Quantum state

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quantum_state 

 

In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system represented by the 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 may be defined differently for different kinds of systems or problems. Two broad categories are

• wave functions describing quantum systems using position or momentum variables and
• the more abstract vector quantum states.

Historical, educational, and application-focused problems typically feature wave functions; modern professional physics uses the abstract vector states. In both categories, quantum states divide into pure versus mixed states, or into coherent states and incoherent states. Categories with special properties include stationary states for time independence and quantum vacuum states in quantum field theory.

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

Main article: Measurement in quantum mechanics

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 ${\displaystyle x}$ 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

See also: Eigenvalues and eigenvectors § Schrödinger equation

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 (see § Formalism in quantum physics below) 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

Main article: Wave function

Wave functions 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 ${\displaystyle x,y,z}$ 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

Probability densities 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 n, the angular momentum quantum number ℓ, the magnetic quantum number m, and the spin z-component s_z. 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 ${\displaystyle (\alpha ,\beta )}$, with a length of one; that is, with $${\displaystyle |\alpha {|}^2+|\beta {|}^2=1,}$$ where ${\displaystyle |\alpha |}$ and ${\displaystyle |\beta |}$ are the absolute values of ${\displaystyle \alpha }$ and ${\displaystyle \beta }$.

The postulates of quantum mechanics state that pure states, at a given time t, 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: $${\displaystyle |\mathrm{\Psi }(t)⟩=\sum _n{C}_n(t)|{\mathrm{\Phi }}_n⟩.}$$ 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 ${\displaystyle 2\times 2}$ 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: $${\displaystyle |\psi ⟩=\frac{1}{\sqrt{2}}(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩),}$$ 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 in 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 ${\displaystyle {\mathrm{\Phi }}_n}$. A number ${\displaystyle P_n}$ represents the probability of a randomly selected system being in the state ${\displaystyle {\mathrm{\Phi }}_n}$. Unlike the linear combination case each system is in a definite eigenstate.

The expectation value ${\displaystyle {⟨A⟩}_{\sigma }}$ 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 that 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.^[a] 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, A and B, where A corresponds to a measurement earlier in time than B. Suppose that the system is in an eigenstate of B at the experiment's beginning. 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 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.

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 $|\mathrm{\Psi }(t)⟩=\sum _n{C}_n(t)|{\mathrm{\Phi }}_n⟩$.) 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.

Formalism in quantum physics

See also: Mathematical formulation of quantum mechanics

Pure states as rays in a complex Hilbert space

See also: Wigner's theorem § Rays and ray 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, because the unit sphere is defined as the set of all vectors with norm 1.

Multiplying a pure state by a scalar is physically inconsequential (as long as the state is considered by itself). If a vector in a complex Hilbert space ${\displaystyle H}$ can be obtained from another vector by multiplying by some non-zero complex number, the two vectors in ${\displaystyle H}$ are said to correspond to the same ray in the projective Hilbert space ${\displaystyle \mathbf{P}(H)}$ of ${\displaystyle H}$. Note that although the word ray is used, properly speaking, a point in the projective Hilbert space corresponds to a line passing through the origin of the Hilbert space, rather than a half-line, or ray in the geometrical sense.

Spin

Main article: Mathematical formulation of quantum mechanics § Spin

The angular momentum has the same dimension (M·L²·T⁻¹) as the Planck constant and, at quantum scale, behaves as a discrete degree of freedom of a quantum system. 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 the reduced Planck 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 $${\displaystyle \{-S,-S+1,\dots ,S-1,S\}}$$

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, corresponding to 3 spatial coordinates and spin, e.g. $${\displaystyle |\psi ({\mathbf{r}}_1,m_1;\dots ;{\mathbf{r}}_N,m_N)⟩.}$$

Here, the spin variables m_ν assume values from the set $${\displaystyle \{-S_{\nu },-S_{\nu }+1,\dots ,S_{\nu }-1,S_{\nu }\}}$$ where ${\displaystyle S_{\nu }}$ is the spin of νth particle. ${\displaystyle S_{\nu }=0}$ 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 Schrödinger 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

