Fundamental Principles

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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

\[i \hbar \odv{\ket{\psi(t)}}{t} = \hat{H} \ket{\psi(t)}\]

\(\hat{H}\) 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:

  • The Postulates: section 5.5 [B1], section 2.2 [B3]