Quantum state

Quantum mechanics
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In quantum physics, quantum state refers to the state of a quantum system.

A quantum state can be either pure or mixed. A pure quantum state corresponds^[1]^[3] to a vector, called a state vector, in a Hilbert space. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant state vectors are identified by the principal quantum number, written $\{ n \}$ . For a more complicated case, consider Bohm's formulation of the EPR experiment in bra–ket notation, where the state vector

\left|\psi\right\rang = \frac{1}{\sqrt{2}}\bigg(\left|\uparrow\downarrow\right\rang - \left|\downarrow\uparrow\right\rang \bigg)

involves superposition of joint spin states for two particles. Mathematically, a pure quantum state corresponds to a state vector in a Hilbert space over complex numbers, which is a generalization of our more usual three-dimensional space.^[4]^:93–96 If this Hilbert space is represented as a function space, then its elements are called 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. Mixed states are described by so-called density matrices. A pure state can also be recast as a density matrix; in this way, pure states can be represented as a subset of the more general mixed states.

For 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. The Hilbert space for the electron's spin is therefore two-dimensional. 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 $\beta$ . A mixed state, in this case, is a $2 \times 2$ matrix that is Hermitian, positive-definite, and has trace 1.

Before a particular measurement is performed on a quantum system, the theory usually gives only a probability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and the observable describing the measurement. These probability distributions arise for both mixed states and pure states: it is impossible in quantum mechanics (unlike classical mechanics) to prepare a state in which all properties of the system are fixed and certain. This is exemplified by the uncertainty principle, and reflects a core difference between classical and quantum physics. Even in quantum theory, however, for every observable there are some states that have an exact and determined value for that observable.^[4]^:4–5^[5]

Conceptual description

Pure states

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

In the mathematical formulation of quantum mechanics, pure quantum states correspond to vectors in a Hilbert space, while each observable quantity (such as the energy or momentum of a particle) is associated with a mathematical operator. The operator serves as a linear function which acts on the states of the system. The eigenvalues of the operator correspond to the possible values of the observable, i.e. 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 system in a linear combination of multiple different eigenstates does in general have quantum uncertainty for the given observable. We can represent this linear combination of eigenstates as:

|\Psi(t)\rangle = \sum_n C_n(t) |\Phi_n\rang

The coefficient which corresponds to a particular state in the linear combination is complex thus allowing interference effects between states. The coefficients are time dependent. How a quantum system changes in time is governed by the time evolution operator. The symbols $|$ and $\rangle$ ^{[lower-alpha 1]} surrounding the $\Psi$ are part of bra–ket notation.

Statistical mixtures of states are different from a 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.^[6]^[7]

The expectation value $\langle A \rangle _\sigma$ of an observable A 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 which is simultaneously an eigenstate for all observables. For example, we cannot prepare a state such that both the position measurement Q(t) and the momentum measurement P(t) (at the same time t) are known exactly; at least one of them will have a range of possible values.^{[lower-alpha 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 generally changes its state.^[8]^[9]^[10] 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,^{[lower-alpha 3]} then they will produce the same results. This has some strange consequences, however, as follows.

Consider two incompatible observables, A and B, where A corresponds to a measurement earlier in time than B.^{[lower-alpha 4]} Suppose that the system is in an eigenstate of B at the experiment's begin. If we measure only B, all runs of the experiment will yield the same result. If we measure first A and then B in the same run of the experiment, the system will transfer to an eigenstate of A after the first measurement, and we will generally notice that the results of 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 us to distinguish between quantum theory and alternative classical (non-quantum) models.

Schrödinger picture vs. Heisenberg picture

One can take the observables to be dependent on time, while the state σ 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 P(t), Q(t).) 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.^[11]^:65

Formalism in quantum physics

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 finite- or infinite-dimensional Hilbert space. The pure states correspond to vectors of norm 1. Thus the set of all pure states corresponds to the unit sphere in the Hilbert space.

Multiplying a pure state by a scalar is physically inconsequential (as long as the state is considered by itself). If one vector is obtained from the other by multiplying by a scalar of unit magnitude, the two vectors are said to correspond to the same "ray" in Hilbert space^[12] and also to the same point in the projective Hilbert space.

Bra–ket notation

Main article: Bra–ket notation

Calculations in quantum mechanics make frequent use of linear operators, scalar products, dual spaces and Hermitian conjugation. In order to make such calculations flow smoothly, 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, some consequences of this are:

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 bold, lower-case letters, or letters with arrows on top.
Dirac defined two kinds of vector, bra and ket, dual to each other.^[13]
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 adjoint 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 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 $|\psi\rangle$ .
Scalar products^[14]^[15] (also called brackets) are written so as to look like a bra and ket next to each other: $\lang \psi_1|\psi_2\rang$ . (The phrase "bra-ket" is supposed to resemble "bracket".)