See also: Dirac delta function § Quantum mechanics

A state ${\displaystyle |\psi ⟩}$ belonging to a separable complex Hilbert space ${\displaystyle H}$ can always be expressed uniquely as a linear combination of elements of an orthonormal basis of ${\displaystyle H}$. Using bra–ket notation, this means any state ${\displaystyle |\psi ⟩}$ can be written as $${\displaystyle \begin{array}{rl}|\psi ⟩& =\sum _i{c}_i|k_i⟩,\\ & =\sum _i|k_i⟩⟨k_i|\psi ⟩,\end{array}}$$ with complex coefficients ${\displaystyle c_i=⟨k_i|\psi ⟩}$ and basis elements ${\displaystyle |k_i⟩}$. In this case, the normalization condition translates to $${\displaystyle ⟨\psi |\psi ⟩=\sum _i⟨\psi |k_i⟩⟨k_i|\psi ⟩=\sum _i{\left|c_i\right|}^2=1.}$$ In physical terms, ${\displaystyle |\psi ⟩}$ has been expressed as a quantum superposition of the "basis states" ${\displaystyle |k_i⟩}$, i.e., the eigenstates of an observable. In particular, if said observable is measured on the normalized state ${\displaystyle |\psi ⟩}$, then $${\displaystyle |c_i{|}^2=|⟨k_i|\psi ⟩{|}^2,}$$ is the probability that the result of the measurement is ${\displaystyle k_i}$.

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 ${\displaystyle k_i}$ belong to the point spectrum. Likewise, the wave function is just the eigenfunction of the Hamiltonian operator with corresponding eigenvalue(s) ${\displaystyle E}$; 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 ${\displaystyle \psi }$ is given by: $${\displaystyle \mathrm{P}\mathrm{r}(x\in B|\psi )={\int }_{B\subset \mathbb{R}}|\psi (x){|}^{2}dx,}$$ where ${\displaystyle |\psi (x){|}^2}$ 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 ${\displaystyle \psi }$ is a pure state belonging to ${\displaystyle H}$, the (generalized) eigenvectors of the position operator do not.

Pure states vs. bound states

See also: Decomposition of spectrum (functional analysis) § Quantum mechanics

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 ${\displaystyle |\varphi ⟩}$ is called a bound state if and only if for every ${\displaystyle \epsilon >0}$ there is a compact set ${\displaystyle K\subset {\mathbb{R}}^3}$ such that $${\displaystyle {\int }_K|\varphi (\mathbf{r},t){|}^2{\mathrm{d}}^3\mathbf{r}\ge 1-\epsilon }$$ for all ${\displaystyle t\in \mathbb{R}}$. The integral represents the probability that a particle is found in a bounded region ${\displaystyle K}$ at any time ${\displaystyle t}$. If the probability remains arbitrarily close to ${\displaystyle 1}$ then the particle is said to remain in ${\displaystyle K}$.

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.

Superposition of pure states

Main article: Quantum superposition

As mentioned above, quantum states may be superposed. If ${\displaystyle |\alpha ⟩}$ and ${\displaystyle |\beta ⟩}$ are two kets corresponding to quantum states, the ket $${\displaystyle c_{\alpha }|\alpha ⟩+c_{\beta }|\beta ⟩}$$ is also a quantum state of the same system. Both ${\displaystyle c_{\alpha }}$ and ${\displaystyle c_{\beta }}$ can be complex numbers; their relative amplitude and relative phase will influence the resulting quantum state.

Writing the superposed state using $${\displaystyle c_{\alpha }=A_{\alpha }{e}^{i{\theta }_{\alpha }}{c}_{\beta }=A_{\beta }{e}^{i{\theta }_{\beta }}}$$ and defining the norm of the state as: $${\displaystyle |c_{\alpha }{|}^2+|c_{\beta }{|}^2=A_{\alpha }^2+A_{\beta }^2=1}$$ and extracting the common factors gives: $${\displaystyle e^{i{\theta }_{\alpha }}\left(A_{\alpha }|\alpha ⟩+\sqrt{1-A_{\alpha }^2}{e}^{i{\theta }_{\beta }-i{\theta }_{\alpha }}|\beta ⟩\right)}$$ 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.

Mixed states

Main article: 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).

Mixed states arise in quantum mechanics in two different situations: first, when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations; and second, when one wants to describe a physical system which is entangled with another, as its state cannot be described by a pure state. In the first case, there could theoretically be another person who knows the full history of the system, and therefore describe the same system as a pure state; in this case, the density matrix is simply used to represent the limited knowledge of a quantum state. In the second case, however, the existence of quantum entanglement theoretically prevents the existence of complete knowledge about the subsystem, and it's impossible for any person to describe the subsystem of an entangled pair as a pure state.

Mixed states inevitably arise from pure states when, for a composite quantum system ${\displaystyle H_1\otimes H_2}$ with an entangled state on it, the part ${\displaystyle H_2}$ is inaccessible to the observer. The state of the part ${\displaystyle H_1}$ is expressed then as the partial trace over ${\displaystyle H_2}$.

