Quantum Mechanics for Mathematicians

The time-dependent state will be

|ψ(t)〉=

∑

k

ckeik·qe−it~

|k|^2

2 m

Since each momentum eigenstate evolves in time by the phase factor

e−it~

| 2 km|^2

the probabilities of observing a momentum value stay constant in time.

10.4 For further reading

Every book about quantum mechanics covers this example of the free quantum
particle somewhere very early on, in detail. Our discussion here is unusual
just in emphasizing the role of the spatial translation group and its unitary
representations.