In quantum mechanics, wave function collapse occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an observation, and is the essence of a measurement in quantum mechanics, which connects the wave function with classical observables such as position and momentum. Collapse is one of the two processes by which quantum systems evolve in time; the other is the continuous evolution governed by the Schrödinger equation. Collapse is a black box for a thermodynamically irreversible interaction with a classical environment.

Calculations of quantum decoherence show that when a quantum system interacts with the environment, the superpositions apparently reduce to mixtures of classical alternatives. Significantly, the combined wave function of the system and environment continue to obey the Schrödinger equation throughout this apparent collapse. More importantly, this is not enough to explain actual wave function collapse, as decoherence does not reduce it to a single eigenstate.

Historically, Werner Heisenberg was the first to use the idea of wave function reduction to explain quantum measurement.

Mathematical description
Before collapsing, the wave function may be any square-integrable function, and is therefore associated with the probability density of a quantum-mechanical system. This function is expressible as a linear combination of the eigenstates of any observable. Observables represent classical dynamical variables, and when one is measured by a classical observer, the wave function is projected onto a random eigenstate of that observable. The observer simultaneously measures the classical value of that observable to be the eigenvalue of the final state.

Mathematical background

The quantum state of a physical system is described by a wave function (in turn—an element of a projective Hilbert space). This can be expressed as a vector using Dirac or bra–ket notation :

The kets  specify the different quantum "alternatives" available—a particular quantum state. They form an orthonormal eigenvector basis, formally

where  represents the Kronecker delta.

An observable (i.e. measurable parameter of the system) is associated with each eigenbasis, with each quantum alternative having a specific value or eigenvalue, , of the observable. A "measurable parameter of the system" could be the usual position  and the momentum  of (say) a particle, but also its energy ,  components of spin (), orbital () and total angular () momenta, etc. In the basis representation these are respectively 

The coefficients  are the probability amplitudes corresponding to each basis . These are complex numbers. The moduli square of , that is  (where  denotes complex conjugate), is the probability of measuring the system to be in the state 

For simplicity in the following, all wave functions are assumed to be normalized; the total probability of measuring all possible states is one:

The process of collapse 
With these definitions it is easy to describe the process of collapse. For any observable, the wave function is initially some linear combination of the eigenbasis  of that observable. When an external agency (an observer, experimenter) measures the observable associated with the eigenbasis  the wave function collapses from the full  to just one of the basis eigenstates,  that is:

 

The probability of collapsing to a given eigenstate  is the Born probability, . Immediately post-measurement, other elements of the wave function vector, , have "collapsed" to zero, and  

More generally, collapse is defined for an operator  with eigenbasis . If the system is in state , and  is measured, the probability of collapsing the system to eigenstate  and measuring the eigenvalue  of  with respect to  would be . Note that this is not the probability that the particle is in state ; it is in state  until cast to an eigenstate of .

However, we never observe collapse to a single eigenstate of a continuous-spectrum operator (e.g. position, momentum, or a scattering Hamiltonian), because such eigenfunctions are non-normalizable. In these cases, the wave function will partially collapse to a linear combination of "close" eigenstates (necessarily involving a spread in eigenvalues) that embodies the imprecision of the measurement apparatus. The more precise the measurement, the tighter the range. Calculation of probability proceeds identically, except with an integral over the expansion coefficient . This phenomenon is unrelated to the uncertainty principle, although increasingly precise measurements of one operator (e.g. position) will naturally homogenize the expansion coefficient of wave function with respect to another, incompatible operator (e.g. momentum), lowering the probability of measuring any particular value of the latter.

Quantum decoherence

Quantum decoherence explains why a system interacting with an environment transitions from being a pure state, exhibiting superpositions, to a mixed state, an incoherent combination of classical alternatives.  This transition is fundamentally reversible, as the combined state of system and environment is still pure, but for all practical purposes irreversible, as the environment is a very large and complex quantum system, and it is not feasible to reverse their interaction. Decoherence is thus very important for explaining the classical limit of quantum mechanics, but cannot explain wave function collapse, as all classical alternatives are still present in the mixed state, and wave function collapse selects only one of them.

History and context
The concept of wavefunction collapse was introduced by Werner Heisenberg in his 1927 paper on the uncertainty principle, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik", and incorporated into the mathematical formulation of quantum mechanics by John von Neumann, in his 1932 treatise Mathematische Grundlagen der Quantenmechanik. Heisenberg did not try to specify exactly what the collapse of the wavefunction meant. However, he emphasized that it should not be understood as a physical process.  Niels Bohr also repeatedly cautioned that we must give up a "pictorial representation", and perhaps also interpreted collapse as a formal, not physical, process.

