types of systems. One-particle systems In general, the one-particle state could be described by a complete set of quantum numbers denoted by . For example, the three quantum numbers associated to an electron in a coulomb potential, like the hydrogen atom, form a complete set (ignoring spin). Hence, the state is called and is an eigenvector of the Hamiltonian operator. One can obtain a state function representation of the state using . All eigenvectors of a Hermitian operator form a complete basis, so one can construct any state obtaining the completeness relation: Many have felt that all the properties of the particle could be known using this vector basis, which is expressed here using the Dirac Bra–ket notation. However this need not be true. Many-particle systems When turning to N-particle systems, i.e., systems containing N identical particles i.e. particles characterized by the same physical parameters such as mass, charge and spin, an extension of the single-particle state function to the N-particle state function is necessary. A fundamental difference between classical and quantum mechanics concerns the concept of indistinguishability of identical particles. Only two species of particles are thus possible in quantum physics, the so-called bosons and fermions which obey the rules: (bosons), (fermions). Where we have interchanged two coordinates of the state function. The usual wave function is obtained using the Slater determinant and the identical particles theory. Using this basis, it is possible to solve any many-particle problem that can be clearly and accurately described by a single wave function single system-wide diagonalizable state. From this perspective, first quantization is not a truly multi-particle theory but the notion of "system" need not consist of a single particle either. See also Canonical quantization Geometric quantization Quantization Second quantization Notes References Quantum mechanics