The mean value in Quantum Mechanics is the average result obtained when a large number of measurements are made on identical systems, i.e. systems in the same state |Ψ(t)>.
For the harmonic oscillator
<X> = <Φ_n|X|Φ_n> = 0,  <P> = <Φ_n|P|Φ_n> = 0.
The mean value of X and P in an eigenstate of H is always zero.

The classical mean value can be defined as the average result obtained when a large number of measurements are made on the same system at different times, or when a large number of measurements are made at the same time on different systems, whose phases are randomly chosen.
x_avg = (x_M/T)∫₀^Tcos(ωt + φ)dt = 0,  T = 2π/ω = period,
or
x_avg = (x_M/2π)∫₀^2πcos(ωt + φ)dφ = 0.
p_avg = -mv = -(mωx_M/T)∫₀^Tsin(ωt + φ)dt = 0,  or  p_avg = -(mωx_M/2π)∫₀^2πsin(ωt + φ)dφ = 0.

For a harmonic oscillator of well-defined energy <X> = x_avg = 0,  <P> = p_avg = 0.
But we have to remember that the classical and quantum mechanical mean value are interpreted differently.

In a similar way the quantum mechanical and the classical root-mean-square deviation are interpreted differently.
In Quantum Mechanics we define
(ΔX)² = <X²> - <X>² = <X²>,  (ΔP)² = <P²> - <P>² = <P²>.
X² = (ħ/(2mω))(a^† + a)(a^† + a) = (ħ/(2mω))(aa + aa^† + a^†a + a^†a^†),
P² = -(mħω/2)(a^† - a)(a^† - a) = (mħω/2)(-aa + aa^† + a^†a - a^†a^†).
<Φ_n|a^†a^†|Φ_n> = <aΦ_n|a^†Φ_n> ∝ <Φ_n-1|Φ_n+1> = 0.
<Φ_n|aa|Φ_n> = <a^†Φ_n|aΦ_n> ∝ <Φ_n+1|Φ_n-1> = 0.
<Φ_n|a^†a|Φ_n> = <aΦ_n|aΦ_n> = n.
<Φ_n|aa^†|Φ_n> = <a^†Φ_n|a^†Φ_n> = n + 1.
<X²> = (ħ/(2mω))(2n + 1) = (ħ/(mω))(n + ½).
<P²> = (mħω/2)(2n + 1) = mħω(n + ½).
ΔXΔP = (n + ½)ħ.
ΔXΔP = ½ħ for n = 0, i.e. for the ground state.

For a classical harmonic oscillator with energy E = (n + ½)ħω we have
E = ½mω²x_M² = (n + ½)ħω,  x_M² = 2(n + ½)ħ/(mω) = 2(ΔX)².
ΔX = x_M/√2.
p_M =  mωx_M = √2mωΔX =  √2ΔP.
ΔP = P_M/√2.
The quantum mechanical root mean square deviations ΔX and ΔP are proportional to the classical maximum amplitude and maximum momentum.
The classical root-mean-square deviation is given by
x²_avg= (x_M²/T)∫₀^Tcos²(ωt + φ)dt = (ΔX)².
p²_avg = (p_M²/T)∫₀^Tsin²(ωt + φ)dt = (ΔP)².
For a harmonic oscillator of well defined energy we obtain the same root-mean-square deviations.  But again we have to remember that the classical and quantum mechanical root-mean-square deviation are interpreted differently.

Proof:
|Ψ(0)> = ∑_nc_n|Φ_n>,  |Ψ(t)> = ∑_nexp(-iE_nt/ħ)c_n|Φ_n>.
<x(t)> =  ∑_n∑_n'c_n*c_nexp(-i(E_n - E_n)t/ħ)<Φ_n|X|Φ_n'>
<Φ_n|X|Φ_n'> = (ħ/(2mω))^1/2[(n'+1)^1/2δ_n,n'+1 + (n+1)^1/2δ_n+1,n'].
<x(t)> = (ħ/(2mω))^1/2[ ∑_n'c_n'+1*c_n(n'+1)^1/2exp(iωt) + ∑_nc_n*c_n+1(n+1)^1/2exp(-iωt)]
=  (ħ/(2mω))^1/2 ∑_n(n+1)^1/2(c_n+1*c_n exp(iωt) + c_n*c_n+1exp(-iωt))
=  (ħ/(2mω))^1/2 ∑_n(n+1)^1/2(Re(c_n+1*c_n)2cos(ωt) - Im(c_n*c_n+1)2sin(ωt))
= A(cosφ cos(ωt) - sinφ sin(ωt)) = A cos(ωt + φ).
where Acosφ = (ħ/(2mω))^1/2∑_n√(n+1) 2Re(c_nc_n+1*), 
Asinφ = (ħ/(2mω))^1/2∑_n√(n+1) 2Im(c_nc_n+1*).

