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{{Quantum book backlink|Conceptual and interpretations}}

The '''Copenhagen interpretation''' is the historically earliest and most widely taught interpretation of [[Wikipedia:Quantum mechanics|quantum mechanics]]. It was developed primarily by [[Wikipedia:Niels Bohr|Niels Bohr]] and [[Wikipedia:Werner Heisenberg|Werner Heisenberg]] in the 1920s.<ref>{{cite book |last=Heisenberg |first=Werner |title=Physics and Philosophy: The Revolution in Modern Science |publisher=Harper & Row |year=1958}}</ref>

In this interpretation, the state of a quantum system is described by a wavefunction <math>\psi</math>, which encodes the probabilities of measurement outcomes. Physical quantities do not have definite values prior to measurement; instead, they are defined only in terms of the experimental context.<ref>{{cite book |last=Bohr |first=Niels |title=Atomic Theory and the Description of Nature |publisher=Cambridge University Press |year=1934}}</ref>
[[File:Interpretations of Quantum Mechanics2.jpg|thumb|450px|Interpretations of Quantum Mechanics: From Observation to Reality]]
== Wavefunction and probability ==

The wavefunction provides probability amplitudes. The probability of observing a result associated with a state <math>\phi</math> is given by

<math>
|\langle \phi \mid \psi \rangle|^2,
</math>

known as the [[Wikipedia:Born rule|Born rule]].<ref>{{cite journal |last=Born |first=Max |title=Zur Quantenmechanik der Stoßvorgänge |journal=Zeitschrift für Physik |volume=37 |year=1926}}</ref>

== Measurement and collapse ==

A central postulate of the Copenhagen interpretation is the '''collapse of the wavefunction'''. Upon measurement, the system transitions from a superposition of states to a single outcome:

<math>
\psi \rightarrow \psi_n,
</math>

with probability determined by the coefficients in the expansion of <math>\psi</math>.

This collapse is not described by the Schrödinger equation and is introduced as an additional postulate.<ref>{{cite book |last=Dirac |first=P. A. M. |title=The Principles of Quantum Mechanics |publisher=Oxford University Press |year=1930}}</ref>

== Complementarity ==

Bohr introduced the principle of '''complementarity''', which states that quantum systems exhibit mutually exclusive properties depending on the experimental setup. For example, light may display wave-like or particle-like behavior, but not both simultaneously.<ref>{{cite book |last=Bohr |first=Niels |title=Atomic Physics and Human Knowledge |publisher=John Wiley & Sons |year=1958}}</ref>

== Classical–quantum divide ==

The Copenhagen interpretation assumes a distinction between:

* the quantum system (described by a wavefunction), and
* the classical measuring apparatus (described by classical physics).

This division is not sharply defined and is one of the conceptual challenges of the interpretation.

== Criticism and significance ==

The Copenhagen interpretation has been criticized for:

* its reliance on measurement as a fundamental concept,
* the lack of a precise definition of wavefunction collapse,
* the ambiguity of the classical–quantum boundary.

Despite these issues, it remains the standard framework used in most practical applications of quantum mechanics.

=Many-worlds=

The '''many-worlds interpretation''' (MWI) of quantum mechanics was proposed by [[Wikipedia:Hugh Everett III|Hugh Everett III]] in 1957 as an alternative to the Copenhagen interpretation.<ref>{{cite journal |last=Everett |first=Hugh |title="Relative State" Formulation of Quantum Mechanics |journal=Reviews of Modern Physics |volume=29 |year=1957}}</ref>

In this interpretation, the wavefunction <math>\psi</math> is taken to be a complete description of reality and evolves at all times according to the Schrödinger equation, without collapse.

== No wavefunction collapse ==

Unlike the Copenhagen interpretation, the many-worlds interpretation does not introduce a collapse postulate. Instead, all possible outcomes of a quantum measurement are realized in different branches of the universal wavefunction.

If a system is in a superposition

<math>
\psi = \sum_n c_n \psi_n,
</math>

then after interaction with a measuring apparatus, the combined system evolves into

<math>
\Psi = \sum_n c_n \, \psi_n \otimes A_n,
</math>

where <math>A_n</math> represents the apparatus recording outcome <math>n</math>.

