Table of Contents
- Introduction
- The Role of Observables in Quantum Theory
- Mathematical Representation of Observables
- Hermitian Operators and Their Properties
- Measurement Postulate
- Eigenvalues as Measurement Outcomes
- Projection Postulate and State Collapse
- Expectation Values and Probabilities
- Compatible and Incompatible Observables
- Commutators and the Uncertainty Principle
- Degeneracy and Simultaneous Measurements
- Measurement in Discrete vs Continuous Systems
- Generalized Measurements and POVMs
- Measurement in Quantum Circuits
- Interpretations and Measurement Problem
- Conclusion
1. Introduction
In quantum mechanics, observables represent physical quantities we can measure, such as position, momentum, or energy. Unlike classical physics, the act of measurement in quantum theory plays a central and non-trivial role: it not only reveals information but also affects the system being measured.
2. The Role of Observables in Quantum Theory
Quantum mechanics does not assign definite values to observables until they are measured. Instead, it provides probabilities for different outcomes based on the system’s state vector. Observables are the bridge between mathematical states and experimental results.
3. Mathematical Representation of Observables
Observables are represented by Hermitian operators acting on a Hilbert space:
\[
\hat{A}^\dagger = \hat{A}
\]
Key properties:
- Real eigenvalues (physical measurements must be real)
- Orthogonal eigenstates
- Complete spectral decomposition
4. Hermitian Operators and Their Properties
Let \( \hat{A} \) be an observable. Then:
- \( \hat{A} \) is self-adjoint: \( \hat{A}^\dagger = \hat{A} \)
- If \( \hat{A}|a\rangle = a|a\rangle \), then \( a \in \mathbb{R} \)
- Eigenstates form a complete orthonormal basis
This ensures measurable outcomes are consistent and interpretable.
5. Measurement Postulate
Upon measurement of observable \( \hat{A} \) in state \( |\psi\rangle \):
- The result is one of the eigenvalues \( a \)
- The system collapses into the corresponding eigenstate \( |a\rangle \)
- The probability of outcome \( a \) is:
\[
P(a) = |\langle a | \psi \rangle|^2
\]
This is the Born rule.
6. Eigenvalues as Measurement Outcomes
Only the eigenvalues of \( \hat{A} \) are physically observable. All other values are excluded. The spectrum of the operator determines the possible outcomes.
7. Projection Postulate and State Collapse
After measuring \( \hat{A} \) and finding result \( a \), the system is projected into:
\[
|\psi\rangle \rightarrow \frac{\hat{P}_a |\psi\rangle}{|\hat{P}_a |\psi\rangle|}
\]
Where \( \hat{P}_a = |a\rangle\langle a| \) is the projector. This process is known as wavefunction collapse.
8. Expectation Values and Probabilities
The expectation value of \( \hat{A} \) in state \( |\psi\rangle \) is:
\[
\langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle
\]
It represents the average of many measurements over identically prepared systems.
9. Compatible and Incompatible Observables
Observables \( \hat{A} \) and \( \hat{B} \) are:
- Compatible if they commute:
\[
[\hat{A}, \hat{B}] = 0
\]
⇒ They share a common eigenbasis, and can be measured simultaneously. - Incompatible if \( [\hat{A}, \hat{B}] \ne 0 \):
⇒ Subject to uncertainty relations
10. Commutators and the Uncertainty Principle
For position and momentum:
\[
[\hat{x}, \hat{p}] = i\hbar
\]
This gives rise to the Heisenberg uncertainty principle:
\[
\Delta x \, \Delta p \ge \frac{\hbar}{2}
\]
Incompatible observables cannot be simultaneously known with arbitrary precision.
11. Degeneracy and Simultaneous Measurements
- Degenerate eigenvalues: multiple eigenstates share the same eigenvalue
- Measurement of a degenerate observable does not fully specify the post-measurement state
- Additional compatible observables can be used to resolve degeneracy
12. Measurement in Discrete vs Continuous Systems
Discrete:
- E.g., spin, energy levels
- Finite or countable spectrum
Continuous:
- E.g., position, momentum
- Eigenstates are delta-normalized distributions
\[
\langle x’ | x \rangle = \delta(x – x’)
\]
13. Generalized Measurements and POVMs
In generalized quantum measurements:
- Use POVMs (Positive Operator-Valued Measures)
- Broader framework than projective measurements
- Useful in quantum information and open quantum systems
Each outcome \( i \) is associated with operator \( E_i \), such that:
\[
\sum_i E_i = \hat{I}, \quad E_i \ge 0
\]
14. Measurement in Quantum Circuits
- Measured qubits collapse into \( |0\rangle \) or \( |1\rangle \)
- Probability dictated by amplitudes
- Used to extract information from quantum algorithms
Example:
If \( |\psi\rangle = \alpha |0\rangle + \beta |1\rangle \), then:
\[
P(0) = |\alpha|^2, \quad P(1) = |\beta|^2
\]
15. Interpretations and Measurement Problem
- Copenhagen: collapse is fundamental
- Many-worlds: all outcomes occur in separate branches
- Objective collapse models: propose dynamical collapse mechanisms
- QBism: collapse is Bayesian updating of observer knowledge
Measurement remains the most debated aspect of quantum theory.
16. Conclusion
Observables and measurements lie at the core of quantum mechanics. They define the interaction between abstract state vectors and empirical outcomes. While measurements provide the link to experiment, they also introduce profound conceptual challenges, from wavefunction collapse to the nature of quantum reality. Mastery of this topic is vital for understanding, designing, and interpreting quantum systems.
