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When I was learning quantum theory at the OU, I was confused for a long time about observables, operators and measurements. I could see the trees: Hermitian operator, eigenvectors, orthonormal basis, eigenvalues, quantum state - expressed in terms of the operator basis with amplitudes projected onto each eigenvector, the application of the operator to the system quantum state, the application of the Born Rule.
Possible measurement values with their probabilities.
That's a lot of trees - but where was the wood?
When I was at school, in the sixth form, I wanted to study mathematical physics at university. Maths by itself was too abstract and purposeless for me; physics too sloppy and hand-wavy. In the end I drifted to philosophy and politics, showing how useless Warwick University was in engaging my youthful intellectual passions.
The OU course also mixed minimal maths with less-than-compelling intuitions (and a fair share of conceptual confusions resulting from inadequate maths - Hilbert Space was a space too far, it seemed).
This is not a criticism of the OU: all undergraduate physics is like that: sloppy and hand-wavy, remember?
So it takes a mathematician to do it right: thank you Michel Talagrand (above). Despite the QFT of the title, it's aimed at undergraduates and does QM first. Properly.
The following is not from the book, but it paraphrases (via Gemini) the section I am currently reading there.
Quantum Mechanics: Observables, Operators, and Probabilities
In quantum mechanics, observables—measurable properties of a system—are represented by Hermitian operators. Here's a mini-tutorial on how we connect these operators to the possible measurement outcomes and their probabilities:
1. Hermitian Operators and Eigenvalues
Every observable is associated with a Hermitian operator (let's call it Â). Hermitian operators have a crucial property: their eigenvalues are always real numbers, and their eigenvectors form a complete, orthonormal basis for the system's Hilbert space (assume its dimension is n).
The eigenvalue equation is fundamental: Â|i⟩ = λi|i⟩
Where:
- Â is the Hermitian operator.
- |i⟩ is the i-th eigenvector of  (i ranging from 1 to n).
- λi is the corresponding eigenvalue for that eigenvector.
The eigenvalues λi represent the possible outcomes of a measurement of the observable represented by Â.
2. Representing Quantum States
The quantum state of the system, represented by a state vector |α⟩, can be expressed as a linear combination of the eigenvectors of Â:
|α⟩ = Σi ci|i⟩ where ci are complex coefficients (i from 1 to n).
3. Applying the Operator and Finding Probabilities
To understand the probabilities of measurement outcomes, consider applying the operator  to the state ∣α⟩ mathematically - we get:
Â|α⟩ = Â(Σi ci|i⟩) = Σi ciÂ|i⟩ = Σi ciλi|i⟩
The probability P of measuring the eigenvalue λi is given by the squared magnitude of the corresponding coefficient ci:
P(λi) = |ci|2
Where ci = ⟨i|α⟩ is the projection of the state |α⟩ onto the eigenvector |i⟩ - its amplitude in that direction.
Remember that the state |α⟩ must be normalized, meaning:
Σi |ci|2 = 1
This ensures that the probabilities of all possible outcomes sum to 1.
4. Measurement and 'State Collapse' (Copenhagen)
During a measurement, the system "collapses" into one of the eigenvectors |i⟩. The measured value is the corresponding eigenvalue λi. The state after the measurement is the eigenstate corresponding to the measured eigenvalue. (This is the Copenhagen interpretation).
5. Expected Value
The expected value (average) of the observable represented by the operator  in the state |α⟩ is:
⟨Â⟩ = ⟨ α|Â|α⟩ = Σi |ci|2 λi (the weighted average of possible values).
In essence, this is how we connect the mathematical representation of observables (Hermitian operators) to the probabilistic outcomes we observe in quantum measurements.
Question: So another way to say it? We have an operator Â. We form a basis for the Hilbert space from the eigenvectors of Â. Each eigenvector now corresponds to (is tagged by) a possible measurement (eigen)value.
If we now project the abstract quantum state ψ onto each of these basis eigenvectors, the projected amplitude gives us - via Born's rule - the probability of observing that particular (eigen)value?
