Quantum state

In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a quantum-mechanical prediction for the system represented by the state. Knowledge of the quantum state, and the quantum mechanical rules for the system's evolution in time, exhausts all that can be known about a quantum system.

Quantum states may be defined differently for different kinds of systems or problems. Two broad categories are

Historical, educational, and application-focused problems typically feature wave functions; modern professional physics uses the abstract vector states. In both categories, quantum states divide into pure versus mixed states, or into coherent states and incoherent states. Categories with special properties include stationary states for time independence and quantum vacuum states in quantum field theory.

From the states of classical mechanics[edit]

As a tool for physics, quantum states grew out of states in classical mechanics. A classical dynamical state consists of a set of dynamical variables with well-defined real values at each instant of time.^[1]^: 3 For example, the state of a cannon ball would consist of its position and velocity. The state values evolve under equations of motion and thus remain strictly determined. If we know the position of a cannon and the exit velocity of its projectiles, then we can use equations containing the force of gravity to predict the trajectory of a cannon ball precisely.

Similarly, quantum states consist of sets of dynamical variables that evolve under equations of motion. However, the values derived from quantum states are complex numbers, quantized, limited by uncertainty relations,^[1]^: 159 and only provide a probability distribution for the outcomes for a system. These constraints alter the nature of quantum dynamic variables. For example, the quantum state of an electron in a double-slit experiment would consist of complex values over the detection region and, when squared, only predict the probability distribution of electron counts across the detector.

Role in quantum mechanics[edit]

The process of describing a quantum system with quantum mechanics begins with identifying a set of variables defining the quantum state of the system.^[1]^: 204 The set will contain compatible and incompatible variables. Simultaneous measurement of a complete set of compatible variables prepares the system in a unique state. The state then evolves deterministically according to the equations of motion. Subsequent measurement of the state produces a sample from a probability distribution predicted by the quantum mechanical operator corresponding to the measurement.

The fundamentally statistical or probabilisitic nature of quantum measurements changes the role of quantum states in quantum mechanics compared to classical states in classical mechanics. In classical mechanics the initial state of one or more bodies is measured; the state evolves according to the equations of motion; measurements of the final state are compared to predictions. In quantum mechanics, ensembles of identically prepared quantum states evolve according to the equations of motion and many repeated measurements are compared to predicted probability distributions.^[1]^: 204

Representations[edit]

The same physical quantum state can be expressed mathematically in different ways called representations.^[1] The position wave function is one representation often seen first in introductions to quantum mechanics. The equivalent momentum wave function is another wave function based representation. Representations are analogous to coordinate systems^[1]^: 244 or similar mathematical devices like parametric equations. Selecting a representation will make some aspects of a problem easier at the cost of making other things difficult.

In formal quantum mechanics (see below) the theory develops in terms of abstract 'vector space', avoiding any particular representation. This allows many elegant concepts of quantum mechanics to be expressed and to be applied even in cases where no classical analog exists.^[1]^: 244

The expression used to denote a state vector (which corresponds to a pure quantum state) takes the form $|\psi \rangle$ (where the " $\psi$ " can be replaced by any other symbols, letters, numbers, or even words). This can be contrasted with the usual mathematical notation, where vectors are usually lower-case Latin letters, and it is clear from the context that they are indeed vectors.

Dirac defined two kinds of vector, bra and ket, dual to each other.

[e]

Each ket $|\psi \rangle$ is uniquely associated with a so-called bra, denoted $\langle \psi |$ , which corresponds to the same physical quantum state. Technically, the bra is the of the ket. It is an element of the dual space, and related to the ket by the Riesz representation theorem. In a finite-dimensional space with a chosen orthonormal basis, writing $|\psi \rangle$ as a column vector, $\langle \psi |$ is a row vector; to obtain it just take the transpose and entry-wise complex conjugate of $|\psi \rangle$ .

adjoint

Scalar products^[g] (also called brackets) are written so as to look like a bra and ket next to each other: $\langle \psi _{1}|\psi _{2}\rangle$ . (The phrase "bra-ket" is supposed to resemble "bracket".)

[f]

Mathematical generalizations[edit]

States can be formulated in terms of observables, rather than as vectors in a vector space. These are positive normalized linear functionals on a C*-algebra, or sometimes other classes of algebras of observables. See State on a C*-algebra and Gelfand–Naimark–Segal construction for more details.

Isham, Chris J (1995). . Imperial College Press. ISBN 978-1-86094-001-9.

Lectures on Quantum Theory: Mathematical and Structural Foundations

The concept of quantum states, in particular the content of the section Formalism in quantum physics above, is covered in most standard textbooks on quantum mechanics.

For a discussion of conceptual aspects and a comparison with classical states, see:

For a more detailed coverage of mathematical aspects, see:

For a discussion of purifications of mixed quantum states, see Chapter 2 of John Preskill's lecture notes for Physics 219 at Caltech.

For a discussion of geometric aspects see: