Here is a short summary of what we have learned.
- Quantum mechanics is a theory that allows us to calculate the probabilities of fundamental interactions taking place in the microscopic world of atoms and molecules.
- To calculate these probabilities, and to use quantum mechanics we consider the state of the system which is a vector in a state space associated with the system. We write this vector as a ket (or bra) vector in Dirac bra-ket notation.
- The expansion coefficients appearing in the expansion of the state (ket) vector in some basis are the probability amplitudes whose modulus square is the probability that the system on measurement will result in a value associated with the basis vector. More technically these expansion coefficients are the scalar or inner product of system ket with the basis vectors.
- The basis vectors are generally the eigenvectors of an observable (which is a Hermitian operator) associated with the dynamical variable we wish to measure.