Consistent with Heisenberg, von Neumann postulated that there were two processes of wave function change:

 The probabilistic, non-unitary, non-local, discontinuous change brought about by observation and measurement, as outlined above.
 The deterministic, unitary,  continuous time evolution of an isolated system that obeys the Schrödinger equation (or a relativistic equivalent, i.e. the Dirac equation).

In general, quantum systems exist in superpositions of those basis states that most closely correspond to classical descriptions, and, in the absence of measurement, evolve according to the Schrödinger equation. However, when a measurement is made, the wave function collapses—from an observer's perspective—to just one of the basis states, and the property being measured uniquely acquires the eigenvalue of that particular state, . After the collapse, the system again evolves according to the Schrödinger equation.

By explicitly dealing with the interaction of object and measuring instrument, von Neumann has attempted to create consistency of the two processes of wave function change.

He was able to prove the possibility of a quantum mechanical measurement scheme consistent with wave function collapse. However, he did not prove the necessity of such a collapse. Although von Neumann's projection postulate is often presented as a normative description of quantum measurement, it was conceived by taking into account experimental evidence available during the 1930s (in particular the Compton-Simon experiment was paradigmatic), but many important present-day measurement procedures do not satisfy it (so-called measurements of the second kind).

The existence of the wave function collapse is required in
 the Copenhagen interpretation
 the objective collapse interpretations
 the transactional interpretation
 the von Neumann–Wigner interpretation in which consciousness causes collapse.

On the other hand, the collapse is considered a redundant or optional approximation in
 the consistent histories approach, self-dubbed "Copenhagen done right"
 the Bohm interpretation
 the many-worlds interpretation
 the ensemble interpretation
 the relational quantum mechanics interpretation

The cluster of phenomena described by the expression wave function collapse is a fundamental problem in the interpretation of quantum mechanics, and is known as the measurement problem.

In the Copenhagen Interpretation collapse is postulated to be a special characteristic of interaction with classical systems (of which measurements are a special case). Mathematically it can be shown that collapse is equivalent to interaction with a classical system modeled within quantum theory as systems with Boolean algebras of observables and equivalent to a conditional expectation value.

Everett's many-worlds interpretation deals with it by discarding the collapse-process, thus reformulating the relation between measurement apparatus and system in such a way that the linear laws of quantum mechanics are universally valid; that is, the only process according to which a quantum system evolves is governed by the Schrödinger equation or some relativistic equivalent.

A general description of the evolution of quantum mechanical systems is possible by using density operators and quantum operations. In this formalism (which is closely related to the C*-algebraic formalism) the collapse of the wave function corresponds to a non-unitary quantum operation. Within the C* formalism this non-unitary process is equivalent to the algebra gaining a non-trivial centre or centre of its centralizer corresponding to classical observables.

The significance ascribed to the wave function varies from interpretation to interpretation, and varies even within an interpretation (such as the Copenhagen Interpretation). If the wave function merely encodes an observer's knowledge of the universe then the wave function collapse corresponds to the receipt of new information. This is somewhat analogous to the situation in classical physics, except that the classical "wave function" does not necessarily obey a wave equation. If the wave function is physically real, in some sense and to some extent, then the collapse of the wave function is also seen as a real process, to the same extent.

See also

 Arrow of time
 Interpretations of quantum mechanics
 Quantum decoherence
 Quantum interference
 Quantum Zeno effect
 Schrödinger's cat
 Stern–Gerlach experiment

Notes

References

External links 
 

Concepts in physics
Quantum measurement
양자역학에서 파동 함수 붕괴는 처음에는 여러 고유 상태의 중첩에서 파동 함수가 외부 세계와의 상호 작용으로 인해 단일 고유 상태로 될 때 발생한다. 이 상호 작용을 관찰이라고 하며 위치 및 운동량 과 같은 고전적인 관찰 가능한 항목과 파동 함수를 연결하는 양자 역학 측정 의 본질이다. 붕괴는 양자 시스템 이 시간에 따라 진화하는 두 가지 과정 중 하나이다. 다른 하나는 슈뢰딩거 방정식에 의해 지배되는 연속적인 진화이다. 붕괴는 고전적인 환경과 열역학적으로 비가역적인 상호작용을 위한 블랙박스이다.

양자 결어긋남 계산은 양자 시스템이 환경과 상호 작용할 때 중첩이 고전적 대안의 합으로 된다는 것을 보여준다. 중요하게도 시스템과 환경의 결합된 파동 함수는 이 명백한 붕괴 내내 슈뢰딩거 방정식을 계속 따른다. 더 중요한 것은 결어긋남이 단일 고유 상태로 감소하지 않기 때문에 실제 파동 함수 붕괴를 설명하기에 충분하지 않다는 것이다.

