The Postulates of Quantum Mechanics

Contents

The conventional postulates
Density-matrix postulates
On Planck's constant
Pictures of time evolution
Canonical quantization rules
The Heisenberg uncertainty principle

Quantum mechanics is not derived from classical mechanics; it rests on its own postulates, which state what a physical state is, which quantities can be measured, what the possible outcomes of a measurement are and with what probabilities they occur, how a measurement alters the system, and how composite systems are described. This chapter collects the conventional postulates in the state-vector formulation, together with the canonical quantization rules and the notion of uncertainty on which Heisenberg's principle is built. The mathematical arena — Hilbert spaces and the operators acting on them — is developed in Hilbert Spaces. Standard modern treatments of this material are found in [Sakurai:2017] [CohenTannoudji:1977] [Griffiths:2018]. The source for this chapter is an early outline: its completed content is ported in full below, and the headings it reserves without content are marked as pending rather than filled in with material the source does not contain.

The conventional postulates

The state postulate

Let \(\mathcal{H}\) be a Hilbert space over \(\C\) and let \(\ket{\psi}\in\mathcal{H}\). The relation

\begin{equation*} \ee^{\ii\phi}\ket{\psi}\sim\ee^{\ii\gamma}\ket{\psi} \iff\phi,\gamma\in\R \end{equation*}

is an equivalence relation. We say that an equivalence class is a ray in \(\mathcal{H}\).

Postulate 77.1 (Quantum physical states).

A quantum physical state is represented by a ray of normalized vectors, that is, vectors satisfying

\begin{equation}\tag{77.1} \braket{\psi}{\psi}=1\ep \end{equation}

If no ambiguity is possible, we shall say that the state of a system is one of its representatives with vanishing global phase; that is, the state will be \(\ket{\psi}\).

The observable postulate

Let \(O\in\mathcal{L}(\mathcal{H})\) be a linear operator on \(\mathcal{H}\).

Postulate 77.2 (Observable quantities).

To every measurable quantity there corresponds a Hermitian operator \(O\).

We shall call such an operator an observable. It must be noted that not every Hermitian operator is an observable.

The measurement postulate

Consider the spectral decomposition of an observable \(O\),

\begin{equation}\tag{77.2} O=\sum_{i}\lambda_i\ketbra{\lambda_i}{\lambda_i}\ep \end{equation}

Postulate 77.3 (Measurement outcomes).

The possible results of a measurement of \(O\) are its eigenvalues \(\lambda_i\).

The probability postulate: the Born rule

Consider a quantum system in the state \(\ket{\psi}\), and let \(O\) be an observable with eigenvalues \(\lambda_i\).

Postulate 77.4 (Born rule).

The probability of obtaining \(\lambda_i\) in a measurement of \(O\) is given by

\begin{equation}\tag{77.3} P(\lambda_i,\ket{\psi})=\abs{\braket{\lambda_i}{\psi}}^2\ep \end{equation}

We must note that the probability is determined on the whole ray of \(\ket{\psi}\), since \(\abs{\ee^{\ii\phi}}=1\). Moreover, by the theory of probability, it must hold that

\begin{equation}\tag{77.4} \sum_{i}P(\lambda_i,\ket{\psi})=1\ep \end{equation}

Expanding \(\ket{\psi}\) in the eigenbasis \(\set{\ket{\lambda_i}}_{i=1}^{\dim\mathcal{H}}\) as \(\ket{\psi}=\sum_{j}a_j\ket{\lambda_j}\), we have

\begin{align*} P(\lambda_i,\ket{\psi}) &=\abs{\braket{\lambda_i}{\psi}}^2\\ &=\abs{\bra{\lambda_i}\left(\sum_{j}a_j\ket{\lambda_j}\right)}^2\\ &=\abs{\sum_{j}a_j\braket{\lambda_i}{\lambda_j}}^2\\ &=\abs{\sum_{j}a_j\delta_{ij}}^2\\ &=\abs{a_i}^2\ec \end{align*}

so the probability of obtaining \(\lambda_i\) in a measurement is the squared modulus of the \(i\)-th component of \(\ket{\psi}\) in the eigenbasis.

The collapse postulate

Consider a quantum system in the state \(\ket{\psi}\), and let \(O\) be an observable with eigenvalues \(\lambda_i\).

Postulate 77.5 (Alteration of quantum systems).

