Momentum Eigenstates

Being in a momentum eigenstate in quantum mechanics means that the wave function of the particle is a plane wave of the form \(\psi(x) = e^{ikx}\), where \(k\) is the wave number related to the momentum of the particle by \(p = \hbar k\). This form is a solution to the time-independent Schrödinger equation for a free particle, where the potential energy \(V(x)\) is zero.

However, plane waves such as \(e^{ikx}\) are not square integrable over all space, which means they do not belong to the space of \(L^2(\mathbb{R})\) functions. Physically, this implies that a particle in a pure momentum eigenstate cannot be localized in space; the probability of finding the particle at any specific location is constant everywhere.

Despite not being normalizable, momentum eigenstates are still useful. They form a basis for the space of square integrable functions due to the completeness of the set of plane waves. This means any physical wave function \(\psi(x)\) that describes a quantum state can be expressed as a superposition (integral) of these plane waves, known as a Fourier transform. This superposition, or wave packet, is square integrable and localizable, making it a more physically realistic state of the particle. 

The wave packet itself is not an eigenstate of the momentum operator \( \hat{p} = -i\hbar \frac{\partial}{\partial x} \) since it is a combination of multiple momentum eigenstates with different \(k\) values. Consequently, a wave packet has a spread in momentum and, due to the Heisenberg Uncertainty Principle, also a spread in position, which allows for the particle to be localized to a region in space.

The spectrum of the momentum operator in this context is continuous, which means that the eigenvalues \(k\) can take any real value, leading to the continuous nature of possible momentum values for a quantum state in free space.