Spin

The angular momentum has the same dimension as the Planck constant and, at quantum scale, behaves as a discrete degree of freedom. Most particles possess a kind of intrinsic angular momentum that does not appear at all in classical mechanics and arises from Dirac's relativistic generalization of the theory. Mathematically it is described with spinors. In non-relativistic quantum mechanics the group representations of the Lie group SU(2) are used to describe this additional freedom. For a given particle, the choice of representation (and hence the range of possible values of the spin observable) is specified by a non-negative number S that, in units of Planck's reduced constant ħ, is either an integer (0, 1, 2 ...) or a half-integer (1/2, 3/2, 5/2 ...). For a massive particle with spin S, its spin quantum number m always assumes one of the 2S + 1 possible values in the set

\{ -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).

Many-body states and particle statistics

Further information: Particle statistics

The quantum state of a system of N particles, each potentially with spin, is described by a complex-valued function with four variables per particle, e.g.

|\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 S = 1/2, photons (quanta of light) are bosons with S = 1 (although in the vacuum they are massless and can't be described with Schrödingerian mechanics).

When symmetrization or anti-symmetrization is unnecessary, N-particle spaces of states can be obtained simply by tensor products of one-particle spaces, to which we will return later.

Basis states of one-particle systems

As with any Hilbert 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 $|{k_i}\rang$ , any ket $|\psi\rang$ can be written

| \psi \rang = \sum_i c_i |{k_i}\rangle

where c_i are complex numbers. In physical terms, this is described by saying that $|\psi\rang$ has been expressed as a quantum superposition of the states $|{k_i}\rang$ . If the basis kets are chosen to be orthonormal (as is often the case), then $c_i=\lang {k_i} | \psi \rang$ .

One property worth noting is that the normalized states $|\psi\rang$ are characterized by

\lang\psi|\psi\rang = 1,

and for orthonormal basis this translates to

\sum_i \left | c_i \right | ^2 = 1.

Expansions of this sort play an important role in measurement in quantum mechanics. In particular, if the $|{k_i}\rang$ are eigenstates (with eigenvalues k_i) of an observable, and that observable is measured on the normalized state $|\psi\rang$ , 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 $|\mathbf{r}\rang$ with eigenvalues $\mathbf{r}$ of the observable which corresponds to measuring position.^{[lower-alpha 5]} If these eigenstates are nondegenerate (for example, if the system is a single, spinless particle), then any ket $|\psi\rang$ is associated with a complex-valued function of three-dimensional space

\psi(\mathbf{r}) \equiv \lang \mathbf{r} | \psi \rang.

^{[lower-alpha 6]} i.e.

\lang \mathbf{r} | \mathbf{r}' \rang = \delta(\mathbf{r}' - \mathbf{r})

Dirac delta function), which means that

\lang \mathbf{r} | \mathbf{r} \rang = \infty.

This function is called the wave function corresponding to $|\psi\rang$ . Similarly to the discrete case above, the probability density of the particle being found at position $\mathbf{r}$ is $|\psi(\mathbf{r})|^2$ and the normalized states have

\int \mathrm{d}^3 \mathbf{r} |\psi(\mathbf{r})|^2 = 1

In terms of the continuous set of position basis $|\mathbf{r}\rang$ , the state $|\psi \rang$ is:

|\psi \rang = \int \mathrm{d}^3 \mathbf{r} \text{ } \psi (\mathbf{r}) |\mathbf{r}\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 $|\alpha\rangle$ and $|\beta\rangle$ are two kets corresponding to quantum states, the ket

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 $c_\alpha$ and $c_\beta$ . In other words, for example, even though $|\psi\rang$ and $e^{i\theta}|\psi\rang$ (for real θ) correspond to the same physical quantum state, they are not interchangeable, since for example $|\phi\rang+|\psi\rang$ and $|\phi\rang+e^{i\theta}|\psi\rang$ do not (in general) correspond to the same physical state. However, $|\phi\rang+|\psi\rang$ and $e^{i\theta}(|\phi\rang+|\psi\rang)$ 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. By the analogy with coherence in other wave phenomena, a superposed state can be referred to as a coherent superposition.

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

Interpretation

Main article: Interpretations of quantum mechanics

Although theoretically, for a given quantum system, a state vector provides the full information about its evolution, it is not easy to understand what information about the "real world" it carries. Due to the uncertainty principle, a state, even if it has the value of one observable exactly defined (i.e. the observable has this state as an eigenstate), cannot exactly define values of all observables.

For state vectors (pure states), probability amplitudes offer a probabilistic interpretation. It can be generalized for all states (including mixed), for instance, as expectation values mentioned above.

Mathematical generalizations

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.