If a measurement of \(O\) yields the eigenvalue \(\lambda_i\), then immediately after the measurement the state of the system is \(\ket{\lambda_i}\).

The time-evolution postulate

The source records for this postulate only the stationary eigenvalue problem of the Hamiltonian,

\begin{equation}\tag{77.5} H\ket{\psi}=E\ket{\psi}\ep \end{equation}

[Heading reserved in the source outline; the statement of the time-dependent Schrödinger equation governing the evolution of quantum states is pending port from a completed source.]

The composite-system postulate

Consider a quantum system composed of \(N\) individual systems whose states are rays in the Hilbert spaces \(\mathcal{H}_{1},\ldots,\mathcal{H}_{N}\).

Postulate 77.6 (Composite quantum systems).

The state of the composite system is a ray in the Hilbert space

\begin{equation}\tag{77.6} \mathcal{H}_T=\mathcal{H}_{1}\otimes\ldots\otimes\mathcal{H}_{N}\ep \end{equation}

Density-matrix postulates

[Heading reserved in the source outline; the reformulation of the postulates in terms of density matrices, covering statistical mixtures as well as pure states, is pending port from a completed source.]

On Planck's constant

[Heading reserved in the source outline; the discussion of Planck's constant, its role as the quantum of action, and its measured value is pending port from a completed source.]

Pictures of time evolution

The Schrödinger picture

[Heading reserved in the source outline; no content to port.]

The Heisenberg picture

[Heading reserved in the source outline; no content to port.]

Canonical quantization rules

From the Poisson bracket to the commutator

[Heading reserved in the source outline, together with its subheadings on conjugate coordinates and conserved quantities; the rule mapping classical Poisson brackets to quantum commutators, and with it the canonical commutation relations between conjugate coordinates, is pending port from a completed source.]

Derivation pending.

From Poisson brackets to commutators: statement of the canonical quantization rule and derivation of the canonical commutation relations between conjugate coordinates, which are used throughout the following chapters.

Linear momentum as a differential operator

In the coordinate representation, the components of the linear momentum act on wave functions as the differential operators

\begin{equation}\tag{77.7} p_i=-\ii\hbar\,\pp_i\ep \end{equation}

Derivation pending.

Justification of the coordinate-representation rule \(p_i=-\ii\hbar\,\pp_i\) from the canonical commutation relations; the source asserts the rule without derivation.

The Heisenberg uncertainty principle

Mean value of an observable

Definition 77.7 (Mean value).

Consider an observable \(A\). We define the mean (or expected) value of the operator in the state \(\ket{\psi}\) as

\begin{equation}\tag{77.8} \gen{A}_{\psi}=\bra{\psi}A\ket{\psi}\ep \end{equation}

Uncertainty

Definition 77.8 (Uncertainty).

In the same context as above, we define the uncertainty of the observable \(A\) in the state \(\ket{\psi}\) as

\begin{equation}\tag{77.9} \Delta A =\sqrt{\gen{\left(A-\gen{A}_{\psi}\identity\right)^2}_{\psi}}\ep \end{equation}

Let us observe the following. We have

\begin{align*} (\Delta A)^2 &=\gen{\left(A-\gen{A}_{\psi}\identity\right)^2}_{\psi}\\ &=\gen{\left(A-\gen{A}_{\psi}\identity\right) \left(A-\gen{A}_{\psi}\identity\right)}_{\psi}\\ &=\gen{A^2+\gen{A}^2_{\psi}\identity-2\gen{A}_{\psi}A}_{\psi}\\ &=\ev{\left(A^2+\gen{A}^2_{\psi}\identity -2\gen{A}_{\psi}A\right)}{\psi}\\ &=\ev{A^2}{\psi}+\gen{A}^2_{\psi}\braket{\psi}{\psi} -2\gen{A}_{\psi}\ev{A}{\psi}\\ &=\gen{A^2}_{\psi}+\gen{A}^2_{\psi}-2\gen{A}^2_{\psi}\ec \end{align*}

where we used the normalization Equation (77.1). Hence the uncertainty of \(A\) is given by

\begin{equation}\tag{77.10} \Delta A=\sqrt{\gen{A^2}_{\psi}-\gen{A}^2_{\psi}}\ep \end{equation}

The generalized uncertainty principle

[Heading reserved in the source outline.]

Derivation pending.

Statement and derivation of the generalized uncertainty relation bounding the product of the uncertainties of two non-commuting observables by the mean value of their commutator.