First Quantization - Theoretical Background

Theoretical Background

The starting point is the notion of quantum states and the observables of the system under consideration. Quantum theory postulates that all quantum states are represented by state vectors in a Hilbert space, and that all observables are represented by Hermitian operators acting on that space. Parallel state vectors represent the same physical state, and therefore one mostly deals with normalized state vectors. Any given Hermitan operator has a number of eigenstates that are left invariant by the action of the operator up to a real scale factor, i. e., . The scale factors are denoted the eigenvalues of the operator. It is a fundamental theorem of Hilbert space theory that the set of all eigenvectors of any given Hermitian operator forms a complete basis set of the Hilbert space.

In general the eigenstates and of two different Hermitian operators and are not the same. By measurement of the type the quantum state can be prepared to be in an eigenstate . This state can also be expressed as a superposition of eigenstates as . If one measures the dynamical variable associated with the operator in this state, one cannot in general predict the outcome with certainty. It is only described in probabilistic terms. The probability of having any given as the outcome is given as the absolute square of the associated expansion coefficient. This non-causal element of quantum theory is also known as the Wave function collapse. However, between collapse events the time evolution of quantum states is perfectly deterministic.

The time evolution of a state vector is governed by the central operator in quantum mechanics, the Hamiltonian (the operator associated with the total energy of the system), through the Schrödinger's equation:

Each state vector is associated with an adjoint state vector, can form inner products, "bra(c)kets" between adjoint "bra" states and "ket" states, and use the standard geometrical terminology; e.g. the norm squared of is given by and and are said to be orthogonal if . If is an orthonormal basis of the Hilbert space, the above-mentioned expansion coefficient is found forming inner products: . A further connection between the direct and the adjoint Hilbert space is given by the relation, which also leads to the definition of adjoint operators. For a given operator the adjoint operator is defined by demanding for any and .