The wave function is a mathematical function that describes the state of a quantum system. It contains all the information about the system, such as its energy, momentum, and position.

The Hamiltonian is the operator that corresponds to the energy of a quantum system. It is a function of the system's position and momentum.

The momentum operator is the operator that corresponds to the momentum of a quantum system. It is a function of the system's position and momentum.

The position operator is the operator that corresponds to the position of a quantum system. It is a function of the system's position and momentum.

The angular momentum operator is the operator that corresponds to the angular momentum of a quantum system. It is a function of the system's position and momentum.

The wave function and the state vector are the same thing. They are just different ways of representing the state of a quantum system.

The energy of a quantum system is a function of the Hamiltonian. The Hamiltonian is an operator that corresponds to the energy of the system.

The momentum of a quantum system is a function of the momentum operator. The momentum operator is an operator that corresponds to the momentum of the system.

The position of a quantum system is a function of the position operator. The position operator is an operator that corresponds to the position of the system.

The angular momentum of a quantum system is a function of the angular momentum operator. The angular momentum operator is an operator that corresponds to the angular momentum of the system.

The time evolution operator is the operator that corresponds to the time evolution of a quantum system. It is a function of the system's Hamiltonian.

The time evolution operator is a solution to the Schrödinger equation. The Schrödinger equation is a differential equation that describes the time evolution of a quantum system.

Stone's theorem states that the time evolution operator is unitary. This means that it preserves the inner product between two state vectors.

von Neumann's theorem states that the expectation value of an operator is a real number. This means that the operator is Hermitian.

Heisenberg's theorem states that the commutator of two operators is equal to the imaginary unit times the Poisson bracket of the corresponding classical quantities. This is a fundamental result in quantum mechanics that has many important implications.