Operators, measurements and probabilities

 

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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.

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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⟩ = λ_i|i⟩

Where:

•  is the Hermitian operator.
• |i⟩ is the i-th eigenvector of Â (i ranging from 1 to n).
• λ_i is the corresponding eigenvalue for that eigenvector.

The eigenvalues λ_i 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⟩ = Σ_i c_i|i⟩     where c_i 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 Â to the state ∣α⟩ mathematically - we get:

Â|α⟩ = Â(Σ_i c_i|i⟩) = Σ_i c_iÂ|i⟩ = Σ_i c_iλ_i|i⟩

The probability P of measuring the eigenvalue λ_i is given by the squared magnitude of the corresponding coefficient c_i:

P(λ_i) = |c_i|²

Where c_i = ⟨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:

Σ_i |c_i|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 λ_i. 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 Â in the state |α⟩ is:

⟨Â⟩ = ⟨ α|Â|α⟩ = Σ_i |c_i|² λ_i    (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.

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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 Â representing a physical observable:

1. Eigenbasis Formation: We can find the set of its eigenvectors {∣a_i​​⟩} which form a complete orthonormal basis for the Hilbert space of the system. Each eigenvector ∣a_i​⟩ is associated with a specific real eigenvalue a_i​, representing a possible outcome of measuring the observable A.

2. Projection and Amplitude: When we project the abstract quantum state ∣ψ⟩ onto one of these basis eigenvectors ∣a_i​⟩, we then obtain the complex amplitude ⟨a_i​​∣ψ⟩. This amplitude encodes the "overlap" of the state with that particular eigenstate.

3. Born's Rule and Probability: According to Born's rule, the probability of observing the eigenvalue a_i​ when the observable A is measured on the system in state ∣ψ⟩ is given by the squared magnitude of this projection amplitude:

   P(a_i​​)=∣⟨a_i​∣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.

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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 λ_i.
• The probability of measuring λ_i 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.

The ghost in the machine

About half way through my OU quantum mechanics course I realised I had no idea what was going on. I could perfectly well follow the maths: the problem was that I didn't understand how the entities I was learning about connected to 'reality'. The subject might as well have been magick or theology.

I know I was not the only mystified one.

The trouble is that you need to be well-acquainted with the basic framework of QM before you can get its relationship to physical reality straight (it never ceases to be incomprehensible).

So, safe in the knowledge that it makes no sense, here is my current understanding of the basic mechanisms of quantum mechanics.

The quantum state (aka the wavefunction or state vector) is what we study in QM; we are not to suppose that this objectively exists out there in spacetime: mathematically, it lives somewhere else, in Hilbert space.

When we propose to make an observation of a quantum system (for example, the energy level of an electron in an atom) we choose the quantum operator corresponding to the property of interest (energy, momentum, position, etc). Call the operator A and the quantum state ψ.

To capture the notion of a measurement we form A(ψ) which we may also write as a sum (superposition) of A's eigenvectors (eigenfunctions), each multiplied by its specific coefficient (the projection of ψ onto the specific eigenvector/eigenfunction and the corresponding eigenvalue).

The eigenvalues are the values of possible observations, and the modulus of their expansion coefficients gives the probability of observing that value (Born's Rule).

What does it mean to apply an operator to a quantum state?

Suppose the operator is the Hamiltonian for the infinite square well potential, H. Initially the electron had some prior quantum state - maybe it was a free particle. Applying Schrodinger's equation for the infinite square well case to the quantum state psi is equivalent to putting the electron into the box. So applying the operator defines a constraint upon psi which effectively sets up the experimental situation. 

However, once we have registered the observation of a specific eigenvalue - say of position - the state vector changes again to the corresponding eigenvector (in this spatial example, a 'spike' eigenfunction centred at that position, which would rapidly spread out...).

So here, in all its incomprehensibility and weirdness, is the reason why quantum mechanics is so hard to grasp, and why no-one really understands what's going on .. even though it all works so brilliantly in practice.

The Calculus of Variations

I had to drop out of my maths MSc course with the Open University back in 2010 - yet another busy-busy client contract. I do have all the material, though and will do the course on my own account starting next week. No doubt you'll be hearing more here soon.

I had the material out this afternoon for review so Alex and Clare were at least subliminally aware of my plans. Towards the end of the afternoon they both decided to go for a walk on our local part of the Mendips, to take the air.

Top of the hill, they were discussing my folly when they were overtaken by a hobbity sort of guy - big beard, bush hat, short and dumpy, hairy feet .. well, you know. He hears the word OU and strikes up a conversation. Alex mentions I'm doing M820 and he immediately recognises the Calculus of Variations. Turns out the hobbity guy has almost finished his OU maths MSc, finishing this year with the formidable Functional Analysis module.

Only in Wells, huh?

Pictured below, the author with SF books from Adrian (thanks!) and backed by a van Gogh church, a present to Alex from the Musee d'Orsay (via us).