Born Rule and Projection Postulate

Introduction

In our previous lesson, we established the axiomatic framework of quantum mechanics, understanding how each postulate—from the nature of states and observables to the rules of time evolution—is underpinned by deep mathematical theorems. We saw that the abstract machinery of Hilbert spaces is connected to physical reality through these foundational principles.

Today, we will take a closer look at the postulates governing measurement. Our focus will shift from their axiomatic justification to their practical application. You will learn how to use the Born rule to calculate the probabilities of different measurement outcomes and how to apply the projection postulate to determine the state of a system after a measurement has occurred. We will then combine these tools to analyze the fascinating consequences of performing measurements in sequence, a scenario that highlights the crucial distinction between compatible and incompatible observables.

The Born Rule: From Amplitudes to Probabilities

In our axiomatic overview, we identified the Born rule as the link between a state vector $∣ ψ ⟩$ and the probability of measuring a specific eigenvalue $a_{n}$ of an observable $A^$ . The rule states $P (a_{n}) = ∣ ⟨ a_{n} ∣ ψ ⟩ ∣^{2}$ . While we saw this is formally justified by Gleason's theorem, it's illuminating to understand why this specific squared-magnitude form is necessary from more direct physical considerations.

The following video provides an elegant argument for the Born rule, starting from two simple requirements: the total probability must sum to one, and the length of the state vector must be a constant, independent of the basis used to represent it. This approach nicely demonstrates why simpler guesses (like using the absolute value of the coefficient) lead to inconsistencies.

Ch 8 Why is probability equal to amplitude squared? Maths of Quantum Mechanics

This video, "Ch 8: Why is probability equal to amplitude squared?" from the Quantum Sense channel, offers a derivation of the Born rule that will appeal to your preference for understanding the 'why' behind the formalism. It shows that the squared-amplitude rule is a unique solution under reasonable physical assumptions.

Please watch the entire video. Pay close attention to these key stages of the argument: The problem with naively taking the probability to be the absolute value of the amplitude (P(c_n) = |c_n|). Note how this would require the length of the state vector to change depending on the measurement basis. The derivation of the probability function f(c). This is the core of the video. Follow the steps that use the independence of coefficients in a superposition to derive a differential equation that f(c) must satisfy, leading to the conclusion that f(c) \propto c^2. The brief mention of Gleason's theorem and its limitation to dimensions three and higher, which connects back to the logic of the derivation requiring at least two independent coefficients.

As the video concludes, the Born rule gives us a clear prescription for calculating measurement probabilities. For a system in a normalized state $∣ ψ ⟩$ , to find the probability of measuring eigenvalue $a_{n}$ for an observable $A^$ :

Find the corresponding normalized eigenvector $∣ a_{n} ⟩$ .
Calculate the probability amplitude, which is the inner product $c_{n} = ⟨ a_{n} ∣ ψ ⟩$ .
The probability is the squared magnitude of this amplitude: $P (a_{n}) = ∣ c_{n} ∣^{2}$ .

If the state $∣ ψ ⟩$ is written as an expansion in the eigenbasis of $A^$ , $∣ ψ ⟩ = \sum_{i} c_{i} ∣ a_{i} ⟩$ , the rule simply states that the probability of measuring $a_{n}$ is $∣ c_{n} ∣^{2}$ . The normalization of the state, $⟨ ψ ∣ ψ ⟩ = 1$ , ensures the probabilities sum to one: $\sum_{i} ∣ c_{i} ∣^{2} = 1$ .

For a continuous spectrum, like measuring position, the interpretation is analogous. If a particle is in state $∣ ψ ⟩$ , its position-space wave function is $ψ (x) = ⟨ x ∣ ψ ⟩$ . The Born rule then implies that $∣ ψ (x) ∣^{2}$ is the probability density of finding the particle at position $x$ . The probability of finding it in a small interval $d x$ is $∣ ψ (x) ∣^{2} d x$ .

The Projection Postulate: State Collapse

The Born rule addresses the "what" and "how likely" of a measurement outcome. The projection postulate addresses the "what next." It describes how the state of the system changes as a result of the measurement.

Projection Postulate: If the measurement of an observable $A^$ on a system in state $∣ ψ ⟩$ yields the result $a_{n}$ , the state of the system immediately after the measurement is the normalized projection of $∣ ψ ⟩$ onto the eigenspace corresponding to $a_{n}$ .

