Observable

In quantum mechanics, an observable refers to any physical property of a system that can be experimentally measured, including position, momentum, energy, or spin. Conceptually, an observable represents a question we can ask about the state of a system, with the outcome of a measurement providing the answer.

Fundamentals of Quantum Mechanics

Article headings

1. In Depth Insights

Measurement outcomes of physical quantities are always real and distinguishable values. For instance, when measuring the position of a particle, real and distinct values

x ∈ ℝ

are observed. In quantum mechanics, according to the second postulate, every physically measurable dynamical variable (observable) is associated with a Hermitian operator. The solutions to the eigenvalue equation of this operator determine both the allowed outcomes of the measurement and the corresponding quantum states. The requirement that operators representing physical observables

A

be Hermitian directly reflects the physical necessity that measurement results must be real and distinguishable.

2. Mathematical Representation

Consider the eigenvalue eigenvector equation for a Hermitian operator

Â

in Dirac notation:

\hat{A}|a_n\rangle=a_n|a_n\rangle

where the eigenvectors

|aₙ⟩

and their corresponding eigenvalues

a

ₙ represent the solutions to this equation. In quantum mechanics, the eigenvalues

aₙ

correspond to the possible measurement outcomes, and the associated eigenvectors

|aₙ⟩

represent the quantum state of the particle immediately following the observation of result

aₙ.

It can be proven that the Hermiticity of operator

Â

ensures that its eigenvalues

aₙ

are strictly real and its eigenvectors

|aₙ⟩

are mutually orthogonal. Consequently, the Hermiticity of operators corresponding to physical observables

A

is mathematically equivalent to the reality and distinguishability of experimental measurement outcomes.

A comprehensive understanding of the measurement process in quantum mechanics emerges from combining the first, second, and third postulates. If the physical state of a quantum system is described by a vector

|ψ⟩

in Hilbert space, then upon measuring observable

A

, the outcome

aₙ

is observed with probability:

P(a_n)=|\langle a_n|\psi\rangle|^2

Following the measurement, the state collapses to

|aₙ⟩.

The set of all possible values

aₙ

constitutes the spectrum of operator

Â

, and the matrix representation of

Â

in its eigenbasis takes the following diagonal form:

\hat{A}=\sum_n a_n|a_n\rangle\langle a_n|

3. Key Principles

First Postulate of Quantum Mechanics

The physical states of a quantum system are represented by state vectors residing in a complex Hilbert space.

Second Postulate of Quantum Mechanics

Every physical observable in quantum mechanics is represented by a linear Hermitian operator acting on the Hilbert space.

Third Postulate of Quantum Mechanics

If the state of a quantum system is represented by the vector

|ψ⟩

, a measurement of observable

A

yields the eigenvalue

aₙ

with probability:

P(a_n)=|\langle a_n|\psi\rangle|^2

Immediately upon measurement, the state vector collapses into the corresponding eigenvector

|aₙ⟩.

4. References

McIntyre, D. H. (2012). Quantum mechanics: A paradigms approach. Pearson Education, Inc., 1, 97.

Related concepts

The Schrödinger Equation

The Schrödinger equation represents the mathematical formulation governing the time evolution of a quantum system. It establishes that the total energy of a system uniquely determines its physical dynamics and state trajectory. In conceptual terms, the equation serves as a governing rule indicating that knowledge of a quantum system's energy allows the determination of its future quantum states.

Heisenberg's Uncertainty Principle

اصل عدم قطعیت هایزنبرگ بیان می‌کند که اندازه‌گیری دقیق همزمان مکان و تکانه یک ذره امکان‌پذیر نیست. هرچه یکی از این دو را با دقت بیشتری بدانیم، دقت دانسته ما درباره دیگری کمتر خواهد بود. اصل عدم قطعیت هایزنبرگ مفهومی بنیادی در مکانیک کوانتومی است و نشان‌دهنده محدودیت‌های اندازه‌گیری و دانش در مقیاس‌های بسیار کوچک است.

