent a physically observable system, the wavefunction must satisfy certain constraints:

Index Schrodinger equation concepts  

Go Back 
Probability in Quantum MechanicsThe wavefunction represents the probability amplitude for finding a particle at a given point in space at a given time. The actual probability of finding the particle is given by the product of the wavefunction with it's complex conjugate (like the square of the amplitude for a complex function). Since the probability must be = 1 for finding the particle somewhere, the wavefunction must be normalized. That is, the sum of the probabilities for all of space must be equal to one. This is expressed by the integral
Part of a working solution to the Schrodinger equation is the normalization of the solution to obtain the physically applicable probability amplitudes. 
Index Schrodinger equation concepts  

Go Back 
Normalization ExamplesIn order to use the wavefunction calculated from the Schrodinger equation to determine the value of any physical observable, it must be normalized so that the probability integrated over all space is equal to one. 
Index Schrodinger equation concepts  

Go Back 