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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Complementarity (physics) | 2/3 | https://en.wikipedia.org/wiki/Complementarity_(physics) | reference | science, encyclopedia | 2026-05-05T13:31:23.741981+00:00 | kb-cron |
In the traditional view, it is assumed that there exists a reality in space-time and that this reality is a given thing, all of whose aspects can be viewed or articulated at any given moment. Bohr was the first to point out that quantum mechanics called this traditional outlook into question. To him the "indivisibility of the quantum of action" [...] implied that not all aspects of a system can be viewed simultaneously. By using one particular piece of apparatus only certain features could be made manifest at the expense of others, while with a different piece of apparatus another complementary aspect could be made manifest in such a way that the original set became non-manifest, that is, the original attributes were no longer well defined. For Bohr, this was an indication that the principle of complementarity, a principle that he had previously known to appear extensively in other intellectual disciplines but which did not appear in classical physics, should be adopted as a universal principle.
=== Debate following the lectures ===
Complementarity was a central feature of Bohr's reply to the EPR paradox, an attempt by Albert Einstein, Boris Podolsky and Nathan Rosen to argue that quantum particles must have position and momentum even without being measured and so quantum mechanics must be an incomplete theory. The thought experiment proposed by Einstein, Podolsky and Rosen involved producing two particles and sending them far apart. The experimenter could choose to measure either the position or the momentum of one particle. Given that result, they could in principle make a precise prediction of what the corresponding measurement on the other, faraway particle would find. To Einstein, Podolsky and Rosen, this implied that the faraway particle must have precise values of both quantities whether or not that particle is measured in any way. Bohr argued in response that the deduction of a position value could not be transferred over to the situation where a momentum value is measured, and vice versa. Later expositions of complementarity by Bohr include a 1938 lecture in Warsaw and a 1949 article written for a festschrift honoring Albert Einstein. It was also covered in a 1953 essay by Bohr's collaborator Léon Rosenfeld.
== Mathematical formalism == For Bohr, complementarity was the "ultimate reason" behind the uncertainty principle. All attempts to grapple with atomic phenomena using classical physics were eventually frustrated, he wrote, leading to the recognition that those phenomena have "complementary aspects". But classical physics can be generalized to address this, and with "astounding simplicity", by describing physical quantities using non-commutative algebra. This mathematical expression of complementarity builds on the work of Hermann Weyl and Julian Schwinger, starting with Hilbert spaces and unitary transformation, leading to the theorems of mutually unbiased bases. In the mathematical formulation of quantum mechanics, physical quantities that classical mechanics had treated as real-valued variables become self-adjoint operators on a Hilbert space. These operators, called "observables", can fail to commute, in which case they are called "incompatible":
[
A
^
,
B
^
]
:=
A
^
B
^
−
B
^
A
^
≠
0
^
.
{\displaystyle \left[{\hat {A}},{\hat {B}}\right]:={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}\neq {\hat {0}}.}
Incompatible observables cannot have a complete set of common eigenstates; there can be some simultaneous eigenstates of
A
^
{\displaystyle {\hat {A}}}
and
B
^
{\displaystyle {\hat {B}}}
, but not enough in number to constitute a complete basis. The canonical commutation relation
[
x
^
,
p
^
]
=
i
ℏ
{\displaystyle \left[{\hat {x}},{\hat {p}}\right]=i\hbar }
implies that this applies to position and momentum. In a Bohrian view, this is a mathematical statement that position and momentum are complementary aspects. Likewise, an analogous relationship holds for any two of the spin observables defined by the Pauli matrices; measurements of spin along perpendicular axes are complementary. The Pauli spin observables are defined for a quantum system described by a two-dimensional Hilbert space; mutually unbiased bases generalize these observables to Hilbert spaces of arbitrary finite dimension. Two bases
{
|
a
j
⟩
}
{\displaystyle \{|a_{j}\rangle \}}
and
{
|
b
k
⟩
}
{\displaystyle \{|b_{k}\rangle \}}
for an
N
{\displaystyle N}
-dimensional Hilbert space are mutually unbiased when
|
⟨
a
j
|
b
k
⟩
|
2
=
1
N
for all
j
,
k
=
1
,
.
.
.
N
−
1.
{\displaystyle |\langle a_{j}|b_{k}\rangle |^{2}={\frac {1}{N}}\ {\text{for all}}\ j,k=1,...N-1.}
Here the basis vector
a
1
{\displaystyle a_{1}}
, for example, has the same overlap with every
b
k
{\displaystyle b_{k}}
; there is equal transition probability between a state in one basis and any state in the other basis. Each basis corresponds to an observable, and the observables for two mutually unbiased bases are complementary to each other. This leads to a definition of 'Principle of Complementarity' as:
For each degree of freedom the dynamical variables are a pair of complementary observables.The concept of complementarity has also been applied to quantum measurements described by positive-operator-valued measures (POVMs).
== Continuous complementarity ==
While the concept of complementarity can be discussed via two experimental extremes, continuous tradeoff is also possible. In 1979 Wooters and Zurek introduced an information-theoretic treatment of the double-slit experiment providing on approach to a quantiative model of complementarity. The wave-particle relation, introduced by Daniel Greenberger and Allaine Yasin in 1988, and since then refined by others, quantifies the trade-off between measuring particle path distinguishability,
D
{\displaystyle D}
, and wave interference fringe visibility,
V
{\displaystyle V}
:
D
2
+
V
2
≤
1
{\displaystyle D^{2}+V^{2}\ \leq \ 1}
The values of
D
{\displaystyle D}
and
V
{\displaystyle V}
can vary between 0 and 1 individually, but any experiment that combines particle and wave detection will limit one or the other, or both. The detailed definition of the two terms vary among applications, but the relation expresses the verified constraint that efforts to detect particle paths will result in less visible wave interference.