Observable

Key takeaways

Observable belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Observable to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Observable from memory before moving on to harder problems.

Reference excerpt

In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum mechanics, an observable is described by a linear operator. For example, these operators might represent submitting the system to various electromagnetic fields and eventually reading a value. Physically meaningful observables must also satisfy transformation laws that relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations that preserve certain mathematical properties of the space in question.

Quantum mechanics Every observable quantity in a quantum system is represented by a linear operator. John Archibald Wheeler used the analogy of a machine to describe operators: a quantum state goes in to the machine and the result state comes out. The result state will be one of the eigenstates of the operator. If the input was an eigenstate, the output will also be that eigenstate. In all other cases the output will be non-deterministic: one of the eigenstates will result with a probability depending on the operator and input. In quantum mechanics, observables correspond to linear self-adjoint operators on a separable complex Hilbert space representing the quantum state space. Observables assign values to outcomes of particular measurements, corresponding to the eigenvalue of the operator. If these outcomes represent physically allowable states (i.e. those that belong to the Hilbert space) the eigenvalues are real; however, the converse is not necessarily true. As a consequence, only certain measurements can determine the value of an observable for some state of a quantum system. In classical mechanics, any measurement can be made to determine the value of an observable. The relation between the state of a quantum system and the value of an observable requires some linear algebra for its description. In the mathematical formulation of quantum mechanics, up to a phase constant, pure states are given by non-zero vectors in a Hilbert space V. Two vectors v and w are considered to specify the same state if and only if w = c v {\displaystyle \mathbf {w} =c\mathbf {v} } for some non-zero c ∈ C {\displaystyle c\in \mathbb {C} } . Observables are given by self-adjoint operators on V. Not every self-adjoint operator corresponds to a physically meaningful observable. Also, not all physical observables are associated with non-trivial self-adjoint operators. For example, in quantum theory, mass appears as a parameter in the Hamiltonian, not as a non-trivial operator. In the case of transformation laws in quantum mechanics, the requisite automorphisms are unitary (or antiunitary) linear transformations of the Hilbert space V. Under Galilean relativity or special relativity, the mathematics of frames of reference is particularly simple, considerably restricting the set of physically meaningful observables. In quantum mechanics, measurement of observables exhibits some seemingly unintuitive properties. Specifically, if a system is in a state described by a vector in a Hilbert space, the measurement process affects the state in a non-deterministic but statistically predictable way. In particular, after a measurement is applied, the state description by a single vector may be destroyed, being replaced by a statistical ensemble. The irreversible nature of measurement operations in quantum physics is sometimes referred to as the measurement problem and is described mathematically by quantum operations. By the structure of quantum operations, this description is mathematically equivalent to that offered by the relative state interpretation where the original system is regarded as a subsystem of a larger system and the state of the original system is given by the partial trace of the state of the larger system. In quantum mechanics, dynamical variables A {\displaystyle A} such as position, translational (linear) momentum, orbital angular momentum, spin, and total angular momentum are each associated with a self-adjoint operator A ^ {\displaystyle {\hat {A}}} that acts on the state of the quantum system. The eigenvalues of operator A ^ {\displaystyle {\hat {A}}} correspond to the possible values that the dynamical variable can be observed as having. For example, suppose | ψ a ⟩ {\displaystyle |\psi _{a}\rangle } is an eigenket (eigenvector) of the observable A ^ {\displaystyle {\hat {A}}} , with eigenvalue a {\displaystyle a} , and exists in a Hilbert space. Then

A ^ | ψ a ⟩ = a | ψ a ⟩ . {\displaystyle {\hat {A}}|\psi _{a}\rangle =a|\psi _{a}\rangle .}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Observable

Start with the simplest possible case. Write down what Observable claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Observable before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Observable ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Observable

In research: Observable appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Observable in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Observable is common in secondary-school and first-year university syllabi. It links to neighbouring topics Control theory, Mathematical physics, Measurement, so understanding it makes those chapters shorter.
In everyday life: Look for Observable outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Observable in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Observable means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Observable out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Observable in simple terms?

In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum.

Why does Observable matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Observable?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Observable.