In a nutshell, a Hamiltonian matrix (ie, the energy “Operator” matrix) must be Hermitian to have real eigenvalues. A Hermetian matrix means it’s equal to the co

In a nutshell, a Hamiltonian matrix (ie, the energy “Operator” matrix) must be Hermitian to have real eigenvalues. A Hermetian matrix means it’s equal to the complex conjugate of its transpose. Scott mentions in this article that his work on Quantum Computing and Quantum Information deals mostly with real quantum states, not so much complex. However these are primarily discrete states. Ie, matrix operators and vector wavestates. When working with position and momentum spaces, you need to deal with continuous wavefunctions, and the presence of complex numbers becomes more immediate. Eg, the momentum operator, as written in position space, has a factor of i, when showing that momentum is the generator of positional translations (p = -i hbar d/dx). This is also why the gradient term of Schrodinger’s equation is negative (kinetic energy is proportional to momentum squared, so the i^2 makes it negative). Similarly complex numbers appear with the Quantum Harmonic Oscillator when rearranging the Hamiltonian suit the ladder operators (or creation/annihilation operators).