Eigen Values and Eigen Functions

A look at eqn. shows that for a stationary (or standing wave)wave in a stretches tring,t he amplitude functioncan have significance only for certain definite values of  Obviously, these functions will be those which must satisfy the following conditions:

(i)  must be equal to zero at each end of the string, as the string is fixed at these points and the amplitude of vibration is zero.
(ii)  must be single valued and finite i.e., at every point on the vibrating string, the amplitude has a definite value at any given instant.

Similarly, Schrodinger wave equation, being a second order differential equation, can have many solutions for the amplitude function but only those values of are acceptable (or have significance) which correspond to some definite values of the total energy E. Such values of the total energy E are called Eigen values (which correspond to discrete energy levels as postulates by bohr). These values are also called proper values or characteristic vales. The corresponding values of the function are called Eigen functions. Just as in the case of a stretched string, the Eigen functions of the Schrodinger equation will be those which must satisfy the following conditions:

(i) must be single valued and finite i.e., for each value of the variables x.y.z there is only one definite value of the function   .

(ii) must be continuous i.e., there must not exist any sudden chase in as its variables are changed.

(iii)must become zero at infinity.

Functions of the above type are called well-behead functions or proper functions. Thus a satisfactory salutation () of the wave equation must be a well-behaved function or a proper function.

Conclusion. Only those wave-functions are acceptable in quantum mechanics, which are single valued, continuous and finite in the given range.
For example:
(i)    Sin x in the range x=0 to is acceptable because it is single valued continuous and finite in the given range (as its value varies form-1 to+1 in the range 0 to ).Similarly cos x is acceptable and sin x+cos x is also acceptable.
(ii)    tan x is not acceptable in the range x=0 to because when
            
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