Quantum state

In quantum physics, a quantum state is a mathematical object that fully describes a quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus. Quantum states can be statistically mixed, corresponding to an experiment involving a random change of the parameters. States obtained in this way are called mixed states, as opposed to pure states which cannot be described as a mixture of others. When performing a certain measurement on a quantum state, the result is in general described by a probability distribution, and the form that this distribution takes is completely determined by the quantum state and the observable describing the measurement. However, unlike in classical mechanics, the result of a measurement on even a pure quantum state is only determined probabilistically. This reflects a core difference between classical and quantum physics.

Mathematically, a pure quantum state is typically represented by a vector in a Hilbert space. In physics, bra-ket notation is often used to denote such vectors. Linear combinations (superpositions) of vectors can describe interference phenomena. Mixed quantum states are described by density matrices.

In a more general mathematical context, quantum states can be understood as positive normalized linear functionals on a C* algebra; see GNS construction.

Conceptual description

The state of a physical system

The state of a physical system is a complete description of the parameters of the experiment. To understand this rather abstract notion, it is useful to first explore it in an example from classical mechanics.

Consider an experiment with a (non-quantum) particle of mass Failed to parse (Missing texvc executable; please see math/README to configure.): m=1

which moves freely, and without friction,

in one spatial direction.

We start the experiment at time Failed to parse (Missing texvc executable; please see math/README to configure.): t=0

by pushing the particle with some speed into some direction.

Doing this, we determine the initial position Failed to parse (Missing texvc executable; please see math/README to configure.): q

and the initial momentum[1]

Failed to parse (Missing texvc executable; please see math/README to configure.): p

of the particle. These initial conditions are what characterizes the state Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma

of the system, 

formally denoted as Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma = (p,q) . We say that we prepare the state of the system by fixing its initial conditions.

At a later time Failed to parse (Missing texvc executable; please see math/README to configure.): t>0 , we conduct measurements on the particle. The measurements we can perform on this simple system are essentially its position Failed to parse (Missing texvc executable; please see math/README to configure.): Q(t)

at time Failed to parse (Missing texvc executable; please see math/README to configure.): t

, its momentum Failed to parse (Missing texvc executable; please see math/README to configure.): P(t) , and combinations of these. Here Failed to parse (Missing texvc executable; please see math/README to configure.): P(t)

and Failed to parse (Missing texvc executable; please see math/README to configure.): Q(t)
refer to the measurable quantities (observables)

of the system as such, not the specific results they produce in a certain run of the experiment.

However, knowing the state Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma

of the system, we can compute the

value of the observables in the specific state, i.e., the results that our measurements will produce, depending on Failed to parse (Missing texvc executable; please see math/README to configure.): p

and Failed to parse (Missing texvc executable; please see math/README to configure.): q

. We denote these values as Failed to parse (Missing texvc executable; please see math/README to configure.): \langle P(t) \rangle _\sigma

and Failed to parse (Missing texvc executable; please see math/README to configure.): \langle Q(t) \rangle _\sigma

. In our simple example, it is well known that the particle moves with constant velocity; therefore,

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle P(t) \rangle _\sigma = p, \quad \langle Q(t) \rangle _\sigma = pt+q.

Now suppose that we start the particle with a random initial position and momentum. (For argument's sake, we may suppose that the particle is pushed away at Failed to parse (Missing texvc executable; please see math/README to configure.): t=0

by some apparatus which is controlled by a random number generator.) The state Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma

of the system is now not described by two numbers 

Failed to parse (Missing texvc executable; please see math/README to configure.): p

and Failed to parse (Missing texvc executable; please see math/README to configure.): q

, but rather by two probability distributions. The observables Failed to parse (Missing texvc executable; please see math/README to configure.): P(t)

and Failed to parse (Missing texvc executable; please see math/README to configure.): Q(t)
will produce random results now;

they become random variables, and their values in a single measurement cannot be predicted. However, if we repeat the experiment sufficiently often, always preparing the same state Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma , we can predict the expectation value of the observables (their statistical mean) in the state Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma . The expectation value of Failed to parse (Missing texvc executable; please see math/README to configure.): P(t)

is again denoted by Failed to parse (Missing texvc executable; please see math/README to configure.): \langle P(t) \rangle _\sigma

, etc.

These "statistical" states of the system are called mixed states, as opposed to the pure states Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma=(p,q)

discussed further above.

Abstractly, mixed states arise as a statistical mixture of pure states.

