All observable physical quantities, such as energy, are associated with hermitian operators. In the case of energy and the corresponding Hamiltonian, H {\displaystyle H} this means H i j = ⟨ i | H | j ⟩ = ⟨ j | H | i ⟩ ∗ = H j i ∗ {\displaystyle H_{ij}=\langle i|H|j\rangle =\langle j|H|i\rangle ^{*}=H_{ji}^{*}} , i.e. H 11 {\displaystyle H_{11}} and H 22 {\displaystyle H_{22}} are real, and H 12 = H 21 ∗ {\displaystyle H_{12}=H_{21}^{*}} . Thus these four matrix elements H i j {\displaystyle H_{ij}} produce a 2 × {\displaystyle \times } 2 hermitian matrix.