Each term corresponds to a different “branch” of reality.

== Branching and worlds ==

In the many-worlds interpretation, measurement leads to a branching of the universe into non-interacting components. Each branch contains a definite outcome, and observers within each branch perceive a single result.

This branching is a consequence of unitary evolution and does not require any additional postulates.<ref>{{cite book |last=DeWitt |first=Bryce S. |last2=Graham |first2=Neill |title=The Many-Worlds Interpretation of Quantum Mechanics |publisher=Princeton University Press |year=1973}}</ref>

== Probability and the Born rule ==

A major question for the many-worlds interpretation is how to recover probabilities, since all outcomes occur.

The standard approach is to interpret the coefficients <math>|c_n|^2</math> as measures of branch weight, leading effectively to the [[Wikipedia:Born rule|Born rule]].<ref>{{cite book |last=Wallace |first=David |title=The Emergent Multiverse |publisher=Oxford University Press |year=2012}}</ref>

== Decoherence ==

The apparent classical behavior of measurement outcomes is explained by [[Wikipedia:Quantum decoherence|decoherence]], which suppresses interference between different branches of the wavefunction.<ref>{{cite book |last=Schlosshauer |first=Maximilian |title=Decoherence and the Quantum-to-Classical Transition |publisher=Springer |year=2007}}</ref>

Decoherence explains why branches evolve independently and why observers do not perceive superpositions at the macroscopic level.

== Interpretation and criticism ==

The many-worlds interpretation is conceptually appealing because it:

* removes the need for wavefunction collapse,
* treats quantum evolution as universally valid,
* provides a deterministic description of quantum processes.

However, it has been criticized for:

* introducing a large (possibly infinite) number of unobservable branches,
* difficulties in interpreting probability,
* questions about the ontology of the wavefunction.

Despite these issues, it is widely studied in foundations of quantum mechanics and quantum cosmology.
=Bohmian mechanics=

'''Bohmian mechanics''', also known as the '''pilot-wave theory''' or '''de Broglie–Bohm theory''', is a deterministic interpretation of quantum mechanics in which particles have well-defined positions at all times, guided by a wavefunction.<ref>{{cite journal |last=Bohm |first=David |title=A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables |journal=Physical Review |volume=85 |year=1952}}</ref>

The theory was originally proposed by [[Wikipedia:Louis de Broglie|Louis de Broglie]] in 1927 and later developed in detail by [[Wikipedia:David Bohm|David Bohm]].<ref>{{cite book |last=Holland |first=Peter R. |title=The Quantum Theory of Motion |publisher=Cambridge University Press |year=1993}}</ref>

== Deterministic dynamics ==

In Bohmian mechanics, a system is described by:

* a wavefunction <math>\psi(x,t)</math>, evolving according to the Schrödinger equation, and
* particle positions <math>x(t)</math>, which evolve according to a guiding equation.

The velocity of a particle is given by

<math>
\frac{dx}{dt} = \frac{\hbar}{m} \, \mathrm{Im} \left( \frac{\nabla \psi}{\psi} \right),
</math>

which depends on the wavefunction.

Thus, unlike standard quantum mechanics, the theory provides definite trajectories for particles.

== Quantum potential ==

An equivalent formulation introduces the '''quantum potential'''. Writing the wavefunction in polar form

<math>
\psi = R e^{iS/\hbar},
</math>

one obtains a modified Hamilton–Jacobi equation with an additional term:

<math>
Q = -\frac{\hbar^2}{2m} \frac{\nabla^2 R}{R}.
</math>

This quantum potential governs non-classical behavior.

== Hidden variables ==

Bohmian mechanics is a '''hidden-variable theory''', meaning that it supplements the wavefunction with additional variables (particle positions) that determine measurement outcomes.

These variables are not directly observable but evolve deterministically.