Quantum states

Probability densities for the electron of a hydrogen atom in different quantum states

In quantum systems, the conceptual distinction between observables and states persists just as described above. The state Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma

of the system is fixed by the way the physicist prepares his experiment

(e.g., how he adjusts his particle source). As above, there is a distinction between pure states and mixed states, the latter being statistical mixtures of the former. However, some important differences arise in comparison with classical mechanics.

In quantum theory, even pure states show statistical behaviour. Regardless of how carefully we prepare the state Failed to parse (Missing texvc executable; please see math/README to configure.): \rho

of the system,

measurement results are not repeatable in general, and we must understand the expectation value Failed to parse (Missing texvc executable; please see math/README to configure.): \langle A \rangle _\sigma

of an observable Failed to parse (Missing texvc executable; please see math/README to configure.): A
as a statistical mean.

It is this mean that is predicted by physical theories.

For any fixed observable Failed to parse (Missing texvc executable; please see math/README to configure.): A , it is generally possible to prepare a pure state Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma_A

such that Failed to parse (Missing texvc executable; please see math/README to configure.): A
has a fixed 

value in this state: If we repeat the experiment several times, each time measuring Failed to parse (Missing texvc executable; please see math/README to configure.): A , we will always obtain the same measurement result, without any random behaviour. Such pure states Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma_A

are called eigenstates of Failed to parse (Missing texvc executable; please see math/README to configure.): A

.

However, it is generally impossible to prepare a simultaneous eigenstate for all observables. For example, we cannot prepare a state such that both the position measurement Failed to parse (Missing texvc executable; please see math/README to configure.): Q(t)

and the momentum measurement Failed to parse (Missing texvc executable; please see math/README to configure.): P(t)

(at the same time Failed to parse (Missing texvc executable; please see math/README to configure.): t ) produce "sharp" results; at least one of them will exhibit random behaviour.^[2] 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 changes its state. More precisely: After measuring an observable Failed to parse (Missing texvc executable; please see math/README to configure.): A , the system will be in an eigenstate of Failed to parse (Missing texvc executable; please see math/README to configure.): A . This expresses a kind of logical consistency: If we measure Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Consider two observables, Failed to parse (Missing texvc executable; please see math/README to configure.): A

and Failed to parse (Missing texvc executable; please see math/README to configure.): B

, where Failed to parse (Missing texvc executable; please see math/README to configure.): A

corresponds

to a measurement earlier in time than Failed to parse (Missing texvc executable; please see math/README to configure.): B .^[3] Suppose that the system is in an eigenstate of Failed to parse (Missing texvc executable; please see math/README to configure.): B . If we measure only Failed to parse (Missing texvc executable; please see math/README to configure.): B , we will not notice statistical behaviour. If we measure first Failed to parse (Missing texvc executable; please see math/README to configure.): A

and then Failed to parse (Missing texvc executable; please see math/README to configure.): B
in the same run of the experiment,

the system will transfer to an eigenstate of Failed to parse (Missing texvc executable; please see math/README to configure.): A

after the first measurement,

and we will generally notice that the results of Failed to parse (Missing texvc executable; please see math/README to configure.): B

are statistical.

Thus, quantum mechanical measurements influence one another, and it is important in which order they are performed.

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 entanglement. These entangled states lead to experimentally testable properties (Bell's theorem) that allow to distinguish between quantum theory and alternative classical (non-quantum) models.

Schrödinger picture vs. Heisenberg picture

In the discussion above, we have taken the observables Failed to parse (Missing texvc executable; please see math/README to configure.): P(t) , Failed to parse (Missing texvc executable; please see math/README to configure.): Q(t)

to be dependent on time, while the state Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma

was fixed once at the beginning of the experiment.

This approach is called the Heisenberg picture. One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the Schrödinger picture. Conceptually (and mathematically), both approaches are equivalent; choosing one of them is a matter of convention.

Both viewpoints are used in quantum theory. While non-relativistic quantum mechanics is usually formulated in terms of the Schrödinger picture, the Heisenberg picture is often preferred in a relativistic context, that is, for quantum field theory.

Formalism in quantum physics

See also: Mathematical formulation of quantum mechanics

Pure states as rays in a Hilbert space

Quantum physics is most commonly formulated in terms of linear algebra, as follows. Any given system is identified with some Hilbert space, such that each vector in the Hilbert space (apart from the origin) corresponds to a pure quantum state. In addition, two vectors that differ only by a nonzero complex scalar correspond to the same state (in other words, each pure state is a ray in the Hilbert space).

