Quantum state


In quantum physics, a quantum state is a mathematical entity that represents a physical system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. Knowledge of the quantum state, and the rules for the system's evolution in time, exhausts all that can be known about a quantum system.
Quantum states are either pure or mixed, and have several possible representations. Pure quantum states are commonly represented as a vector in a Hilbert space. Mixed states are statistical mixtures of pure states and cannot be represented as vectors on that Hilbert space, and instead are commonly represented as density matrices.
Common examples of quantum states are the wave functions describing position and momentum, finite-dimensional vectors describing spin such as the singlet, and states describing many-body quantum systems in a Fock space.

From the states of classical mechanics

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. 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, 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

The process of describing a quantum system with quantum mechanics begins with identifying a set of variables defining the quantum state of the system. 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.

Measurements

Measurements, macroscopic operations on quantum states, filter the state. Whatever the input quantum state might be, repeated identical measurements give consistent values. For this reason, measurements 'prepare' quantum states for experiments, placing the system in a partially defined state. Subsequent measurements may either further prepare the system – these are compatible measurements – or it may alter the state, redefining it – these are called incompatible or complementary measurements. For example, we may measure the momentum of a state along the axis any number of times and get the same result, but if we measure the position after once measuring the momentum, subsequent measurements of momentum are changed. The quantum state appears unavoidably altered by incompatible measurements. This is known as the uncertainty principle.

Eigenstates and pure states

The quantum state after a measurement is in an eigenstate corresponding to that measurement and the value measured. Other aspects of the state may be unknown. Repeating the measurement will not alter the state. In some cases, compatible measurements can further refine the state, causing it to be an eigenstate corresponding to all these measurements. A full set of compatible measurements produces a pure state. Any state that is not pure is called a mixed state as discussed in more depth [|below].
The eigenstate solutions to the Schrödinger equation can be formed into pure states. Experiments rarely produce pure states. Therefore statistical mixtures of solutions must be compared to experiments.

Representations

The same physical quantum state can be expressed mathematically in different ways called representations. 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 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 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.

Wave function representations

s represent quantum states, particularly when they are functions of position or of momentum. Historically, definitions of quantum states used wavefunctions before the more formal methods were developed. The wave function is a complex-valued function of any complete set of commuting or compatible degrees of freedom. For example, one set could be the spatial coordinates of an electron.
Preparing a system by measuring the complete set of compatible observables produces a pure quantum state. More common, incomplete preparation produces a mixed quantum state. Wave function solutions of Schrödinger's equations of motion for operators corresponding to measurements can readily be expressed as pure states; they must be combined with statistical weights matching experimental preparation to compute the expected probability distribution.

Pure states of wave functions

Numerical or analytic solutions in quantum mechanics can be expressed as pure states. These solution states, called eigenstates, are labeled with quantized values, typically quantum numbers.
For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant pure states are identified by the principal quantum number, the angular momentum quantum number, the magnetic quantum number, and the spin z-component. For another example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. A pure state here is represented by a two-dimensional complex vector, with a length of one; that is, with
where and are the absolute values of and.
The postulates of quantum mechanics state that pure states, at a given time, correspond to vectors in a separable complex Hilbert space, while each measurable physical quantity is associated with a mathematical operator called the observable. The operator serves as a linear function that acts on the states of the system. The eigenvalues of the operator correspond to the possible values of the observable. For example, it is possible to observe a particle with a momentum of 1 kg⋅m/s if and only if one of the eigenvalues of the momentum operator is 1 kg⋅m/s. The corresponding eigenvector with eigenvalue 1 kg⋅m/s would be a quantum state with a definite, well-defined value of momentum of 1 kg⋅m/s, with no quantum uncertainty. If its momentum were measured, the result is guaranteed to be 1 kg⋅m/s.
On the other hand, a pure state described as a superposition of multiple different eigenstates does in general have quantum uncertainty for the given observable. Using bra–ket notation, this linear combination of eigenstates can be represented as:
The coefficient that corresponds to a particular state in the linear combination is a complex number, thus allowing interference effects between states. The coefficients are time dependent. How a quantum state changes in time is governed by the time evolution operator.

Mixed states of wave functions

A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent mixed states. A mixture of quantum states is again a quantum state.
A mixed state for electron spins, in the density-matrix formulation, has the structure of a matrix that is Hermitian and positive semi-definite, and has trace 1. A more complicated case is given by the singlet state, which exemplifies quantum entanglement:
which involves superposition of joint spin states for two particles with spin 1/2. The singlet state satisfies the property that if the particles' spins are measured along the same direction then either the spin of the first particle is observed up and the spin of the second particle is observed down, or the first one is observed down and the second one is observed up, both possibilities occurring with equal probability.
A pure quantum state can be represented by a ray, an element of a projective Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.
The Schrödinger–HJW theorem classifies the multitude of ways to write a given mixed state as a convex combination of pure states.
Before a particular measurement is performed on a quantum system, the theory gives only a probability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and the linear operators describing the measurement. Probability distributions for different measurements exhibit tradeoffs exemplified by the uncertainty principle: a state that implies a narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another.
Statistical mixtures of states are a different type of linear combination. A statistical mixture of states is a statistical ensemble of independent systems. Statistical mixtures represent the degree of knowledge whilst the uncertainty within quantum mechanics is fundamental. Mathematically, a statistical mixture is not a combination using complex coefficients, but rather a combination using real-valued, positive probabilities of different states. A number represents the probability of a randomly selected system being in the state. Unlike the linear combination case each system is in a definite eigenstate.
The expectation value of an observable is a statistical mean of measured values of the observable. It is this mean, and the distribution of probabilities, that is predicted by physical theories.
There is no state that is simultaneously an eigenstate for all observables. For example, we cannot prepare a state such that both the position measurement and the momentum measurement are known exactly; at least one of them will have a range of possible values. This is the content of the Heisenberg uncertainty relation.
Moreover, in contrast to classical mechanics, it is unavoidable that performing a measurement on the system generally changes its state. More precisely: After measuring an observable A, the system will be in an eigenstate of A; thus the state has changed, unless the system was already in that eigenstate. This expresses a kind of logical consistency: If we measure A twice in the same run of the experiment, the measurements being directly consecutive in time, then they will produce the same results. This has some strange consequences, however, as follows.
Consider two incompatible observables, and, where corresponds to a measurement earlier in time than.
Suppose that the system is in an eigenstate of at the experiment's beginning. If we measure only, all runs of the experiment will yield the same result.
If we measure first and then in the same run of the experiment, the system will transfer to an eigenstate of after the first measurement, and we will generally notice that the results of are statistical. Thus: Quantum mechanical measurements influence one another, and the order in which they are performed is important.
Another feature of quantum states becomes relevant if we consider a physical system that consists of multiple subsystems; for example, an experiment with two particles rather than one. Quantum physics allows for certain states, called entangled states, that show certain statistical correlations between measurements on the two particles which cannot be explained by classical theory. For details, see Quantum entanglement. These entangled states lead to experimentally testable properties
that allow us to distinguish between quantum theory and alternative classical models.