The 7 Basic Rules of Quantum Mechanics

For reference purposes and to help focus discussions on Physics Forums in interpretation questions on the real issues, there is a need for fixing the common ground. There is no consensus about the interpretation of quantum mechanics, and – not surprisingly – there is disagreement even among the mentors and science advisors here on Physics Forums. But the following formulation in terms of 7 basic rules of quantum mechanics was agreed upon among the science advisors of Physics Forums in a long and partially heated internal discussion on ”Best Practice to Handle Interpretations in Quantum Physics”, September 24 – October 29, 2017, based on a first draft by @atyy and several improved versions by  @tom.stoer. Other significant contributors to the discussions included @fresh_42,  @kith,  @stevendaryl, and @vanhees71.

I slightly expanded the final version and added headings and links to make it suitable as an Insight article. A revised version of this article is published as Section 1.1 of my recent book

• Coherent Quantum Physics: A Reinterpretation of the Tradition, de Gruyter, Berlin 2019.

The 7 Basic Rules

The basic rules reflect what is almost generally taught as the basics in quantum physics courses around the world. Often they are stated in terms of axioms or postulates, but this is not essential for their practical validity. In some interpretations, some of these rules are not considered fundamental rules but only valid as empirical or effective rules for practical purposes.

These rules describe the basis of quantum formalism and are found in almost all introductory quantum mechanics textbooks, among them: Basdevant 2016; Cohen-Tannoudji, Diu and Laloe 1977; Dirac 1930, 1967; Gasiorowicz 2003; Greiner 2008; Griffiths and Schroeter 2018; Landau and Lifshitz 1958, 1977; Liboff 2003; McIntyre 2012; Messiah 1961; Peebles 1992; Rae and Napolitano 2015; Sakurai 2010; Shankar 2016; Weinberg 2013. [Even Ballentine 1998, who rejects rule (7) = his process (9.9) as fundamental, derives it at the bottom of p.243 as an effective rule.] There are generalizations of these rules (e.g., Auletta, Fortunato, and Parisi 2009; Busch, Grabowski, and Lahti 2001; Nielsen and Chuang 2011) for degenerate eigenvalues, for mixed states, and for measurements not defined by self-adjoint operators but by POVMs. These generalizations are necessary to be able to apply quantum mechanics to all situations encountered in practice. The basic rules are carefully formulated so that they are correct as they stand and at the same time fully compatible with these generalizations.

When stating the rules, italic text corresponds to the physical systems, their preparation, measurement, measured values, etc.; non-italic text corresponds to mere mathematical objects that represent the physical system, etc.

1. A quantum system is described using a Hilbert space $\mathcal{H}$. Often, this Hilbert space is assumed to be separable.
2. A pure state of a quantum system is represented by a normalized vector $|\psi \rangle$ in $\mathcal{H}$; state vectors differing only by a phase factor of absolute value 1 represent the same state. In the position representation, where the Hilbert space is the space of square-integrable functions of a position vector $x$, $\psi(x)$ is called the wave function of the system.
3. The time evolution of an isolated quantum system represented by the state vector $|\psi(t)\rangle$ is given by
   $$\mathrm{i} \hbar\frac{\mathrm{d}}{\mathrm{d} t} |\psi(t) \rangle = H \, |\psi(t) \rangle$$
   where $H$ is the Hamilton operator and $\hbar$ is Planck’s constant. This is the Schrödinger equation.
   This rule is valid in the formulation of quantum mechanics called the Schrödinger picture. There are other, equivalent formulations of the time evolution, especially the Heisenberg picture and the Dirac (interaction) pictures, where time evolution is entirely or partially shifted from the state vector to the operators.
4. An observable of a quantum system is represented by a Hermitian operator $A$ with real spectrum acting on a dense subspace of $\mathcal{H}$.
5. The possible measured values of a measurement of an observable are the spectral values of the corresponding operator $A$. In the case of a discrete spectrum, these are the eigenvalues $a$ satisfying $A\, |a\rangle = a\, |a\rangle$.
6. Let $\{|a\rangle\}$ be a complete set of (generalized) eigenvectors of the self-adjoint operator $A$ with spectral values $a$. Let the quantum system be prepared in a state represented by the state vector $|\psi\rangle$. If a measurement of the observable corresponding to $A$ is performed, the probability $p_\psi(a)$ to find the measured value $a$ is given by
   $$p_\psi(a) = |\langle a | \psi\rangle|^2$$
   This is the Born rule, in a formulation that assumes that all eigenvalues are nondegenerate.
7. For successive, non-destructive projective measurements with discrete results, each measurement with measuring value $a$ can be regarded as preparation of a new state whose state vector is the corresponding eigenvector $|a\rangle$, to be used for the calculation of subsequent time evolution and further measurements. This is the von Neumann projection postulate.

Formal Comments

(2) To be precise, a pure state is not represented by a unit vector but by a unit ray, i.e. the equivalence class $$[\psi] = e^{\mathrm{i} \varphi}  |\psi \rangle$$ with $\varphi \in \mathbb{R}$ and $|\psi \rangle$ being a normalized vector in $\mathcal{H}$.