붕괴되기 전에 파동 함수는 제곱 적분 함수가 될 수 있으므로 양자 역학 시스템의 확률 밀도와 관련이 있다. 이 함수는 관찰 가능한 고유 상태 의 선형 조합으로 표현할 수 있다. 관찰가능량은 고전적인 동적 변수를 나타내며, 고전적인 관찰자에 의해 측정될 때 파동 함수는 해당 관찰 가능한 임의의 고유 상태에 투영 된다. 관찰자는 관찰 가능한 것의 고전적 값을 최종 상태의 고유값으로 동시에 측정한다.

물리적 시스템의 양자 상태는 파동 함수로 설명된다. 이것은 Dirac 또는 bra-ket 표기법 을 사용하여 벡터로 표현할 수 있다. :

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…{\displaystyle |\phi _{1}\rangle ,|\phi _{2}\rangle ,|\phi _{3}\rangle ,\dots }는 사용 가능한 다른 양자 "대안"(특정 양자 상태)을 지정한다. 그들은 공식적으로 직교 고유 벡터 기저를 형성한다.

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\delta _는 크로네커 델타를 나타낸다.

관찰 가능한(즉, 시스템의 측정 가능한 매개변수)는 각 고유 기준과 연관되며, 각 양자 대안은 특정 값 또는 고유값을 가진다.

다음의 단순화를 위해 모든 파동 함수는 정규화되었다고 가정한다. 가능한 모든 상태를 측정할 총 확률은 다음과 같다.

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이러한 정의를 통해 붕괴 과정을 쉽게 설명할 수 있다. 관측 가능한 모든 것에 대해 파동 함수는 초기에 고유 기저 
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{\displaystyle \{|\phi _{i}\rangle \}}의 일부 선형 조합이다. 외부 기관(관찰자, 실험자)이 고유 기준과 관련된 관찰 가능 항목을 측정할 때 
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{\displaystyle \{|\phi _{i}\rangle \}} 중 하나로 파동 함수는 붕괴한다.

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주어진 고유 상태로 붕괴될 확률 
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{\displaystyle P_{k}=|c_{k}|^{2}} . 측정 직후, 파동 함수 벡터의 다른 요소, 
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{\displaystyle c_{i\neq k}}, 0으로 "붕괴"되었다.

그러나 연속 스펙트럼 연산자(예: 위치, 운동량 또는 산란 해밀토니언)의 단일 고유 상태로의 붕괴는 관찰하지 않는다. 이러한 고유 함수는 정규화할 수 없기 때문이다. 이러한 경우 파동 함수는 측정 장치의 부정확성을 구현하는 "가까운" 고유 상태(필연적으로 고유값의 확산을 포함)의 선형 조합으로 부분적으로 붕괴된다.

양자 결어긋남은 환경과 상호작용하는 시스템이 중첩을 나타내는 순수 상태에서 고전적 대안의 일관성 없는 조합인 혼합 상태 로 전환하는 이유를 설명한다. 이 전환은 시스템과 환경의 결합 상태가 여전히 순수하기 때문에 근본적으로 가역적이지만 , 환경이 매우 크고 복잡한 양자 시스템이고 이들의 상호 작용을 되돌릴 수 없기 때문에 모든 실제적인 목적을 위해 되돌릴 수 없다. 따라서 결어긋남은 양자 역학의 고전적 한계를 설명하는 데 매우 중요하지만 모든 고전적 대안이 여전히 혼합 상태로 존재하고 파동 함수 붕괴는 그 중 하나만 선택하기 때문에 파동 함수 붕괴를 설명할 수 없다.

코펜하겐 해석에서 붕괴는 고전 시스템과의 상호 작용의 특별한 특성으로 가정된다. 수학적으로 붕괴는 관측 가능 항목의 부울 대수가 있는 시스템으로 양자 이론 내에서 모델링된 고전 시스템과의 상호 작용과 동일하고 조건부 기대 값과 동일하다는 것을 보여줄 수 있다.

에버렛 의 다세계 해석은 붕괴 과정을 버리고 양자 역학의 선형 법칙이 보편적으로 유효하도록 측정 장치와 시스템 사이의 관계를 재구성함으로써 이를 처리한다. 즉, 양자 시스템이 진화하는 유일한 과정은 슈뢰딩거 방정식 또는 일부 상대론적 등가물의 지배를 받는다.

파동 함수에 부여된 의미는 해석에 따라 다르며 해석(예: 코펜하겐 해석) 내에서도 다르다. 파동 함수가 우주에 대한 관찰자의 지식을 단순히 인코딩한다면 파동 함수 붕괴는 새로운 정보의 수신에 해당한다. 이것은 고전적인 "파동 함수"가 반드시 파동 방정식을 따르지 않는다는 점을 제외하고는 고전 물리학의 상황과 다소 유사하다. 파동 함수가 어떤 의미에서 어느 정도 물리적으로 실제라면 파동 함수의 붕괴도 같은 정도로 실제 과정으로 간주된다.