== Nonlocality ==

A key feature of Bohmian mechanics is '''nonlocality'''. The motion of one particle can depend instantaneously on the configuration of other distant particles through the wavefunction.<ref>{{cite book |last=Bell |first=John S. |title=Speakable and Unspeakable in Quantum Mechanics |publisher=Cambridge University Press |year=1987}}</ref>

This nonlocality is consistent with [[Wikipedia:Bell's theorem|Bell’s theorem]], which shows that no local hidden-variable theory can reproduce all quantum predictions.

== Agreement with quantum mechanics ==

Bohmian mechanics reproduces all standard predictions of quantum mechanics when the distribution of particle positions satisfies

<math>
\rho(x) = |\psi(x)|^2.
</math>

This condition is known as '''quantum equilibrium'''.

== Interpretation and criticism ==

Bohmian mechanics provides:

* a clear ontology (particles with definite positions),
* deterministic evolution,
* an explicit account of measurement without collapse.

However, it is often criticized for:

* requiring nonlocal interactions,
* introducing additional (hidden) variables,
* being less compatible with relativistic quantum field theory.

Despite these issues, it remains an important alternative interpretation and is widely studied in the foundations of quantum mechanics.
=Measurement problem=

The '''measurement problem''' is a central conceptual issue in [[Wikipedia:Quantum mechanics|quantum mechanics]], concerning how and why definite outcomes arise from quantum systems described by superpositions.<ref>{{cite book |last=von Neumann |first=John |title=Mathematical Foundations of Quantum Mechanics |publisher=Princeton University Press |year=1955}}</ref>

According to the standard formalism, a system evolves deterministically according to the Schrödinger equation, yet measurements yield single, definite results.

== Superposition and outcome ==

A quantum system may exist in a superposition of states,

<math>
\psi = \sum_n c_n \psi_n,
</math>

where each <math>\psi_n</math> corresponds to a possible outcome. However, when a measurement is performed, only one outcome is observed.

This raises the question:

> How does a single outcome emerge from a superposition?

== Collapse postulate ==

In the Copenhagen interpretation, this is addressed by introducing the '''collapse of the wavefunction''':

<math>
\psi \rightarrow \psi_n,
</math>

with probability <math>|c_n|^2</math>.

However, this collapse is not described by the Schrödinger equation and appears as an additional, non-dynamical postulate.<ref>{{cite book |last=Dirac |first=P. A. M. |title=The Principles of Quantum Mechanics |publisher=Oxford University Press |year=1930}}</ref>

== System–apparatus interaction ==

When a quantum system interacts with a measuring device, the combined system evolves into an entangled state:

<math>
\Psi = \sum_n c_n \, \psi_n \otimes A_n,
</math>

where <math>A_n</math> represents different states of the apparatus.

This evolution alone does not select a single outcome, leading to the core of the measurement problem.

== Decoherence ==

[[Wikipedia:Quantum decoherence|Decoherence]] provides a partial resolution by explaining how interference between different components of a superposition becomes negligible due to interaction with the environment.<ref>{{cite book |last=Schlosshauer |first=Maximilian |title=Decoherence and the Quantum-to-Classical Transition |publisher=Springer |year=2007}}</ref>

Decoherence explains:

* why classical behavior emerges,
* why different outcomes do not interfere.

However, it does not explain why a single outcome is observed.

== Interpretational responses ==

Different interpretations of quantum mechanics resolve the measurement problem in different ways:

* '''Copenhagen interpretation''' — introduces wavefunction collapse.
* '''Many-worlds interpretation''' — all outcomes occur in separate branches.
* '''Bohmian mechanics''' — definite particle positions determine outcomes.

Each approach modifies or supplements the standard formalism to account for observed results.

== Significance ==

The measurement problem highlights a tension between:

* the linear, deterministic evolution of quantum states, and
* the probabilistic, definite outcomes observed in experiments.

It remains one of the most fundamental unresolved issues in the foundations of quantum theory and continues to motivate research in quantum foundations, quantum information, and quantum cosmology.
=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}
{{Author|Harold Foppele}}

{{Sourceattribution|Quantum Interpretations of quantum mechanics|1}}

Physics:Quantum Interpretations of quantum mechanics - Revision history

imported>WikiHarold at 00:12, 11 May 2026

imported>WikiHarold at 00:12, 11 May 2026