Time-Independent Energy Levels and Coefficients

Let us think of x as including only electrons, or even a single electron on the x-axis. Such a drastic approximation is justified by the fact that in most processes only one or two electrons are active. The majority of electrons remain in their orbitals. We may think of nnclei as moving classically in the first approximation. If H has no explicit time dependence, its eigenfunctions, Vn(x), are time independent and satisfy 

In other words, is a solution of the time-independent SE. A time-dependent wave function that satisfies the wave equation, may be written as 

An arbitrary time-dependent wave function may be expanded in the set of functions given by Equation 7.3  

Knowing the wave function at time t = 0 means that the coefficients Q are known. From Equation 7.5, we know the wave function at all times t 0. 

The energy expectation value over the time-dependent wave function T is a constant, since it follows from Equation 7.4 that 

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