A mixed state cannot be described with a single ket vector. Instead, it is described by its associated density matrix (or density operator), usually denoted ρ. Density matrices can describe both mixed and pure states, treating them on the same footing. Moreover, a mixed quantum state on a given quantum system described by a Hilbert space ${\displaystyle H}$ can be always represented as the partial trace of a pure quantum state (called a purification) on a larger bipartite system ${\displaystyle H\otimes K}$ for a sufficiently large Hilbert space ${\displaystyle K}$.

The density matrix describing a mixed state is defined to be an operator of the form $${\displaystyle \rho =\sum _s{p}_s|{\psi }_s⟩⟨{\psi }_s|}$$ where p_s is the fraction of the ensemble in each pure state ${\displaystyle |{\psi }_s⟩.}$ 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 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 A is given by $${\displaystyle ⟨A⟩=\sum _s{p}_s⟨{\psi }_s|A|{\psi }_s⟩=\sum _s\sum _i{p}_s{a}_i|⟨{\alpha }_i|{\psi }_s⟩{|}^2=\mathrm{tr}(\rho A)}$$ where ${\displaystyle |{\alpha }_i⟩}$ and ${\displaystyle a_i}$ are eigenkets and eigenvalues, respectively, for the operator A, and "tr" denotes trace. It is important to note that two types of averaging are occurring, one (over ${\displaystyle i}$) being the usual expected value of the observable when the quantum is in state ${\displaystyle |{\psi }_s⟩}$, and the other (over ${\displaystyle s}$) being a statistical (said incoherent) average with the probabilities p_s that the quantum is in those states.

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.

Blind men and an elephant

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Blind_men_and_an_elephant

The parable of the blind men and an elephant is a story of a group of blind men who have never come across an elephant before and who learn and imagine what the elephant is like by touching it. Each blind man feels a different part of the animal's body, but only one part, such as the side or the tusk. They then describe the animal based on their limited experience and their descriptions of the elephant are different from each other. In some versions, they come to suspect that the other person is dishonest and they come to blows. The moral of the parable is that humans have a tendency to claim absolute truth based on their limited, subjective experience as they ignore other people's limited, subjective experiences which may be equally true. The parable originated in the ancient Indian subcontinent, from where it has been widely diffused.

The Buddhist text Tittha Sutta, Udāna 6.4, Khuddaka Nikaya, contains one of the earliest versions of the story. The Tittha Sutta is dated to around c. 500 BCE, during the lifetime of the Buddha. Other versions of the parable describes sighted men encountering a large statue on a dark night, or some other large object while blindfolded.

In its various versions, it is a parable that has crossed between many religious traditions and is part of Jain, Hindu and Buddhist texts of 1st millennium CE or before. The story also appears in 2nd millennium Sufi and Baháʼí Faith lore. The tale later became well known in Europe, with 19th-century American poet John Godfrey Saxe creating his own version as a poem, with a final verse that explains that the elephant is a metaphor for God, and the various blind men represent religions that disagree on something no one has fully experienced. The story has been published in many books for adults and children, and interpreted in a variety of ways.

The parable

The earliest versions of the parable of blind men and the elephant are found in Buddhist, Hindu and Jain texts, as they discuss the limits of perception and the importance of complete context. The parable has several Indian variations, but broadly goes as follows:

A group of blind men heard that a strange animal, called an elephant, had been brought to the town, but none of them were aware of its shape and form. Out of curiosity, they said: "We must inspect and know it by touch, of which we are capable". So, they sought it out, and when they found it they groped about it. The first person, whose hand landed on the trunk, said, "This being is like a thick snake". For another one whose hand reached its ear, it seemed like a kind of fan. As for another person, whose hand was upon its leg, said, the elephant is a pillar like a tree-trunk. The blind man who placed his hand upon its side said the elephant, "is a wall". Another who felt its tail, described it as a rope. The last felt its tusk, stating the elephant is that which is hard, smooth and like a spear.

In some versions, the blind men then discover their disagreements, suspect the others to be not telling the truth and come to blows. The stories also differ primarily in how the elephant's body parts are described, how violent the conflict becomes and how (or if) the conflict among the men and their perspectives is resolved. In some versions, they stop talking, start listening and collaborate to "see" the full elephant. In another, a sighted man enters the parable and describes the entire elephant from various perspectives, the blind men then learn that they were all partially correct and partially wrong. While one's subjective experience is true, it may not be the totality of truth.

The parable has been used to illustrate a range of truths and fallacies; broadly, the parable implies that one's subjective experience can be true, but that such experience is inherently limited by its failure to account for other truths or a totality of truth. At various times the parable has provided insight into the relativism, opaqueness or inexpressible nature of truth, the behavior of experts in fields of contradicting theories, the need for deeper understanding, and respect for different perspectives on the same object of observation. In this respect, it provides an easily understood and practical example that illustrates ontologic reasoning.

