Final answer:
A particle is in a state of equilibrium when it is at the equilibrium position. In classical mechanics, this is when the net force is zero. For a quantum particle in a box, it is when the particle is in a state with a symmetrical probability distribution about the center of the box.
Step-by-step explanation:
The time interval during which a particle is in a state of equilibrium in classical mechanics is when it is at the equilibrium position. This would occur midway between the extreme positions of the particle’s motion in a potential well, where the net force acting on the particle is zero. The classical analogy for a particle in a box model in quantum mechanics is the ground state, where a particle has nonzero energy due to the Heisenberg uncertainty principle but is not in motion because, in a state of definite energy, the average momentum of the particle is zero.
If we consider the behavior of a quantum particle in a box, its equilibrium state would analogously be when it is most likely to be found in the middle of the box, where the classical particle would also be at equilibrium. In accordance with the time-independent Schrödinger equation, this is when the quantum particle is in its ground state or any other such energy state that has symmetrical probability distribution about the center of the box.