# Fundamental Principles¶

## Postulates of Quantum Mechanics¶

### State Space¶

Any isolated system evolves in an abstract Hilbert space,
basically a complex vector space with an inner product,
known as the *state space* \(\mathcal{E}_H\) of the system.
The state of the system is completely described by a *state vector* \(\ket{\psi}\),
a normalized vector of the state space.

### Measurement¶

Every physical quantity of the system is described by a linear Hermitian operator \(\hat{A}\) acting in \(\mathcal{E}_H\): it is called an

*observable*.The only possible result of a measurement of an observable \(\hat{A}\) is one of the eigenvalues \(a_\alpha\) of \(\hat{A}\).

The probability of measuring \(a_\alpha\) when the system is in the generic state \(\psi\) is given by the square of the inner product of \(\ket{\psi}\) with the normalized eigenstate \(\ket{a_\alpha}\): \(\| \bk{\psi}{a_\alpha} \|^2\)

Immediately after the measurement of an observable \(\hat{A}\), the state of the system is the normalized eigenstate \(\ket{a_\alpha}\) corresponding to the eigenvalue \(a_\alpha\).

Different variants of measurements must be considered, see Measurements.

### Time Evolution¶

The time evolution of a *closed* quantum system is given by the Schrödinger equation

\(\hat{H}\) is a Hermitian operator known as the *Hamiltonian* of the system.

The system is *closed* as long as the system does not undergo any obervation,
i.e. has no interaction with its environment.

This evolution is described by a unitary transformation.

## Superposition¶

The state of a quantum system is described by a vector in an abstract space: a Hilbert space, that may be of finite or infinite dimension. A way to describe a measurement is to associate it to one particular basis of this space, e.g. \(\{ \ket{\psi_1}, \dots, \ket{\psi_n} \}\), such that the different outcomes of the measurement correspond to the projection onto one of the elements of the basis.

The relation between a measurement and the concept of superposition is that any state vector can be expressed in the measurement’s basis, and that this expression is basically a weighted sum of the basis’ vectors: e.g. \(\ket{\psi} = c_1 \ket{\psi_1} + \dots + c_n \ket{\psi_n}\). This sum is a superposition of the states \(\psi_\alpha\).

Thus the state of a quantum system can be expressed in an infinite number of different superpositions, each of it associated to a basis of the Hilbert space.

**References:**