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Gemini:
Indeed, if we take a Hermitian operator  representing a physical observable:
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Eigenbasis Formation: We can find the set of its eigenvectors {∣ai⟩} which form a complete orthonormal basis for the Hilbert space of the system. Each eigenvector ∣ai⟩ is associated with a specific real eigenvalue ai, representing a possible outcome of measuring the observable A.
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Projection and Amplitude: When we project the abstract quantum state ∣ψ⟩ onto one of these basis eigenvectors ∣ai⟩, we then obtain the complex amplitude ⟨ai∣ψ⟩. This amplitude encodes the "overlap" of the state with that particular eigenstate.
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Born's Rule and Probability: According to Born's rule, the probability of observing the eigenvalue ai when the observable A is measured on the system in state ∣ψ⟩ is given by the squared magnitude of this projection amplitude:
P(ai)=∣⟨ai∣ψ⟩∣2
So, in essence, the eigenvectors of the operator provide the "labels" (the eigenvalues) for the possible measurement outcomes. The projection of the quantum state onto each of these "labeled" basis vectors gives us the amplitude, which then dictates the probability of observing that specific outcome.
So Why Do Eigenvalues Correspond to Measurement Outcomes?
In quantum mechanics, we are told that the possible outcomes of a measurement correspond to the eigenvalues of a Hermitian operator acting on a Hilbert space. But why? What is the deeper reason that a purely mathematical spectrum of an abstract operator should dictate the real outcomes we see on measuring devices in spacetime?
1. The Short Answer
Because quantum theory is built that way. According to its postulates:
- States are represented by vectors |ψ⟩ in a Hilbert space H.
- Observables are Hermitian operators  on H.
- The possible outcomes of measuring  are its eigenvalues λi.
- The probability of measuring λi is |⟨i|ψ⟩|², where |i⟩ is the corresponding eigenvector.
This structure is postulated — but it is not arbitrary.
2. Symmetries Determine Observables
In physics, observables arise from symmetry principles. Time translation symmetry gives rise to the Hamiltonian Ĥ. Spatial translations give us the momentum operator p̂. Rotations yield angular momentum operators L̂. These symmetries act via unitary transformations on the Hilbert space, and their infinitesimal generators are Hermitian operators.
So the operator structure of quantum theory is not just decoration — it is forced upon us by the demand for symmetry and conservation.
3. Measurement Projects onto Eigenstates
A measuring device interacts with a quantum system. That interaction can be modeled as coupling to a "pointer" system, followed by decoherence. The mathematics of projection captures this: measurement extracts the component of the quantum state aligned with a particular eigenvector of the observable.
This is why eigenvectors are special: they correspond to stable, repeatable outcomes. If a system is in an eigenstate, repeated measurements of the same observable yield the same result.
4. Eigenvalues Are Measurement Invariants
When an operator  acts on its eigenstate |a⟩, it simply returns a multiple of that state:
Â|a⟩ = a|a⟩This means that a measurement corresponding to  does not disturb the system — it remains in the same state. The number "a" is the only value consistent with both the structure of the operator and the stability of the measurement process. So eigenvalues become the only meaningful "answers" the system can give.
5. Representation Theory: Bridging Hilbert Space and Spacetime
The real bridge between abstract Hilbert space and physical spacetime lies in representation theory. In relativistic quantum field theory, the states of a system form representations of the Poincaré group (the group of spacetime symmetries). The observables — energy, momentum, spin, charge — arise as generators of these symmetries.
The Casimir operators of the symmetry group — such as mass and spin — label the irreducible representations. Their eigenvalues classify particles and define measurable quantities. In this way, the algebraic structure of operators in Hilbert space becomes the language of physical reality.
Conclusion
The identification of eigenvalues with measurement outcomes is not an accident. It emerges from the confluence of:
- symmetry principles,
- linear operator theory,
- the mathematical structure of Hilbert space,
- and the decohering nature of measurement interactions.
An eigenvalue is what survives the entanglement of quantum structure with classical observation. It is the fixed point where mathematics and measurement meet.
