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The real generality of the wave-particle duality is that for any system for which we write the equations of motion in terms of a Hamiltonian, with its coordinates and associated momenta, that system may be represented by a wave as a function of the coordinates, and the momentum may be represented b y 6 li times the derivative with respect to the coordinate, operating on that wavefunction. Then all of the quantum effects we have discussed are present for that system. This is one rather precise way to state the wave-particle-duality premise, though there are certainly other ways.

The most important such operator for us is the Hamiltonian operator representing the energy, for which the condition is I We see that this looks like the Schroedinger Equation, Eq. 16), with -Uza/& replaced by the energy eigenvalue, Ej . For this reason it is also called the time-independent Schroedinger Equation. In fact, any wavefunction which satisfies Eq. 21) can be seen from Eq. 16) to have a very simple time dependence given by In this way also this is closely analogous to the normal modes of a violin string, which are distortions which exactly retain their shape, but change in phase, or amplitude, with time as cos(ot + 6) .

We may guess a form of the solution, and confirm that it is correct. The correct guess is the form ~ ( x=) A exp(-x2/(2L2)) . 40) Substituting this form in Eq. 41) The factors ~ ( xcancel ) so this is indeed a solution if @/(ML4) = K. Then E = @/(2ML2) . It is more convenient to write this in terms of the classical = vibrational frequency u (in radians per second), with 0 2 = K/M h'2/(@L4). The first step in this equation is simply the classical expression, the second gives w = fi/(ML2), and the energy is given by 1 E = - h .