Basic Quantum Mechanics by Klaus Ziock

By Klaus Ziock

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10) where TT 2TT 3TT nTT d' . ()( = O,-, - , - , ... , an n IS an even Integer. 11) The energy eigenvalues are, according to Eq. 4, n = 1,2,3, ... , there are infinitely many quadratically spaced energy levels possible. This leaves only the constants A and B to be determined. 13) Applied to the wave functions (Eqs. 14) In Eq. 12 we left out the case n = 0 for good reasons. According to Eqs. 11, n = 0 results in u(x) == O. 15) This sounds startling, but is a direct consequence of the uncertainty principle.

5 Fig. 6 Harmonic oscillator eigenfunctions and their squares. 6 show the normalized wave functions and their squares for the quantum numbers, n = 0 to n = 7. We sum up: The boundary conditions select, just as in the case of the particle in the box, a denumerable set of eigenfunctions belonging to a set of discrete, (in this case, equidistant) energy eigenvalues. , if we change x to -x), and the other changes sign. Functions with this kind of behavior under exchange of x with -x are said to have a definite parity.

70) The normalization as defined in the example (Eq. 71) 10 For a derivation of this expression, see J. L. Powell and B. , Reading, Mass. 1961. 30 0. , ~ c. 20 £ 0. 45 Displacement from equilibrium position Displacement from equilibrium position (b) (bi Fig. 5 Fig. 6 Harmonic oscillator eigenfunctions and their squares. 6 show the normalized wave functions and their squares for the quantum numbers, n = 0 to n = 7. We sum up: The boundary conditions select, just as in the case of the particle in the box, a denumerable set of eigenfunctions belonging to a set of discrete, (in this case, equidistant) energy eigenvalues.

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