The mathematical operation of applying $\hat{A}$ to a ket $| \phi \rangle$, is a *generalized rotation in Hilbert space*, that results in another ket $\hat{A}| \phi \rangle$ which may be useful in further calculations, but is **not** in general the result of measuring the physical quantity $A$.

Therefore, the quantity $\langle \psi |A| \phi \rangle$ has no simple, general interpretation. However, in many concrete problems, a quantity with physical meaning appears in that form, specially when dealing with continuous states. For example, in particle scattering, there is the *scattering matrix* $S$. The quantity $\langle \beta |S| \alpha \rangle$ represents the amplitude for the free-particle state $| \beta \rangle$ to be found from the initial state $| \alpha \rangle$ after the scattering has taken place. This is used to compute scattering cross sections.

Another example is that of the atomic transitions that account for the spectral lines. The different mechanisms involved (being the *electric dipole* the most usual) are represented by operators whose *matrix elements* $\langle k | E | n \rangle$ are directly related to the transition probabilities.

(Apart from that, you surely know that the special case $\langle \psi |A| \psi \rangle$ is the *expectation value for the operator $\hat{A}$ for the state represented by $| \psi \rangle$*, except by a normalization factor $\langle \psi |\psi \rangle$)

As for the other integral $\langle \phi | \psi \rangle$, your interpretation is essentially correct, except that it is not the square root of a probability, but rather a complex number (the difference matters when you have to add two amplitudes, for example)

This post imported from StackExchange Physics at 2014-06-14 13:08 (UCT), posted by SE-user Eduardo Guerras Valera