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.

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.

Pure states of wave functions

for the electron of a hydrogen atom in different quantum states.]] 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 (\alpha, \beta) , with a length of one; that is, with |\alpha|^2 + |\beta|^2 = 1, where |\alpha| and |\beta| are the absolute values of \alpha and \beta . 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 (such as the energy or momentum of a particle) 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 (which physicists call an eigenstate) 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: |\Psi(t)\rangle = \sum_n C_n(t) |\Phi_n\rang. 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 (i.e., physically indistinguishable) 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 2 \times 2 matrix that is Hermitian and positive semi-definite, and has trace 1. A more complicated case is given (in bra–ket notation) by the singlet state, which exemplifies quantum entanglement: \left|\psi\right\rang = \frac{1}{\sqrt{2}}\bigl(\left|\uparrow\downarrow\right\rang - \left|\downarrow\uparrow\right\rang \bigr), 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 \Phi_n . A number P_n represents the probability of a randomly selected system being in the state \Phi_n . Unlike the linear combination case each system is in a definite eigenstate. The expectation value {\langle A \rangle}_\sigma 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 (at the same time ) 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 (Bell's theorem) that allow us to distinguish between quantum theory and alternative classical (non-quantum) models. One can take the observables to be dependent on time, while the state was fixed once at the beginning of the experiment. This approach is called the Heisenberg picture. (This approach was taken in the later part of the discussion above, with time-varying observables , .) One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the Schrödinger picture. (This approach was taken in the earlier part of the discussion above, with a time-varying state |\Psi(t)\rangle = \sum_n C_n(t) |\Phi_n\rang .) Conceptually (and mathematically), the two 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. Compare with Dirac picture. \{ -S, -S+1, \ldots, S-1, S \} As a consequence, the quantum state of a particle with spin is described by a vector-valued wave function with values in C 2S+1 . Equivalently, it is represented by a complex-valued function of four variables: one discrete quantum number variable (for the spin) is added to the usual three continuous variables (for the position in space). |\psi (\mathbf r_1,\, m_1;\; \dots;\; \mathbf r_N,\, m_N)\rangle. Here, the spin variables m ν assume values from the set \{-S_\nu,\, -S_\nu + 1,\, \ldots,\, S_\nu - 1,\, S_\nu \} where S_\nu is the spin of νth particle. S_\nu = 0 for a particle that does not exhibit spin. The treatment of identical particles is very different for bosons (particles with integer spin) versus fermions (particles with half-integer spin). The above N-particle function must either be symmetrized (in the bosonic case) or anti-symmetrized (in the fermionic case) with respect to the particle numbers. If not all N particles are identical, but some of them are, then the function must be (anti)symmetrized separately over the variables corresponding to each group of identical variables, according to its statistics (bosonic or fermionic). Electrons are fermions with , photons (quanta of light) are bosons with (although in the vacuum they are massless and can't be described with Schrödinger mechanics). When symmetrization or anti-symmetrization is unnecessary, -particle spaces of states can be obtained simply by tensor products of one-particle spaces, to which we will return later. \begin{align}| \psi \rang &= \sum_i c_i |{k_i}\rangle,\\ &= \sum_i |{k_i}\rangle\langle k_{i} |\psi\rangle, \end{align} with complex coefficients c_i = \lang {k_i} | \psi \rang and basis elements |k_i\rang . In this case, the normalization condition translates to \lang\psi|\psi\rang = \sum_i \lang \psi |{k_i}\rangle\langle k_{i} |\psi\rangle = \sum_i \left | c_i \right | ^2 = 1. In physical terms, |\psi\rang has been expressed as a quantum superposition of the "basis states" |{k_i}\rang , i.e., the eigenstates of an observable. In particular, if said observable is measured on the normalized state |\psi\rang , then |c_i|^2 = |\lang {k_i} | \psi \rang|^2, is the probability that the result of the measurement is k_i . In general, the expression for probability always consist of a relation between the quantum state and a portion of the spectrum of the dynamical variable (i.e. random variable) being observed. For example, the situation above describes the discrete case as eigenvalues k_i belong to the point spectrum. Likewise, the wave function is just the eigenfunction of the Hamiltonian operator with corresponding eigenvalue(s) E ; the energy of the system. An example of the continuous case is given by the position operator. The probability measure for a system in state \psi is given by: \mathrm{Pr}(x \in B | \psi) = \int_{B \subset \mathbb{R}} |\psi(x)|^2 d x, where |\psi(x)|^2 is the probability density function for finding a particle at a given position. These examples emphasize the distinction in charactertistics between the state and the observable. That is, whereas \psi is a pure state belonging to H , the (generalized) eigenvectors of the position operator do not. Though closely related, pure states are not the same as bound states belonging to the pure point spectrum of an observable with no quantum uncertainty. A particle is said to be in a bound state if it remains localized in a bounded region of space for all times. A pure state |\phi\rangle is called a bound state if and only if for every \varepsilon >0 there is a compact set K\subset \mathbb{R}^3 such that \int_K |\phi(\mathbf{r},t)|^2\,\mathrm{d}^3 \mathbf{r} \geq 1 - \varepsilon for all t\in\mathbb{R} . The integral represents the probability that a particle is found in a bounded region K at any time t . If the probability remains arbitrarily close to 1 then the particle is said to remain in K . For example, non-normalizable solutions of the free Schrödinger equation can be expressed as functions that are normalizable, using wave packets. These wave packets belong to the pure point spectrum of a corresponding projection operator which, mathematically speaking, constitutes an observable. However, they are not bound states. c_\alpha|\alpha\rang + c_\beta|\beta\rang is also a quantum state of the same system. Both c_\alpha and c_\beta can be complex numbers; their relative amplitude and relative phase will influence the resulting quantum state. Writing the superposed state using c_\alpha = A_\alpha e^{i\theta_\alpha} \ \ c_\beta = A_\beta e^{i\theta_\beta} and defining the norm of the state as: |c_\alpha|^2 + |c_\beta|^2 = A_\alpha^2 + A_\beta^2 = 1 and extracting the common factors gives: e^{i\theta_\alpha} \left(A_\alpha|\alpha\rang + \sqrt{1-A_\alpha^2} e^{i\theta_\beta - i\theta_\alpha}|\beta\rang \right) The overall phase factor in front has no physical effect. Only the relative phase affects the physical nature of the superposition. One example of superposition is the double-slit experiment, in which superposition leads to quantum interference. Another example of the importance of relative phase 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. \rho = \sum_s p_s | \psi_s \rangle \langle \psi_s | where is the fraction of the ensemble in each pure state |\psi_s\rangle. The density matrix can be thought of as a way of using the one-particle formalism to describe the behavior of many similar particles by giving a probability distribution (or ensemble) of states that these particles can be found in. A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the trace of 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 is given by \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 = \operatorname{tr}(\rho A) where |\alpha_i\rangle and a_i are eigenkets and eigenvalues, respectively, for the operator , and " " denotes trace. Two types of averaging are occurring, one (over i ) being the usual expected value of the observable when the quantum is in state |\psi_s\rangle , and the other (over s ) being a statistical (said incoherent) average with the probabilities that the quantum is in those states.

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