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Why is harmonic oscillator h ( function of r = r(x) ) can be directly added...

Why is harmonic oscillator h ( function of r = r(x) ) can be directly added into Hamilton system H ? )

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Answer #1

energy of the quantum harmonic oscillator must be at least

Taking the lower limit from the uncertainty principle

Then the energy expressed in terms of the position uncertainty can be written

Minimizing this energy by taking the derivative with respect to the position uncertainty and setting it equal to zero gives

Solving for the position uncertainty gives

Substituting gives the minimum value of energy allowed.

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