The integral of Ψ over all space is invariant, because it is the inner product of Ψ with the state of zero energy, which is a wave with infinite wavelength, a constant function of space. For any energy eigenstate η (x), the inner product, ⟨ ⟨ η η | ψ ψ ⟩ ⟩ = ∫ ∫ η η ( r ) ψ ψ ( r ) d 3 r , only changes in time in a simple way: its phase rotates with a frequency determined by the energy of η. When η has zero energy, like the infinite wavelength wave, it doesn't change at all.