Question

calculate the mean kineticc energy and potential energy of 1- dimensional oscillator in the ground state having angular velocity w

Answer 1

SHM  oscillator:  x(t) = A Sin ωt
                           v(t) = A ω Cos ωt = v₀ Cos ωt ,  v₀ = Aω

We take a spring mass system. mass = m. Spring constant = k.

Total energy of an oscillator = 1/2 k A² = 1/2 mv₀²,
                as at x=0, PE=0, and at x=A, KE  = 0.

   U(x) = 1/2 k x².   KE = 1/2 m v²
       1/2 k A² = 1/2 k x² + 1/2 m v²

KE = 1/2 * m v₀² Cos² ωt
Average mean KE from ωt = 0 to 2π (one period)= 
      = 1/2 m v₀² * average of [1+cos2ωt]/2  over one period
      = 1/2 m v₀² * 1/2 
      = 1/2 * total energy

Average mean PE = 1/2 k A²  Sin² ωt    for ωt= 0 to 2π
          = 1/2  k A² *  [1- Cos2ωt]/2    over one period
          = 1/2  k A² * 1/2
           = 1/2 of total energy