Quantum mechanical description of time-dependent systems

The symmetry properties also enable us to establish the selection rules for transitions between states. For the group C2 , transitions between and U2 states and between bi and 2 states are forbidden. All others are allowed. Thus the transition between Ai is 

Until now in our quantum mechanical discussions we have described the stationary or time-independent states of a system. Furthermore, our language was such as to imply that we had sufficient information about the system to know that it was in a particular state described by a particular set of quantum numbers. For example, in the case of the harmonic oscillator we spoke as though we knew that the oscillator was in the pth state with wave function i// , and energy = (v A- j)hv or, in the case of the hydrogen atom, that it was in a state described by the set of numbers n, /, m. This approach is very useful in a first discussion of quantum mechanical properties of various kinds of systems. However, we do not have reason to presuppose that a system is in a particular quantum state. 

Having obtained the set of particular solutions of the Schrodinger equation, the set of i// , the general solution is a linear combination of the j/ , namely, 

This is the total probability of finding the system in some state. If we use Eq. (25.40) in Eq. (25.41), we have 

Equation (25.43) says that the sum of the squares of the absolute value of the coefficients a in the series in Eq. (25.40) is unity. The manner in which we obtained Eq. (25.43) requires that we interpret the right-hand side as a sum of probabilities. Therefore, we interpret as the probability of finding the system in the state described by. According to Eq. (25.43), 

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Quantum-Mechanical Description of Time-Dependent Systems 647... 

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