Similarly,  <P(t)> = -A mω sin(ωt + φ).

For a one-dimensional simple harmonic oscillator we may define raising and lowering operators
a = (mω/(2ħ))^1/2(X + iP/(mω)),  a^† = (mω/(2ħ))^1/2(X - iP/(mω)),
with properties
a|n> = √(n) |n - 1>,  n ≠ 0, a|n> = 0, b = 0, and
a^†|n> = √(n + 1) |n + 1>.
(a)  Show by direct calculation that the ground state of the oscillator satisfies
(ΔX)²(ΔP)² = ¼|<[x,p]>|²,
and hence is a minimum uncertainty state.
(b)  A coherent state of a one-dimensional oscillator, |b>, may be obtained by applying the finite displacement operator exp(-iPb/ħ) to the ground state eigenket |0>.   Here b is the displacement distance.  Using this definition of a coherent state, prove that the coherent state is also a minimum uncertainty state.

• Solution:
  (a)  Let  X_s = (mω/ħ)^1/2X,  P_s =  (mωħ)^-1/2P.
  a = (2)^-1/2(X_s + iP_s), a^† = (2)^-1/2(X_s - iP_s).
  X_s = (2)^-1/2(a^† + a),  P_S = i(2)^-1/2(a^† - a),  {X_s,P_s] = i.
  (ΔX)² = <X²> - <X>²,  (ΔP)² = <P²> - <P>².
  For the harmonic oscillator in its ground state <X> = <P> = 0.
  <X_s²> = ½<0|aa + aa^† + a^†a + a^†a^†|0> = ½<0|aa^†|0> = ½.
  <P_s²> = -½<0|aa - aa^† - a^†a + a^†a^†|0> = ½<0|aa^†|0> = ½.
  <X_s²><P_s²> = ¼.  <X²><P²> = ¼ħ².  [X,P] = iħ.  |<[X,P]>|² = ħ².
  Therefore (ΔX)²(ΔP)² = ¼|<[X,P]>|²,
  the ground state is a minimum uncertainty state.
  
  (b)  Consider two operators A and B.
  If Each of these operators commutes with the commutatot [A,B], then exp(A + B) = exp(A) exp(B) exp(-½(A,B]).  (Proof)
  The operator exp(-iPb/ħ) = exp(c(a^† - a)).  [a^†,-a] = I.  The commutator is the identity operator and commutes with a and a^†,
  Therefore  exp(ca^†) exp(-ca) exp(-c²/2).
  Here c = -b(mω/(2ħ))^1/2.
  |b> = exp(c(a^† - a))|0> = exp(-|c|²/2)∑_n=0^∞(cⁿ/√(n!))a^†|0>,
  since |n> = (n!)^-1/2(a^†)ⁿ|0>.
  |b> is an eigenstate of the lowering operator a.
   a|b> = exp(-|c|²/2)∑₀^∞(cⁿ/(√n!))a|n> = c exp(-|c|²/2)∑₁^∞(cⁿ(√n)/(√n!))|n-1>
  = c exp(-|c|²/2)∑₁^∞(c^n-1/√((n-1)!))|n-1> = c exp(-|c|²/2)∑₀^∞(cⁿ/(√n!))|n> = c|b)>.
  The eigenvalue of a is c.
  |Ψ(t)> = U(t,0)|b> = exp(-iHt/ħ)|b> = exp(-|c|²/2)∑₀^∞(cⁿ/(√n!))exp(-i(n + ½)ωt)|n>
  = exp(-iωt/2) exp(-|c|²/2)∑₀^∞(cⁿ/(√n!))exp(-inωt)|n> =  exp(-iωt/2) exp(-|c'|²/2)∑₀^∞(c'ⁿ/(√n!))|n>,
  with c' = c exp(-iωt), |c'| = |c|.
  |ψ(t)> is an eigenstate of a  with eigenvalue c’.
  <x(t)> = (ħ/(2mω))^1/2<a^† + a>(t) = (ħ/(2mω))^1/2(<aψ(t)|ψ(t)> + <ψ(t)|aψ(t)>)
  = (ħ/(2mω))^1/2(c'* + c').
  <X(t)>² = (ħ/(2mω))(c'* + c')².
  <X²(t)> = (ħ/(2mω))<a^†2 + a² + 2a^†a + 1>(t) = (ħ/(2mω))((c'* + c')² + 1).
  (ΔX)² = (ħ/(2mω)).
  <P(t)> = i(mωħ/2)^1/2<a^† - a>(t) = i(mωħ/2)^1/2(c'* - c').
  <P(t)>² = -(mωħ/2)(c'* - c')².
  <P²(t)> = -(mωħ/2)<a^†2 + a² - 2a^†a - 1>(t) = -(mωħ/2)((c'* - c')² - 1).
  (ΔP)² = (mωħ/2).
  ΔXΔP = ħ/2.
  The coherent state is also a minimum uncertainty state.