Examples are provided in Chapter 4, Activity 2, and Chapter 5, Activity 1. In this situation, the oscillatory time dependence does not cancel out in calculations, but rather accounts for the time dependence of physical observables. A new derivation of Schrödingers equation is presented, based on Schrödingers original discussions on refraction and the optical-mechanical analogy, but adopting a much simpler formalism: Newtonian mechanics and some basic elements of classical wave theory (such as Snells law). When a system is not is a stationary state, the wavefunction can be represented by a sum of eigenfunctions like those above. Since the Schrödinger equation (that is the quantum wave equation) is linear, the behavior of the original wave function can be computed through the superposition principle. We will introduce quantum tomorrow and the waves will be wavefunctions. A wavefunction with this oscillatory time dependence e-iωt therefore is called a stationary-state function. Everything above is a classical picture of wave, not specifically quantum, although they all apply. We will see that all observable properties of a molecule in an eigenstate are constant or independent of time because the calculation of the properties from the eigenfunction is not affected by the time dependence of the eigenfunction. When molecules are described by such an eigenfunction, they are said to be in an eigenstate of the time-independent Hamiltonian operator.
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