Notes

↑ Sometimes written ">"; see angle brackets.
↑ To avoid misunderstandings: Here we mean that $Q (t)$ and $P (t)$ are measured in the same state, but not in the same run of the experiment.
↑ i.e. separated by a zero delay. One can think of it as stopping the time, then making the two measurements one after the other, then resuming the time. Thus, the measurements occured at the same time, but it is still possible to tell which was first.
↑ For concreteness' sake, suppose that A = Q(t₁) and B = P(t₂) in the above example, with t₂ > t₁ > 0.
↑ Note that a state $|\psi\rang$ is a superposition of different basis states $|\mathbf{r}\rang$ , so $|\psi\rang$ and $|\mathbf{r}\rang$ are elements of the same Hilbert space. A particle in state $|\mathbf{r}\rang$ is located precisely at position $\mathbf{r} = (x,y,z)$ , while a particle in state $|\psi\rang$ can be found at different positions with corresponding probabilities.
↑ In the continuous case, the basis kets $|\mathbf{r}\rang$ are not unit kets (unlike the state $|\psi\rang$ ): They are normalized according to $\textstyle \int \mathrm{d}^3 \mathbf{r}' \lang \mathbf{r} | \mathbf{r}' \rang = 1,$ ^[16] p. 17: " $\int Ψ f' Ψ f * d q = δ(f' - f)$ " (the left side corresponds to $〈 f | f'〉$ ), " $\int δ(f' - f) d f' = 1$ ".
↑ Note that this criterion works when the density matrix is normalized so that the trace of ρ is 1, as it is for the standard definition given in this section. Occasionally a density matrix will be normalized differently, in which case the criterion is $\operatorname{Tr}(\rho^2)=(\operatorname{Tr} \rho)^2$

References

↑ Dirac (1958),^[2] p. 16: "each state of a dynamical system at a particular time corresponds to a ket vector."
1 2 3 4 Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.
↑ Feynman, R.P., Leighton, R.B., Sands, M. (1963). The Feynman Lectures on Physics, Addison-Wesley, Reading MA, available at http://www.feynmanlectures.info/, Volume $III$ , p. 8–2: "The states $χ$ and $φ$ correspond to the two vectors $B$ and $A$ ."
1 2 Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, ISBN 0-13-111892-7
↑ Ballentine, L. E. (1970), "The Statistical Interpretation of Quantum Mechanics", Reviews of Modern Physics 42: 358–381, Bibcode:1970RvMP...42..358B, doi:10.1103/RevModPhys.42.358
↑ Statistical Mixture of States
↑ http://electron6.phys.utk.edu/qm1/modules/m6/statistical.htm
↑ Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys. 43: 172–198. Translation as 'The actual content of quantum theoretical kinematics and mechanics'. Also translated as 'The physical content of quantum kinematics and mechanics' at pp. 62–84 by editors John Wheeler and Wojciech Zurek, in Quantum Theory and Measurement (1983), Princeton University Press, Princeton NJ.
↑ Bohr, N. (1927/1928). The quantum postulate and the recent development of atomic theory, Nature Supplement April 14 1928, 121: 580–590.
↑ Dirac (1958),^[2] p. 4: "If a system is small, we cannot observe it without producing a serious disturbance."
1 2 Gottfried, Kurt; Yan, Tung-Mow (2003). Quantum Mechanics: Fundamentals (2nd, illustrated ed.). Springer. ISBN 9780387955766.
↑ Weinberg, Steven. "The Quantum Theory of Fields", Vol. 1. Cambridge University Press, 1995 p. 50.
↑ Dirac (1958),^[2] p. 20: "The bra vectors, as they have been here introduced, are quite a different kind of vector from the kets, and so far there is no connexion between them except for the existence of a scalar product of a bra and a ket."
↑ Dirac (1958),^[2] p. 19: "A scalar product $〈 B | A 〉$ now appears as a complete bracket expression."
↑ Gottfried (2013),^[11] p. 31: "to define the scalar products as being between bras and kets."
↑ Landau (1965),<ref name='Landau (1965)' group=''>
↑ Blum, Density matrix theory and applications, page 39.
↑ Eugene Wigner (1962). "Remarks on the mind-body question" (PDF). In I.J. Good. The Scientist Speculates. London: Heinemann. pp. 284–302. Footnote 13 on p.180
↑ Lev Landau (1927). "Das Dämpfungsproblem in der Wellenmechanik (The Damping Problem in Wave Mechanics)". Zeitschrift für Physik 45 (5–6): 430–441. Bibcode:1927ZPhy...45..430L. doi:10.1007/bf01343064. English translation reprinted in: D. Ter Haar, ed. (1965). Collected papers of L.D. Landau. Oxford: Pergamon Press. p.8–18
↑ Lev Landau; Evgeny Lifshitz (1965). Quantum Mechanics — Non-Relativistic Theory (PDF). Course of Theoretical Physics 3 (2nd ed.). London: Pergamon Press.