Question

The function f(x)=eikx where k is a constant is an eigenfunction of the linear momentum operator. What is its eigenvalue?
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Answer 1

Answer:

Operators, Eigenvalues and Eigenfunctions

An operator O may be thought as “something” that operates on a function to produce

another function:

Of(x) = g(x)

In most cases, the operators of quantum mechanics are linear. Operators are linear if

they have properties:

O[f(x) + g(x)] = Of(x) + Og(x)

Oc f(x) = cOf(x)

where c is a constant (c can be a complex number: c = a + ib, i = √–

–

1

–

)

Examples:

linear operators:

x (multiplication by x):

x[f(x) + g(x)] = x f(x) + xg(x)

d

dx (differentiation with respect to x):

d

dx [f(x) + g(x)] = d

dx

f(x) +

d

dx g(x)

A nonlinear operator: (square root operator):

f(x) + g(x) ! f(x) + g(x)

The eigenvalues and eigenfunctions of an operator A are those numbers aj and functions

! j which satisfy

Explanation: