An ** eigenfunction** in a vector space of function is a non-zero function, , that when acted upon by an operator , returns the same function but multiplied by a constant , which is known as an

**.**

*eigenvalue*The above equation is called an ** eigenvalue equation**. The time-independent Schrodinger equation

where is a Hermitian operator called the ** Hamiltonian**, is also an eigenvalue equation. We say that the eigenfunction of a stationary state is a solution to the Schrodinger equation. For a one-dimensional, one-particle system in a stationary state, , and

where .

###### Question

Show that an eigenfunction representing a stationary state satisfy eq40.

###### Answer

Stationary states of are * determinate states*, meaning a measurement of the energy of a particle in that state results in the same value of . Hence, the variance of the observable of for a stationary state would be zero, i.e. . Since is also the average value of ,

where is the observable of and denotes the average value of , which is since is a stationary state of . Next, we assume that the operator for the observable is . Our assumption is only valid if is Hermitian, which can be proven as follows:

Using the property of the Hermitian operator and the condition that is real (),

This implies that is a Hermitian operator and so,

Since produces another ket, the property of a Hermitian operator gives

From the positive semi-definiteness property of the inner product space, where , with if ,

Examples of functions that are eigenfunctions of are and , and examples of functions that are not eigenfunctions of are and . We can easily verify whether a function is an eigenfunction or not one by substituting it into eq41.

###### Question

Why is an eigenfunction defined as a non-zero function?

###### Answer

Since , an eigenfunction that is zero would have an infinite number of eigenvalues (any value of multiplied by zero is zero), which contradicts the Born interpretation that an acceptable eigenfunction representing a stationary state in quantum mechanics must be single-valued.