In quantum mechanics, vectors in a vector space describe states of a quantum system. The state vector carries both classical and quantum information. The classical information is accessible to classical measurement, the quantum information is not. The state space of a composite quantum system is the tensor product of individual Hilbert spaces H = H1 H2. If the state of a composite system cannot be written as |Ψ〉 = |Ψ1〉 ⊗ |Ψ2〉 of its components, it is an entangled state.
Finally, when working with vectors in an n-dimensional vector space, it is extremely useful to have a set of n mutually orthogonal unit vectors and use them as the basis (orthonormal basis) to construct any vector in that space. Such a vector will necessarily be a linear combination of the basis vectors. In fact, the dimension of the vector space can be defined as the maximum number n of mutually orthogonal vectors that can be found in that space. An orthonormal basis is not unique, but n is. Any two bases can be related to each other for conversion purposes. For example, one may choose one basis for preparing a quantum system but measure it in another basis. Specifying an orthonormal basis is conceptually similar to choosing a Cartesian coordinate system in physical three-dimensional space.
7.1 The need for observable-operators
Any extraction of a real measurable variable from an abstract entity requires some transformative operator. Such operators acting on |Ψ〉 are called observables in the literature. To make it explicit, in this paper we call them “observable-operators”. In classical physics, the measurable information is directly available, in quantum mechanics it is not. This is a crucial difference. A quantum observable-operator, is an intermediary that produces a classical value when it acts on |Ψ〉. The operator’s unusual action is that it initiates the random “collapse” of |Ψ〉 to one of the operator’s eigenvectors according to postulate 3. Further, the classical value the operator produces is the eigenvector’s corresponding eigenvalue.
Observable-operators arise in quantum mechanics because it deals with probability waves (the state vector) rather than with discrete particles whose motion and dynamics can be deterministically described by Newtonian physics. Part of the development of quantum mechanics is the establishment of the operators associated with the parameters needed to describe the system.
7.2 Observables and Hermitian operators
Recall that Fig. 3.1 could be interpreted in two ways. That picture, a classical object, did not capture the irreversible aspect of quantum measurement, nor can we claim that the state we observed is an eigenstate. The picture collapses to an interpretation (from the observer’s point of view) or flickers only during the observed period and stays superposed at other times. We infer this because we believe the picture is a static object. Thus, any observation is independent of all other observations. Further, more than one observer can observe the picture concurrently, each making his own interpretation. Thus, multiple observers, in principle, can pool their observations and determine all the possible states of the picture. In quantum mechanics, such pooling is not possible.
Modern scientific theories seek confirmation in observations.85 Observations require that some external agency interact with the system being measured. Measurements must follow a strict protocol to minimize disturbance to the measured object. We hope that the measured system can tolerate some minimum disturbance without getting destabilized to enable observations. A classical measuring apparatus always destabilizes a quantum system. Hence, we cannot expect to find any deterministic causal connections using measurements. However, we may assume causality to apply to undisturbed systems, e.g., we may express the evolution of the system (or subsystem) by differential equations that relate past conditions with future ones, but their ability to connect with observations will only be indirect; there will be an unavoidable indeterminacy in the calculation of observational results.
Physical measurements produce real numbers, so we are concerned with real dynamical variables. A complex dynamical variable necessitates two measurements—one each for the real and imaginary parts. This works fine only when measurements introduce negligible disturbances in the measured system (as in classical fluid flows where complex variables are sometimes used).86 In quantum systems, when Heisenberg’s uncertainty principle applies, one cannot make simultaneous measurements if non-commutating observable-operators are involved. Further, the sequence in which the variables are measured, matters. Observables are also associated with a vector space, but they are not state-vectors. They are the things we measure, and they are represented by linear operators.87
In quantum mechanics, measurements are intimately related to the eigenvalues and eigenvectors of observable-operators. In fact, the eigenstates have a significance beyond that of an orthonormal basis. Every measurable quantity, by postulate, has an associated observable-operator represented by a Hermitian operator and not a variable! A Hermitian operator M, has the defining self-adjoint property, i.e., M = M†. (The complex conjugate of a transposed matrix is called its Hermitian conjugate, denoted by a dagger superscript.) The set of eigenvalues of M represents the set of measurement outcomes M can produce. Each eigenvalue has a corresponding eigenstate (or eigenvector), to which the system will collapse after the measurement.
Note also that
- A Hermitian matrix can be unitarily diagonalized generating an orthonormal basis of eigenvectors which spans the state space of the quantum system.
- The eigenvalues of a Hermitian matrix are real. The possible outcomes of a measurement are precisely the eigenvalues of the given observable-operator.
- The matrix representing the product of two linear operators is the product of the matrices representing the two factors.
The above implies that any state of a quantum system can be expressed as a linear combination of the eigenvectors of any Hermitian operator. Equivalently, it means that any quantum state can be expressed as a linear superposition of the eigenstates of an observable-operator. Classical mechanics deals with real variables, quantum mechanics deals with operators whose eigenvalues are real. When the focus is on operators whose fundamental property is to bring about change is it surprising that quantum mechanics needs a separate postulate for measurement, while classical mechanics, which ideally views measurement as a passive activity, does not?
A complicated situation arises when there are two or more eigenstates of an observable with the same eigenvalue. Then any state formed by the linear superposition of the eigenstates will also be an eigenstate of the observable-operator belonging to the same eigenvalue. The implication here is that if two or more states for which a measurement of the observable is certain to belong to the same eigenvalue, then for any state formed by superposition of them will also belong to the same eigenvalue. This means that it is always possible to find two orthogonal vectors to form linear combinations.88 Since two eigenstates of an observable-operator attached to different eigenvalues are orthogonal, it follows that two states for which a measurement is certain to give two different results are orthogonal too.
 Aristotle (384 BC – 322 BC) did not seek confirmation in observations nor did he think it was necessary. Galileo (1564-1642) decidedly thought otherwise while laying the foundation for modern physics.
 See, e.g., Lamb (1932).
 Susskind & Friedman (2015), Chapter 3, Sec. 3.1.1.
 Susskind & Friedman (2015), Chapter 3, Sec. 3.1.5.