Quantum state

Quantum mechanics
Introduction Glossary · History
Background Bra–ket notation Classical mechanics Hamiltonian Interference Old quantum theory
Fundamental concepts Complementarity Decoherence Nonlocality Quantum state Superposition Tunnelling Uncertainty Wave function
Experiments Bell's inequality Davisson–Germer Delayed choice quantum eraser Double-slit Franck–Hertz experiment Mach-Zehnder inter. Elitzur–Vaidman Popper Quantum eraser Schrödinger's cat Stern–Gerlach Wheeler's delayed choice
Formulations Formulations Heisenberg Interaction Matrix mechanics Schrödinger Sum over histories Phase space Relativistic quantum mechanics
Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger
Interpretations Interpretations (overview) Consciousness-caused Consistent histories Copenhagen de Broglie–Bohm Ensemble Hidden variables Many-worlds Objective collapse Pondicherry Quantum logic Relational Stochastic Transactional
Advanced topics Quantum chaos Quantum field theory Quantum information science Scattering theory Fractional quantum mechanics
Scientists Bell Blackett Bogolyubov Bohm Bohr Bardeen Born Bose de Broglie Compton Cooper Dirac Dyson Davisson Duarte Debye Ehrenfest Einstein Everett Fock Fermi Feynman Hertz Heisenberg Hilbert Jordan Klitzing Kusch Kramers von Neumann Pauli Lamb Laue Laughlin Moseley Millikan Onnes Planck Raman Rydberg Salam Schrödinger Störmer Shockley Schrieffer Shull Sommerfeld Thomson Tsui Tomonaga Weyl Ward Wien Wigner Zeeman Zeilinger
v t e

In quantum physics, quantum state refers to the state of a quantum system. A quantum state is given as a vector in a Hilbert space, called the state vector. The state vector theoretically contains statistical information about the quantum system. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant state vector is given by the principal quantum number $\{ n \}$ . For a more complicated case, consider Bohm formulation of EPR experiment, 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 2 different particles.^:47-48

In a more general usage, a quantum state can be either "pure" or "mixed." The above example is pure. Mathematically, a pure quantum state is represented by a state vector in a Hilbert space, which is a generalization of our more usual three dimensional space.^:93-96 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. Quantum states, mixed as well as pure, are described by so-called density matrices, although these give probabilities, not densities.

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 is a two-dimensional complex vector $(\alpha, \beta)$ , with a length of one. That is,

$|\alpha|^2 + |\beta|^2 = 1\,.$

A mixed state 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 states that determine its value exactly.

Conceptual description [edit]

Quantum states [edit]

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. 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 ">" surrounding the $\Psi$ are part of bra-ket notation.

Statistical mixtures of states are separate from a linear combination. A statistical mixture of states occurs with 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 of complex coefficients but by a combination of probabilities of different states $\Phi_n$ . $P_n$ 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.

In general we must understand the expectation value $\langle A \rangle _\sigma$ of an observable A as a statistical mean. 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. 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:

Consider two observables, A and B, where A corresponds to a measurement earlier in time than B. Suppose that the system is in an eigenstate of B. If we measure only B, we will not notice statistical behaviour. 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 [edit]

In the discussion above, we have taken the observables P(t), Q(t) 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. 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. Compare with Dirac picture.

Formalism in quantum physics [edit]

Pure states as rays in a Hilbert space [edit]

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.

If two unit vectors differ only by a scalar of magnitude 1, known as a "global phase factor," then they are indistinguishable. Therefore, distinct pure states can be put in correspondence with "rays" in the Hilbert space, or equivalently points in the projective Hilbert space.

Bra-ket notation [edit]

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 $|\psi\rangle$ (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.
Instead of vector, the term ket is used synonymously.
Each ket $|\psi\rangle$ is uniquely associated with a so-called bra, denoted $\langle\psi|$ , which is also said to correspond 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; just take the transpose and entry-wise complex conjugate of $|\psi\rangle$ .
Inner products (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, many-body states [edit]

Main article: Mathematical formulation of quantum mechanics#Spin

It is important to note that in quantum mechanics besides, e.g., the usual position variable r, a discrete variable m exists, corresponding to the value of the z-component of the spin vector. This can be thought of as a kind of intrinsic angular momentum. However, it does not appear at all in classical mechanics and arises from Dirac's relativistic generalization of the theory. As a consequence, the quantum state of a system of N particles is described by a function with four variables per particle, e.g.

$|\psi (\mathbf r_1,m_1;\dots ;\mathbf r_N,m_N)\rangle.$

Here, the variables m_ν assume values from the set

$\{ -S_\nu, -S_\nu +1 \cdots +S_\nu -1,+S_\nu \}$

where $S_\nu$ (in units of Planck's reduced constant ħ = 1), is either a non-negative integer (0, 1, 2 ... for bosons), or semi-integer (1/2, 3/2, 5/2 ... for fermions). Moreover, in the case of identical particles, 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.

Electrons are fermions with S = 1/2, photons (quanta of light) are bosons with S = 1.

Apart from the symmetrization or anti-symmetrization, N-particle states can thus simply be obtained by tensor products of one-particle states, to which we return herewith.

Basis states of one-particle systems [edit]

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

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

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|. (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 $|\psi\rang$ is associated with a complex-valued function of three-dimensional space:

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

This function is called the wavefunction corresponding to $|\psi\rang$ .

Superposition of pure states [edit]

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.

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 [edit]

Notes [edit]

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.

References [edit]

Ballentine, Leslie (1998). Quantum Mechanics: A Modern Development (2nd, illustrated, reprint ed.). World Scientific. ISBN - get this book.
Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, ISBN - get this book
Ballentine, L. E. (1970), "The Statistical Interpretation of Quantum Mechanics", Reviews of Modern Physics 42: 358–381, doi:10.1103/RevModPhys.42.358
Statistical Mixture of States
http://electron6.phys.utk.edu/qm1/modules/m6/statistical.htm
For concreteness' sake, suppose that A = Q(t₁) and B = P(t₂) in the above example, with t₂ > t₁ > 0.
Gottfried, Kurt; Yan, Tung-Mow (2003). Quantum Mechanics: Fundamentals (2nd, illustrated ed.). Springer. pp. pp. 65. ISBN - get this book.
Blum, Density matrix theory and applications, page 39. 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 $\operatorname{Tr}(\rho^2)=(\operatorname{Tr} \rho)^2$

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