Alternatively, many authors choose to only consider normalized vectors (vectors of norm 1) as corresponding to quantum states. In this case, the set of all pure states corresponds to the unit sphere of a Hilbert space, with the proviso that two normalized vectors correspond to the same state if they differ only by a complex scalar of absolute value 1 (called a phase factor).

Bra-ket notation

Main article: Bra-ket notation

Calculations in quantum mechanics make frequent use of linear operators, inner products, dual spaces, and Hermitian conjugation. In order to make such calculations more straightforward, and to obviate the need (in some contexts) to fully understand the underlying linear algebra, Paul Dirac invented a notation to describe quantum states, known as bra-ket notation. Although the details of this are beyond the scope of this article (see the article Bra-ket notation), some consequences of this are:

• The variable name used to denote a vector (which corresponds to a pure quantum state) is chosen to be of the form Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rangle

(where the "Failed to parse (Missing texvc executable; please see math/README to configure.): \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 bold, lower-case letters, or letters with arrows on top.

• Instead of vector, the term ket is used synonymously.
• Each ket Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rangle

is uniquely associated with a so-called bra, denoted Failed to parse (Missing texvc executable; please see math/README to configure.): \langle\psi|

, which is also said to correspond to the same physical quantum state. Technically, the bra is an element of the dual space, and related to the ket by the Riesz representation theorem.

• Inner products (also called brackets) are written so as to look like a bra and ket next to each other: Failed to parse (Missing texvc executable; please see math/README to configure.): \lang \psi_1|\psi_2\rang

. (Note that the phrase "bra-ket" is supposed to resemble "bracket".)

Basis states

As with any vector space, if a basis is chosen for the Hilbert space of a system, then any ket can be expanded as a linear combination of those basis elements. Symbolically, given basis kets Failed to parse (Missing texvc executable; please see math/README to configure.): |k_i\rang , any ket Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rang

can be written

Failed to parse (Missing texvc executable; please see math/README to configure.): | \psi \rang = \sum_i c_i |k_i \rangle

where c_i are complex numbers. In physical terms, this is described by saying that Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rang

has been expressed as a quantum superposition of the states Failed to parse (Missing texvc executable; please see math/README to configure.): |k_i\rang

. If the basis kets are chosen to be orthonormal (as is often the case), then Failed to parse (Missing texvc executable; please see math/README to configure.): c_i=\lang k_i|\psi\rang .

One property worth noting is that the normalized states Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rang

are characterized by

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_i |c_i|^2 = 1

Expansions of this sort play an important role in measurement in quantum mechanics. In particular, If the Failed to parse (Missing texvc executable; please see math/README to configure.): |k_i\rang

are eigenstates (with eigenvalues Failed to parse (Missing texvc executable; please see math/README to configure.): k_i

) of an observable, and that observable is measured on the normalized state Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rang , then the probability that the result of the measurement is k_i is |c_i|². (The normalization condition above mandates that the total sum of probabilities is equal to one.)

A particularly important example is the position basis, which is the basis consisting of eigenstates of the observable which corresponds to measuring position. If these eigenstates are nondegenerate (for example, if the system is a single, spinless particle), then any ket Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rang

is associated with a complex-valued function of three-dimensional space:

Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(\mathbf{r}) \equiv \lang \mathbf{r} | \psi \rang

. This function is called the wavefunction corresponding to Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rang .

Superposition of pure states

Main article: Quantum superposition

One aspect of quantum states, mentioned above, is that superpositions of them can be formed. If Failed to parse (Missing texvc executable; please see math/README to configure.): |\alpha\rangle

and Failed to parse (Missing texvc executable; please see math/README to configure.): |\beta\rangle
are two kets corresponding to quantum states, the ket

Failed to parse (Missing texvc executable; please see math/README to configure.): c_\alpha|\alpha\rang+c_\beta|\beta\rang

is a different quantum state (possibly not normalized). Note that which quantum state it is depends on both the amplitudes and phases (arguments) of Failed to parse (Missing texvc executable; please see math/README to configure.): c_\alpha

and Failed to parse (Missing texvc executable; please see math/README to configure.): c_\beta