References in religion

Hinduism

The blind men and the elephant
(wall relief in Northeast Thailand)

The Rigveda, dated to have been written down (from earlier oral traditions) between 1500 and 1200 BCE, states "Reality is one, though wise men speak of it variously." According to Paul J. Griffiths, this premise is the foundation of universalist perspective behind the parable of the blind men and an elephant. The hymn asserts that the same reality is subject to interpretations and described in various ways by the wise. In the oldest version, four blind men walk into a forest where they meet an elephant. In this version, they do not fight with each other, but conclude that they each must have perceived a different beast although they experienced the same elephant. The expanded version of the parable occurs in various ancient and Hindu texts. Many scholars refer to it as a Hindu parable.

The parable or references appear in bhasya (commentaries, secondary literature) in the Hindu traditions. For example, Adi Shankara mentions it in his bhasya on verse 5.18.1 of the Chandogya Upanishad as follows:

etaddhasti darshana iva jatyandhah

Translation: That is like people blind by birth in/when viewing an elephant.

— Adi Shankara, Translator: Hans Henrich Hock

Jainism

Seven blind men and an elephant parable at a Jain temple

The medieval era Jain texts explain the concepts of anekāntavāda (or "many-sidedness") and syādvāda ("conditioned viewpoints") with the parable of the blind men and an elephant (Andhgajanyāyah), which addresses the manifold nature of truth. This parable is found in the most ancient Jain agams before 5th century BCE. Its popularity remained till late. For example, this parable is found in Tattvarthaslokavatika of Vidyanandi (9th century) and Syādvādamanjari of Ācārya Mallisena (13th century). Mallisena uses the parable to argue that immature people deny various aspects of truth; deluded by the aspects they do understand, they deny the aspects they don't understand. "Due to extreme delusion produced on account of a partial viewpoint, the immature deny one aspect and try to establish another. This is the maxim of the blind (men) and the elephant." Mallisena also cites the parable when noting the importance of considering all viewpoints in obtaining a full picture of reality. "It is impossible to properly understand an entity consisting of infinite properties without the method of modal description consisting of all viewpoints, since it will otherwise lead to a situation of seizing mere sprouts (i.e., a superficial, inadequate cognition), on the maxim of the blind (men) and the elephant."

Buddhism

Blind monks examining an elephant, an ukiyo-e print by Hanabusa Itchō (1652–1724).

The Buddha twice uses the simile of blind men led astray. The earliest known version was recorded in the one of Buddhist scriptures, known as Tittha Sutta.

In another scripture known as Canki Sutta, the Buddha describes a row of blind men holding on to each other as an example of those who follow an old text that has passed down from generation to generation. In the Udana (68–69) he uses the elephant parable to describe sectarian quarrels. A king invited a group of blind men in the capital to be brought to the palace, where an elephant is brought in and they are asked to describe it.

When the blind men had each felt a part of the elephant, the king went to each of them and said to each: "Well, blind man, have you seen the elephant? Tell me, what sort of thing is an elephant?"

The men assert the elephant is either like a pot (the blind man who felt the elephant's head), a winnowing basket (ear), a plowshare (tusk), a plow (trunk), a granary (body), a pillar (foot), a mortar (back), a pestle (tail) or a brush (tip of the tail).

The men cannot agree with one another and come to blows over the question of what it is like and their dispute delights the king. The Buddha ends the story by comparing the blind men to preachers and scholars who are blind and ignorant and hold to their own views: "Just so are these preachers and scholars holding various views blind and unseeing.... In their ignorance they are by nature quarrelsome, wrangling, and disputatious, each maintaining reality is thus and thus." The Buddha then speaks the following verse:

O how they cling and wrangle, some who claim
For preacher and monk the honored name!
For, quarreling, each to his view they cling.
Such folk see only one side of a thing.

Sufism

The Persian Sufi poet Sanai (1080–1131/1141 CE) of Ghazni (currently, Afghanistan) presented this teaching story in his The Walled Garden of Truth.

Rumi, the 13th Century Persian poet and teacher of Sufism, included it in his Masnavi. In his retelling, "The Elephant in the Dark", some Hindus bring an elephant to be exhibited in a dark room. A number of men touch and feel the elephant in the dark and, depending upon where they touch it, they believe the elephant to be like a water spout (trunk), a fan (ear), a pillar (leg) and a throne (back). Rumi uses this story as an example of the limits of individual perception:

The sensual eye is just like the palm of the hand. The palm has not the means of covering the whole of the beast.

