In quantum mechanics, it is indeed possible for an operator to have an eigenvalue of 0. An eigenvalue represents a possible outcome when measuring a physical quantity associated with an operator. If an operator has an eigenvalue of 0, it implies that there is a corresponding eigenstate (also known as eigenvector) with that eigenvalue.
When an operator has an eigenvalue of 0, it means that the associated observable quantity, upon measurement, will yield the value of 0 for that particular eigenstate. This indicates that the corresponding physical property is not present or not measurable in that state.
To provide a concrete example, consider the momentum operator in one dimension, represented by the operator symbol "p." In this case, the eigenvalue equation for the momentum operator can be written as:
p |ψ⟩ = λ |ψ⟩
If λ = 0 is an eigenvalue, it means there exists a state |ψ⟩ such that the momentum measurement on that state will yield a value of 0. This implies that the corresponding particle has zero momentum in that particular state.
It's important to note that the presence of an eigenvalue of 0 does not imply that all measurements of the associated operator will yield 0. Different eigenstates with different eigenvalues can exist for the same operator, and when measured, the observable quantity will take on the corresponding eigenvalue for the particular state.
In summary, in quantum mechanics, an operator can have an eigenvalue of 0, indicating that the corresponding observable quantity has a value of 0 when measured in the corresponding eigenstate.