. In other words, for example, even though Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rang

and Failed to parse (Missing texvc executable; please see math/README to configure.): e^{i\theta}|\psi\rang
(for real θ) correspond to the same physical quantum state, they are not interchangeable, since for example Failed to parse (Missing texvc executable; please see math/README to configure.): |\phi\rang+|\psi\rang
and Failed to parse (Missing texvc executable; please see math/README to configure.): |\phi\rang+e^{i\theta}|\psi\rang
do not (in general) correspond to the same physical state. However, Failed to parse (Missing texvc executable; please see math/README to configure.): |\phi\rang+|\psi\rang
and Failed to parse (Missing texvc executable; please see math/README to configure.): e^{i\theta}(|\phi\rang+|\psi\rang)
do correspond to the same physical state. This is sometimes described by saying that "global" phase factors are unphysical, but "relative" phase factors are physical and important.

One example of a quantum interference phenomenon that arises from superposition is the double-slit experiment. The photon state is a superposition of two different states, one of which corresponds to the photon having passed through the left slit, and the other corresponding to passage through the right slit. The relative phase of those two states has a value which depends on the distance from each of the two slits. Depending on what that phase is, the interference is constructive at some locations and destructive in others, creating the interference pattern.

Another example of the importance of relative phase in quantum superposition is Rabi oscillations, where the relative phase of two states varies in time due to the Schrödinger equation. The resulting superposition ends up oscillating back and forth between two different states.

Mixed states

See also: Density matrix

A pure quantum state is a state which can be described by a single ket vector, as described above. A mixed quantum state is a statistical ensemble of pure states (see quantum statistical mechanics).

A mixed state cannot be described as a ket vector. Instead, it is described by its associated density matrix (or density operator), usually denoted Failed to parse (Missing texvc executable; please see math/README to configure.): \rho . Note that density matrices can describe both mixed and pure states, treating them on the same footing.

The density matrix is defined as

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho = \sum_s p_s | \psi_s \rangle \langle \psi_s |

where Failed to parse (Missing texvc executable; please see math/README to configure.): p_s

is the fraction of the ensemble in each pure state Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi_s\rangle.

A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the trace of ρ² is equal to 1 if the state is pure, and less than 1 if the state is mixed. Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state.

The rules for measurement in quantum mechanics are particularly simple to state in terms of density matrices. For example, the ensemble average (expectation value) of a measurement corresponding to an observable Failed to parse (Missing texvc executable; please see math/README to configure.): A

is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle A \rangle = \sum_s p_s \langle \psi_s | A | \psi_s \rangle = \sum_s \sum_i p_s a_i | \langle \alpha_i | \psi_s \rangle |^2 = tr(\rho A)

where Failed to parse (Missing texvc executable; please see math/README to configure.): |\alpha_i\rangle, \; a_i

are eigenkets and eigenvalues, respectively, for the operator Failed to parse (Missing texvc executable; please see math/README to configure.): A

, and tr denotes trace. It is important to note that two types of averaging are occurring, one being a quantum average over the basis kets of the pure states, and the other being a statistical average over the ensemble of pure states.

Mathematical formulation

For a mathematical discussion on states as functionals, see Gelfand-Naimark-Segal construction. There, the same objects are described in a C*-algebraic context.

Notes

1. ^  If you are not familiar with the concept of momentum, think of it as being the velocity of the particle. That is fully justified in this context.
2. ^  To avoid misunderstandings: Here we mean that Failed to parse (Missing texvc executable; please see math/README to configure.): Q(t)

and Failed to parse (Missing texvc executable; please see math/README to configure.): P(t)
are measured in the same state, but not in the  same run of the experiment.)

1. ^  For concreteness' sake, you may suppose that Failed to parse (Missing texvc executable; please see math/README to configure.): A=Q(t_1)

and Failed to parse (Missing texvc executable; please see math/README to configure.): B=P(t_2)
in the above example, with Failed to parse (Missing texvc executable; please see math/README to configure.): t_2>t_1>0

.

See also

• Bra-ket notation
• Density matrix
• Quantum harmonic oscillator
• Qubit
• Orthonormal basis
• Probability amplitude
• Wave function

Further reading

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:

• Isham, Chris J (1995). Lectures on Quantum Theory: Mathematical and Structural Foundations. Imperial College Press. ISBN 978-1860940019. 

For a more detailed coverage of mathematical aspects, see:

• Bratteli, Ola; Robinson, Derek W (1987). Operator Algebras and Quantum Statistical Mechanics 1. Springer. 2nd edition. ISBN 978-3540170938.  In particular, see Sec. 2.3.cs:Kvantový stav

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