Rumi does not present a resolution to the conflict in his version, but states:

The eye of the Sea is one thing and the foam another. Let the foam go, and gaze with the eye of the Sea. Day and night foam-flecks are flung from the sea: oh amazing! You behold the foam but not the Sea. We are like boats dashing together; our eyes are darkened, yet we are in clear water.

Rumi ends his poem by stating "If each had a candle and they went in together the differences would disappear."

Meaning as a proverb by language

Japanese

In Japanese, the proverb is used as a simile of circumstance that ordinary men often fail to understand a great man or his great work.

Chinese

In Chinese, the proverb means failure to see the whole picture, for example, due to improper generalization.

Modern treatments

Variants of the story

One of the most famous versions of the 19th century was the poem "The Blind Men and the Elephant" by John Godfrey Saxe (1816–1887):

It was six men of Indostan
    To learning much inclined,
Who went to see the Elephant
    (Though all of them were blind),
That each by observation
    Might satisfy his mind

    Moral:
So oft in theologic wars,
    The disputants, I ween,
Rail on in utter ignorance
    Of what each other mean,
And prate about an Elephant
    Not one of them has seen!

In the poem, each man concluded that the elephant was like a wall, snake, spear, tree, fan or rope, depending upon where they had touched. Their heated debate comes short of physical violence, but the conflict was never resolved.

An elephant joke inverts the story in the following way, with the act of observation severely and fatally altering the subject of investigation:

Six blind elephants were discussing what men were like. After arguing they decided to find one and determine what it was like by direct experience. The first blind elephant felt the man and declared, 'Men are flat.' After the other blind elephants felt the man, they agreed.

Moral:

We have to remember that what we observe is not nature in itself, but nature exposed to our method of questioning.

— Werner Heisenberg

Commentary

"Blind men and elephant", from the Holton-Curry Reader (by Martha Adelaide Holton & Charles Madison Curry, 1914).

Blind men and elephant

Idries Shah commented on an element of self-reference by readers of the story in one of the many interpretations of the story, and its function as a teaching story:

...people address themselves to this story in one or more [...] interpretations. They then accept or reject them. Now they can feel happy; they have arrived at an opinion about the matter. According to their conditioning they produce the answer. Now look at their answers. Some will say that this is a fascinating and touching allegory of the presence of God. Others will say that it is showing people how stupid mankind can be. Some say it is anti-scholastic. Others that it is just a tale copied by Rumi from Sanai – and so on.

Shah adapted the tale in his book The Dermis Probe. This version begins with a conference of scientists, from different fields of expertise, presenting their conflicting conclusions on the material upon which a camera is focused. As the camera slowly zooms out it gradually becomes clear that the material under examination is the hide of an African elephant. The words 'The Parts Are Greater Than The Whole' then appear on the screen. This retelling formed the script for a short four-minute film by the animator Richard Williams. The film was chosen as an Outstanding Film of the Year and was exhibited at the London and New York film festivals.

In science

The story is seen as a metaphor in many disciplines, being pressed into service as an analogy in fields well beyond the traditional. In physics, it has been seen as an analogy for the wave–particle duality.^[26] In biology, the way the blind men hold onto different parts of the elephant has been seen as a good analogy for the polyclonal B cell response or for the reasons why it is challenging to find new drugs to treat diseases such as cancer or Alzheimer's disease. In medicine, the story has also been used to describe situations where diseases such as chronic obstructive pulmonary disease (COPD) are treated as several other diseases instead.

In literature

The Russian preface to a collection of Lewis Carroll's works (including such books as A Tangled Tale) includes the story as an analogy to the impression one gets from reading a few articles about Carroll, with him only being seen as a writer and poet by some, and a mediocre mathematician by others. The full picture, however, is that "Carroll only resembles Carroll the way an elephant only resembles an elephant".

In media

The story enjoys a continuing appeal in media, as shown by the number of illustrated children's books of the fable; for example, the children's book Seven Blind Mice, by Ed Young (1992) and one by Paul Galdone. In the title cartoon of one of his books, cartoonist Sam Gross postulated that one of the blind men, encountering a pile of the elephant feces, concluded that "An elephant is soft and mushy."

Touching the Elephant was a 1997 BBC Radio 4 documentary in which four people of varying ages, all blind from birth, were brought to London Zoo to touch an elephant and describe their response.

Ship of Theseus, a 2012 Indian philosophical drama named after the eponymous thought experiment, also references the parable.

Natalie Merchant sang Saxe's poem in full on her Leave Your Sleep album.

The name of Michael Nesmith's Grammy award-winning 1981 music video special 'Elephant Parts' references the parable.