Note that the Hamiltonian is time independent and hence x ( t ) , p ( t ) are the position and momentum operators in the Heisenberg picture. Differentiating both equations once more and solving for them with proper initial conditions, p ˙ ˙ ( 0 ) = − − m ω ω 2 x 0 , x ˙ ˙ ( 0 ) = p 0 m , leads to x ( t ) = x 0 cos ⁡ ⁡ ( ω ω t ) + p 0 ω ω m sin ⁡ ⁡ ( ω ω t ) , p ( t ) = p 0 cos ⁡ ⁡ ( ω ω t ) − − m ω ω x 0 sin ⁡ ⁡ ( ω ω t ) .