Harmonic oscillator in Heisenberg picture

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SUMMARY

The discussion focuses on the harmonic oscillator in the Heisenberg picture, specifically addressing the second time derivative of the x Heisenberg operator, which is expressed as -ω²x. The solution to this differential equation yields xH(t) = Acos(ωt) + Bsin(ωt), where A and B are time-independent operators. The incorporation of A and B as multiplicative factors rather than additive constants is crucial for maintaining the equality of the left-hand side and right-hand side of the equation when substituted back. The discussion also raises a question about the integration of operators and the placement of constants in such scenarios.

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For the harmonic oscillator in 1-D we get the 2nd time derivative of the x Heisenberg operator = -ω2 x. When that is integrated it gives xH (t) = Acos(ω t) +Bsin (ω t) where A and B are time independent operators. My question is why are the constants A and B incorporated into the terms as a multiplicative factor instead of being additive constants ? And what would happen if the term to be integrated already had an operator in it. you wouldn't know whether to place the multiplicative constant before or after the other operator ?
 
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a = \ddot x = -\omega^2 x

These constants are multiplied because when you sub it back in it should give LHS = RHS.
 

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