Let's denote the projection operator onto the eigenspace of $a_{n}$ as $P^_{n}$ .

If the eigenvalue $a_{n}$ is non-degenerate, its eigenspace is one-dimensional, and the projector is simply $P^_{n} = ∣ a_{n} ⟩ ⟨ a_{n} ∣$ . After the measurement, the new state is $∣ ψ^{'} ⟩ = ∣ a_{n} ⟩$ .
If the eigenvalue $a_{n}$ is $d$ -fold degenerate, the projector is $P^_{n} = \sum_{i = 1}^{d} ∣ a_{n}, i ⟩ ⟨ a_{n}, i ∣$ , where ${∣ a_{n}, i ⟩}$ is an orthonormal basis for the eigenspace. The unnormalized post-measurement state is $P^_{n} ∣ ψ ⟩$ . To get the physical state, we must renormalize it: $∣ ψ^{'} ⟩ = \frac{P ^ _{n} ∣ ψ ⟩}{⟨ ψ ∣ P ^ _{n}^{†} P ^ _{n} ∣ ψ ⟩} = \frac{P ^ _{n} ∣ ψ ⟩}{⟨ ψ ∣ P ^ _{n} ∣ ψ ⟩}$ The term in the denominator, $⟨ ψ ∣ P^_{n} ∣ ψ ⟩$ , is precisely the probability of obtaining the outcome $a_{n}$ , so the normalization is physically consistent.

The following image provides a schematic for this process.

Figure 1: Measurement process for (a) a single qubit and (b) a multi-qubit system. The box 'M' represents a measurement apparatus. The initial state $|\psi\rangle$ enters, and upon measurement of an outcome $|x\rangle$, the probability is given by the Born rule, $p_x = |\alpha_x|^2$, and the system collapses into the measured state.

To see how projection operators work mathematically, the next video walks through a concrete example for a spin-1/2 system.

Projection Operators and Measurement

The video "Projection Operators and Measurement" by ASCPhysicsAndAstronomy clearly demonstrates the mechanics of applying a projection operator to a quantum state.

Watch from the beginning of the video. It explains how to construct a projection operator (e.g., for spin-up) and apply it to a general state. Crucially, it highlights the need to renormalize the resulting state, as the projection itself does not preserve the norm.

Sequential Measurements: The Role of Commutation

With the Born rule and projection postulate in hand, we can now analyze one of the most conceptually rich topics in quantum mechanics: sequential measurements. The outcome depends critically on whether the measured observables are compatible (commuting) or incompatible (non-commuting).

Let's begin by consulting a concise summary of the key principles.

X Consequences of the measurement postulate

These lecture notes from Lancaster University provide a clear and direct explanation of what happens during consecutive measurements.

Please read the sections titled "Consecutive measurements" and "Simultaneous and incompatible variables". Focus on the distinction drawn between commuting and non-commuting observables and the effect of measurement order.

As the notes explain, the commutator is key:

1. Compatible Observables: $[A^, B^] = 0$

If two observables commute, they possess a common set of eigenvectors.

Suppose we first measure $A^$ on a state $∣ ψ ⟩$ and obtain the result $a_{n}$ . The state collapses to a simultaneous eigenstate $∣ a_{n}, b_{m} ⟩$ .
A subsequent measurement of $B^$ will now yield the definite value $b_{m}$ with probability 1.
Crucially, this second measurement does not change the state's status as an eigenstate of $A^$ . A third measurement of $A^$ would still yield $a_{n}$ .
The order of measurement does not affect the joint probability distribution, and the information from the first measurement is preserved.

2. Incompatible Observables: $[A^, B^] \neq = 0$

If two observables do not commute, they do not share a full set of eigenvectors. This leads to the quintessential weirdness of quantum measurement.

