Schrodingers Equation and Stationary States

You have very likely seen stationary states before, although the name might be new to you. The orbitals in a hydrogen atom (Is, 2pz, and so forth) are all stationary states, as we will discuss in Section 6.3. The probability distribution P(x) and the expectation values of all observables are constant in time for stationary states. 

If f is a stationary state, then we can multiply Schrodinger s equation by — 1, and show that — f is also a stationary state  

The probability density P(x) = f(x) 2 is the same for f as it is for —f the expectation values for all observable operators are the same as well. In fact, we can even multiply f by a complex number and the same result holds. The overall phase of the wavefunction is arbitrary, in the same sense that the zero of potential energy is arbitrary. Phase differences at different points in the wavefunction, on the other hand, have very important consequences as we will discuss shortly. 

Equation 6.8 does not always have to be satisfied f(x) does not have to be a stationary state. However, if f(x) does not satisfy Equation 6.8, the probability distribution P(x) and the expectation values of observables will change with time. The stationary states of a system constitute a complete basis set—which just means that any wavefunction f can be written as a superposition of the stationary states  

Note that the label (x) has been dropped in Equation 6.10, and in many other equations in this chapter. It is still understood that the wavefunction depends on position, but eliminating the label simplifies the notation. 

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