Average Values and Other Properties

The exponents on the magnitudes of the energies have been intentionally kept the same, 10 °, to illustrate how the energy changes with quantum number n. Note that whereas the wavefunctions depend on n, the energies depend on rP. You should verify that the units in the above expression do yield units of joules as the unit of energy. 

There are other common observables in addition to energy. One could operate on the wavefunction with the position operator, x, which is simply multiplication by the coordinate x, but multiplying the sine functions of the particle-in-a-box by the coordinate x does not yield an eigenvalue equation. The P s of equation 10.11 are not eigenfunctions of the position operator. 

This should not be cause for concern. The postulates of quantum mechanics do not require that acceptable s be eigenfunctions of the position operator. (They require that P be an eigenfunction of the Hamiltonian operator, but not any other.) This does not imply that we cannot extract any information about the position from the wavefunction, only that we cannot determine eigenvalue observables for position. The same is true for other operators, like momentum. 

The next postulate of quantum mechanics that we will deal with concerns observables like this. It is postulated that although specific values of some observables may not be forthcoming from all wavefunctions, average values of these observables might be determined. In quantum mechanics, the average value or expectation value A) of an observable A whose operator is A is given by the expression. In this expression, A is the operator that operates on the result of which is then multiplied by 

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

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