In simple terms, we say that a Hamiltonian or some other operator/observable will cause a transition from an initial quantum state | i ⟩ ⟩ to a final quantum state | f ⟩ ⟩ if the following holds true: ⟨ ⟨ f | H ^ ^ | i ⟩ ⟩ = M i , f ≠ ≠ 0 | ⟨ ⟨ f | H ^ ^ | i ⟩ ⟩ | 2 = | M i , f | 2 , where the last line is the probability amplitude of transition caused by some operator H ^ ^ and the matrix element M i , f encapsulating this information. In effect, the calculation involves finding the matrix elements of the H-operator which gives this information of transition between two states. Examples of this can be seen in nuclear physics such as in the beta decay transition, neutrinoless double beta decay and double beta decay.