Equivalently, a pure state can be represented by a rank 1 density operator $\rho=|\psi \rangle\langle\psi|$ satisfying $\rho^2=\rho=\rho^*$ and $Tr~\rho=1$. Mixed states are represented by more general (non-idempotent) Hermitian density operators of trace 1.

(3) It is equivalent to define the time evolution of an isolated quantum system by $$|\psi(t)\rangle = U(t)\,|\psi(0)\rangle$$

with the unitary time evolution operator

$$U(t) = e^{-iHt/\hbar}.$$

The evolution according to (3) is therefore also referred to as unitary evolution.

(4) Equivalently, $A$ is self-adjoint.

(6) In case of degenerate subspaces, let $\{|a,\nu \rangle\}$ be a complete set of (generalized) eigenvectors of $A$, indexed by $\nu$. The probability $p_\psi(a)$ to find the measured value $a$ is then given by summing (or integrating) over $\nu$ i.e. over the entire $a$-subspace

$$p_\psi(a) = \sum_\nu |\langle a,\nu | \psi\rangle|^2.$$

(7) The projection postulate is valid only under the assumptions stated; examples are passing barriers with holes or slits, polarization filters, and certain other instruments that modify the state of a quantum system passing through it. This (nonunitary, dissipative) change of the state to an eigenstate in the course of a projective measurement is often referred to as ”state reduction”, ”collapse of the wave function” or ”reduction of the wave packet”. Note that there is no direct conflict with the unitary evolution in (3) since, during a measurement, a system is never isolated.

In other cases, the prepared state may be quite different. (See the discussion in Landau and Lifschitz, Vol. III, Section 7.) The most general kind of quantum measurement and the resulting prepared state is described by so-called positive operator valued measures (POVMs).

Comments on the Interpretation

Not further discussing the foundations of quantum mechanics beyond this is called shut-up-and-calculate. It is the mode of working sufficiently for all who do not want to delve into often highly disputed foundational (and partly philosophical) problems. However, the above-mentioned rules are often considered conceptually unsatisfactory because they introduce not well-defined terms ‘probability’, ‘measurement’, and ‘observer’ to define these basic rules whereas in principle one expects that at least measurement and observation can be regarded as quantum mechanical processes or interactions which follow the same fundamental rules and do not play any special role. The associated issues are treated in different ways by different interpretations of quantum mechanics.

In the Copenhagen Interpretation (also called Standard Interpretation or Orthodox Interpretation; terminology and interpretation details vary), the above rules are simply operational rules that work in practice.  A state vector is a tool that one uses to calculate the probabilities of measurement outcomes, and one is agnostic about whether the state vector represents any object that exists in reality.  Rules (6) and (7) apply only when a measurement has occurred. Thus unlike in classical physics, it is not enough to specify the initial conditions of the state and let the state evolve.  One must also specify when a measurement has occurred: Generally, a measurement is understood to have occurred when a definite (irreversible, i.e., nonunitary) measurement result or outcome has been obtained; e.g., the observer records a mark on a screen. (However passing a Stern-Gerlach magnet – which in modern terminology is a premeasurement only – is frequently but inaccurately considered to be a measurement, although it is described by a unitary process where even in principle no measurement result becomes available.)

A noteworthy aspect of the standard interpretation is that the state vector cannot represent the whole universe, but must exclude an observer or measuring apparatus that decides when a measurement has occurred; this is the so-called Heisenberg cut between the quantum and the classical world. To date, this has not been a problem in making successful experimental predictions, so practitioners are often satisfied with quantum formalism and the standard interpretation.

However, many have suggested that there is a conceptual problem with the standard interpretation because the whole universe presumably obeys the laws of physics. So there should be laws of physics that describe the whole universe, without any need to exclude any observer or measurement apparatus from the quantitative description. Then one must be able to derive the rules (5)-(7) for measuring subsystems of the universe from the dynamics of the universe. The problem of how to do this is called the measurement problem. A related problem, the problem of the emergence of a classical macroscopic world from the microscopic quantum description, is often considered as essentially solved by decoherence.

To solve the measurement problem, other interpretations of quantum formalism or theories have been proposed. These alternative interpretations or theories are based on different postulates than those of the standard interpretation, but seek to explain why the standard interpretation has been so successful (e.g., by deriving the rules of the standard interpretation from other postulates). The major alternative interpretations or theories that have been proposed include Everett’s Relative State Interpretation (“Many-Worlds”), the Ensemble Interpretation (or Minimal Statistical Interpretation), the Transactional Interpretation, and the Consistent Histories Interpretation.

Still, other interpretations (e.g., Bohmian Mechanics, Ghirardi–Rimini–Weber theory, the Cellular Automaton Interpretation, and the Thermal Interpretation) modify one or more of the 7 basic rules and only strive to derive the latter in some approximation, for all practical purposes (FAPP). In particular, rule (7) cannot be fundamental if one wants to interpret the state vector $|\psi\rangle$ in an ontic way, i.e., as some direct and ‘faithful’ representation of ‘externally existing reality independent from any observer, observation or measurement.

None of the interpretations currently available has been able to solve the measurement problem in a way deemed satisfactory by those interested in the foundations. So there are still major open problems both with the standard interpretation of quantum mechanics and with alternative interpretations. Fortunately, none of these problems seems to be of any practical relevance.

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