Final Version of Note 177(3)

This is the final version, giving a result in agreement with computer algebra, and double cross checking the method, both concept and mathematics. The result by computer algebra by co author Horst Eckardt is that the force eigenvalues of the well known harmonic oscillator are all the same, and the same as the classical values. There is therefore a hitherto unknown vacuum force:

F0 = – kx

which does not exist in classical physics and accompanies teh well known zero point or vacuum energy of the harmonic oscillator:

E0 = h bar omega / 2

Without computer algebra this would be a very complicated calculation by hand, even F1 has seven terms. As can be seen in Eq. (11) there are four quantum force terms with h bar appearing in them, but they happen to cancel out for the harmonic oscillator. These results are the first calculations of force eigenvalues in quantum mechanics, they were certainly not known hitherto.

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