Final answer:
In quantum mechanics, the ground state of a free particle has zero momentum, but the momentum operator does not have proper eigenstates and instead has a continuous spectrum. This can be understood mathematically by considering the wave function of the particle, which can have non-zero values at multiple positions in space. The ground state corresponds to a wave function that is constant and non-zero throughout space.
Step-by-step explanation:
In quantum mechanics, the ground state of a free particle is the eigenstate with zero momentum. However, the momentum operator does not have proper eigenstates, but instead its spectrum is described in terms of a projection-valued measure over the real axis. This means that the momentum of a free particle can take on a continuous range of values, rather than being confined to a single eigenvalue.
The coexistence of the ground state with zero momentum and the continuous spectrum of the momentum operator can be understood from a mathematical perspective by considering the wave function of the particle. The wave function represents the state of the particle and can have a non-zero value at any position in space. The momentum operator acts on the wave function to give the momentum of the particle, but since the wave function can have non-zero values at multiple positions, the momentum can be any value within the continuous spectrum.
Thus, the ground state with zero momentum corresponds to a wave function that is constant and non-zero throughout space. This state has the unique property of having zero uncertainty in position, but infinite uncertainty in momentum. It is a special case that arises due to the mathematical formulation of quantum mechanics.