Hilbert Space

One of the essential differences between classical and quantum mechanics stems from the nature of physical properties. Quantum mechanics exhibits discrete quantities and inherently probabilistic measurement results. To mathematically formulate these distinct features, physicists employ Hilbert spaces. The state vectors residing in this space define the physical system, allowing the precise computation of measurement outcome probabilities and the expectation values of physical observables.

Quantum Superposition

Quantum superposition simply means that something can exist in multiple states concurrently, like a light being both on and off at once. When we observe it, we find only one definite state, yet until that observation occurs, both states exist together in potential.

Wave Function

The wave function represents the mathematical formalism used to describe the quantum state of a particle or a collection of particles. Rather than specifying an exact spatial coordinate, the wave function contains probabilistic information that dictates the likelihood of discovering the particle at a particular location or configuration upon measurement.

1. In Depth Insights

Measurement outcomes of physical quantities are always real and distinguishable values. For instance, when measuring the position of a particle, real and distinct values

x ∈ ℝ

A

be Hermitian directly reflects the physical necessity that measurement results must be real and distinguishable.

2. Mathematical Representation

Consider the eigenvalue eigenvector equation for a Hermitian operator

Â

in Dirac notation:

\hat{A}|a_n\rangle=a_n|a_n\rangle

where the eigenvectors

|aₙ⟩

and their corresponding eigenvalues

a

ₙ represent the solutions to this equation. In quantum mechanics, the eigenvalues

aₙ

correspond to the possible measurement outcomes, and the associated eigenvectors

|aₙ⟩

represent the quantum state of the particle immediately following the observation of result

aₙ.

It can be proven that the Hermiticity of operator

Â

ensures that its eigenvalues

aₙ

are strictly real and its eigenvectors

|aₙ⟩

are mutually orthogonal. Consequently, the Hermiticity of operators corresponding to physical observables

A

is mathematically equivalent to the reality and distinguishability of experimental measurement outcomes.

|ψ⟩

in Hilbert space, then upon measuring observable

A

, the outcome

aₙ

is observed with probability:

P(a_n)=|\langle a_n|\psi\rangle|^2

Following the measurement, the state collapses to

|aₙ⟩.

The set of all possible values

aₙ

constitutes the spectrum of operator

Â

, and the matrix representation of

Â

in its eigenbasis takes the following diagonal form:

\hat{A}=\sum_n a_n|a_n\rangle\langle a_n|

3. Key Principles

First Postulate of Quantum Mechanics

The physical states of a quantum system are represented by state vectors residing in a complex Hilbert space.

Second Postulate of Quantum Mechanics

Every physical observable in quantum mechanics is represented by a linear Hermitian operator acting on the Hilbert space.

Third Postulate of Quantum Mechanics

If the state of a quantum system is represented by the vector

|ψ⟩

, a measurement of observable

A

yields the eigenvalue

aₙ

with probability:

P(a_n)=|\langle a_n|\psi\rangle|^2

Immediately upon measurement, the state vector collapses into the corresponding eigenvector

|aₙ⟩.

4. References

McIntyre, D. H. (2012). Quantum mechanics: A paradigms approach. Pearson Education, Inc., 1, 97.

Observable

Article headings

1. In Depth Insights

2. Mathematical Representation

3. Key Principles

First Postulate of Quantum Mechanics

Second Postulate of Quantum Mechanics

Third Postulate of Quantum Mechanics

4. References

Related concepts

The Schrödinger Equation

Heisenberg's Uncertainty Principle

Hilbert Space

Quantum Superposition

Wave Function

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Observable

Article headings

1. In Depth Insights

2. Mathematical Representation

3. Key Principles

First Postulate of Quantum Mechanics

Second Postulate of Quantum Mechanics

Third Postulate of Quantum Mechanics

4. References

Related concepts

The Schrödinger Equation

Heisenberg's Uncertainty Principle

Hilbert Space

Quantum Superposition

Wave Function