Position Operator - Eigenstates

Eigenstates

The eigenfunctions of the position operator, represented in position basis, are dirac delta functions.

To show this, suppose is an eigenstate of the position operator with eigenvalue . We write the eigenvalue equation in position coordinates,

recalling that simply multiplies the function by in position representation. Since is a variable while is a constant, must be zero everywhere except at . The normalized solution to this is

Although such a state is physically unrealizable and, strictly speaking, not a function, it can be thought of as an "ideal state" whose position is known exactly (any measurement of the position always returns the eigenvalue ). Hence, by the uncertainty principle, nothing is known about the momentum of such a state.