Suppose we measure $A^$ and get $a_{n}$ . The state collapses to $∣ a_{n} ⟩$ .
Since $∣ a_{n} ⟩$ is generally not an eigenstate of $B^$ , it can be written as a superposition of $B^$ 's eigenstates: $∣ a_{n} ⟩ = \sum_{m} d_{m} ∣ b_{m} ⟩$ .
A measurement of $B^$ will now yield a probabilistic outcome $b_{m}$ with probability $∣ d_{m} ∣^{2}$ . The state then collapses to $∣ b_{m} ⟩$ .
This second measurement has fundamentally altered the system. The state $∣ b_{m} ⟩$ is no longer an eigenstate of $A^$ . If we measure $A^$ again, we will get a probabilistic outcome. The information about the definite value $a_{n}$ from the first measurement has been destroyed by the intervening measurement of the incompatible observable $B^$ .

A classic physical example of sequential measurements is passing photons through a series of polarizers. A polarizer acts as a measurement device for a specific polarization direction.

3. Axioms of Quantum Mechanics

These MIT OpenCourseWare notes by Prof. Paola Cappellaro provide an excellent illustration of sequential measurements using photon polarization.

Please read the subsection "Photon Polarization" in Section 3.3.3. It shows how sending a horizontally polarized photon through a 45-degree polarizer (an incompatible measurement) changes the state, allowing it to subsequently pass through a vertical polarizer—something that was impossible before.

This example perfectly demonstrates the principles.

Start with a horizontally polarized photon, $∣ h ⟩$ . A measurement with a vertical polarizer $P^_{v} = ∣ v ⟩ ⟨ v ∣$ yields a null result with probability $∣ ⟨ v ∣ h ⟩ ∣^{2} = 0$ .
Now, insert a 45-degree polarizer between the source and the vertical polarizer. The state of a 45-degree polarized photon is $∣4 5^{\circ} ⟩ = \frac{1}{2} (∣ h ⟩ + ∣ v ⟩)$ .
The first measurement on the initial $∣ h ⟩$ state is with the 45-degree polarizer. The probability of the photon passing is $P (4 5^{\circ}) = ∣ ⟨ 4 5^{\circ} ∣ h ⟩ ∣^{2} = ∣ \frac{1}{2} ∣^{2} = 1/2$ .
If it passes, the state collapses to $∣4 5^{\circ} ⟩$ .
Now, this new state encounters the vertical polarizer. The probability of passing is $P (v) = ∣ ⟨ v ∣4 5^{\circ} ⟩ ∣^{2} = ∣ \frac{1}{2} ∣^{2} = 1/2$ .
The total probability of passing through both is $P (total) = P (4 5^{\circ}) \times P (v ∣4 5^{\circ}) = \frac{1}{2} \times \frac{1}{2} = 1/4$ .

The intermediate measurement of an incompatible observable fundamentally changed the system, creating a non-zero probability for an outcome that was previously forbidden. The notes go on to show that if you perform a series of very small rotations (a sequence of many "weak" measurements), you can effectively "drag" the state along, a phenomenon related to the Quantum Zeno Effect.

Conclusion

In this lesson, we have moved from the abstract statement of the measurement postulates to their concrete application. We've seen how the mathematical formalism of inner products and projection operators translates directly into predictions about experimental outcomes.

Key Takeaways:

The Born Rule provides the procedure to calculate the probability of a measurement outcome: $P (a_{n}) = ∣ ⟨ a_{n} ∣ ψ ⟩ ∣^{2}$ . This specific form is required for a consistent probabilistic theory with a basis-independent state vector norm.
The Projection Postulate describes the "collapse" of the state vector onto the eigenstate (or eigenspace) corresponding to the measured eigenvalue. The post-measurement state is the renormalized projection: $∣ ψ^{'} ⟩ = P^_{n} ∣ ψ ⟩ / ⟨ ψ ∣ P^_{n} ∣ ψ ⟩$ .
Sequential measurements of incompatible (non-commuting) observables are order-dependent and destructive. The act of measuring one observable can erase the information gained from a previous measurement of an incompatible one.
Measurements of compatible (commuting) observables are order-independent, and information is preserved through the sequence.

The profound consequences of non-commuting observables lie at the heart of quantum mechanics. The inherent trade-off in our knowledge of such quantities is quantified by the uncertainty principle. In our next lesson, we will derive the generalized uncertainty relation for any two observables directly from their commutator, formalizing the concepts we have explored today.

Recommended resources

Introduction

The Born Rule: From Amplitudes to Probabilities

The Projection Postulate: State Collapse

Sequential Measurements: The Role of Commutation

Conclusion