# Uncertainty principle

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In quantum mechanics, the **uncertainty principle** is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle known as complementary variables, such as position *x* and momentum *p*, can be known simultaneously. For instance, in 1927, Werner Heisenberg stated that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.^{[1]} The formal inequality relating the standard deviation of position σ_{x} and the standard deviation of momentum σ_{p} was derived by Earle Hesse Kennard^{[2]} later that year and by Hermann Weyl^{[3]} in 1928:

(*ħ* is the reduced Planck constant).

Historically, the uncertainty principle has been confused^{[4]}^{[5]} with a somewhat similar effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems. Heisenberg offered such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty.^{[6]} It has since become clear, however, that the uncertainty principle is inherent in the properties of all wave-like systems,^{[7]} and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects. Thus, *the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology*.^{[8]} It must be emphasized that *measurement* does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer.^{[9]}

Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. Certain experiments, however, may deliberately test a particular form of the uncertainty principle as part of their main research program. These include, for example, tests of number-phase uncertainty relations in superconducting^{[10]} or quantum optics^{[11]} systems. Applications dependent on the uncertainty principle for their operation include extremely low noise technology such as that required in gravitational-wave interferometers.^{[12]}

## Introduction

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As a principle, Heisenberg's uncertainty relationship must be something that is in accord with all experience. However, humans do not form an intuitive understanding of this indeterminacy in everyday life, so it may be helpful to demonstrate how it is integral to more easily understood physical situations. Two alternative conceptualizations of quantum physics can be examined with the goal of demonstrating the key role the uncertainty principle plays. A wave mechanics picture of the uncertainty principle provides for a more visually intuitive demonstration, and the somewhat more abstract matrix mechanics picture provides for a demonstration of the uncertainty principle that is more easily generalized to cover a multitude of physical contexts.

Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another (i.e., position and momentum are conjugate variables). A nonzero function and its Fourier transform cannot both be sharply localized. A similar tradeoff between the variances of Fourier conjugates arises in all systems underlain by Fourier analysis, for example in sound waves: A pure tone is a sharp spike at a single frequency, while its Fourier transform gives the shape of the sound wave in the time domain, which is a completely delocalized sine wave. In quantum mechanics, the two key points are that the position of the particle takes the form of a matter wave, and momentum is its Fourier conjugate, assured by the de Broglie relation *p* = *ħk*, where Template:Mvar is the wavenumber.

In matrix mechanics, the mathematical formulation of quantum mechanics, any pair of non-commuting self-adjoint operators representing observables are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the wavefunction for a certain measurement value (the eigenvalue). For example, if a measurement of an observable Template:Mvar is performed, then the system is in a particular eigenstate Template:Mvar of that observable. However, the particular eigenstate of the observable Template:Mvar need not be an eigenstate of another observable Template:Mvar: If so, then it does not have a unique associated measurement for it, as the system is not in an eigenstate of that observable.^{[13]}

### Wave mechanics interpretation

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According to the de Broglie hypothesis, every object in the universe is a wave, a situation which gives rise to this phenomenon. The position of the particle is described by a wave function . The time-independent wave function of a single-moded plane wave of wavenumber *k*_{0} or momentum *p*_{0} is

The Born rule states that this should be interpreted as a probability density function in the sense that the probability of finding the particle between *a* and *b* is

In the case of the single-moded plane wave, is a uniform distribution. In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. Consider a wave function that is a sum of many waves, however, we may write this as

where *A*_{n} represents the relative contribution of the mode *p*_{n} to the overall total. The figures to the right show how with the addition of many plane waves, the wave packet can become more localized. We may take this a step further to the continuum limit, where the wave function is an integral over all possible modes

with representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that is the *Fourier transform* of and that *x* and *p* are conjugate variables. Adding together all of these plane waves comes at a cost, namely the momentum has become less precise, having become a mixture of waves of many different momenta.

One way to quantify the precision of the position and momentum is the standard deviation σ. Since is a probability density function for position, we calculate its standard deviation.

The precision of the position is improved, i.e. reduced σ_{x}, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σ_{p}. Another way of stating this is that σ_{x} and σ_{p} have an inverse relationship or are at least bounded from below. This is the uncertainty principle, the exact limit of which is the Kennard bound. Click the *show* button below to see a semi-formal derivation of the Kennard inequality using wave mechanics.

Proof of the Kennard inequality using wave mechanics |
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We are interested in the variances of position and momentum, defined as
Without loss of generality, we will assume that the means vanish, which just amounts to a shift of the origin of our coordinates. (A more general proof that does not make this assumption is given below.) This gives us the simpler form The function can be interpreted as a vector in a function space. We can define an inner product for a pair of functions where the asterisk denotes the complex conjugate. With this inner product defined, we note that the variance for position can be written as We can repeat this for momentum by interpreting the function as a vector, but we can also take advantage of the fact that and are Fourier transforms of each other. We evaluate the inverse Fourier transform through integration by parts: where the canceled term vanishes because the wave function vanishes at infinity. Often the term is called the momentum operator in position space. Applying Parseval's theorem, we see that the variance for momentum can be written as The Cauchy–Schwarz inequality asserts that The modulus squared of any complex number we let and and substitute these into the equation above to get All that remains is to evaluate these inner products. Plugging this into the above inequalities, we get or taking the square root Note that the only |

### Matrix mechanics interpretation

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In matrix mechanics, observables such as position and momentum are represented by self-adjoint operators. When considering pairs of observables, an important quantity is the *commutator*. For a pair of operators Template:Mvar and Template:Mvar, one defines their commutator as

In the case of position and momentum, the commutator is the canonical commutation relation

The physical meaning of the non-commutativity can be understood by considering the effect of the commutator on position and momentum eigenstates. Let be a right eigenstate of position with a constant eigenvalue *x*_{0}. By definition, this means that Applying the commutator to yields

where Template:Mvar is the identity operator.

Suppose, for the sake of proof by contradiction, that is also a right eigenstate of momentum, with constant eigenvalue Template:Mvar. If this were true, then one could write

On the other hand, the above canonical commutation relation requires that

This implies that no quantum state can simultaneously be both a position and a momentum eigenstate.

When a state is measured, it is projected onto an eigenstate in the basis of the relevant observable. For example, if a particle's position is measured, then the state amounts to a position eigenstate. This means that the state is *not* a momentum eigenstate, however, but rather it can be represented as a sum of multiple momentum basis eigenstates. In other words, the momentum must be less precise. This precision may be quantified by the standard deviations,

As in the wave mechanics interpretation above, one sees a tradeoff between the respective precisions of the two, quantified by the uncertainty principle.

## Robertson–Schrödinger uncertainty relations

The most common general form of the uncertainty principle is the *Robertson uncertainty relation*.^{[14]}

For an arbitrary Hermitian operator we can associate a standard deviation

where the brackets indicate an expectation value. For a pair of operators Template:Mvar and Template:Mvar, we may define their *commutator* as

In this notation, the Robertson uncertainty relation is given by

The Robertson uncertainty relation immediately follows from a slightly stronger inequality, the *Schrödinger uncertainty relation*,^{[15]}

where we have introduced the *anticommutator*,

Proof of the Schrödinger uncertainty relation |
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The derivation shown here incorporates and builds off of those shown in Robertson,^{[14]} Schrödinger^{[15]} and standard textbooks such as Griffiths.^{[16]} For any Hermitian operator , based upon the definition of variance, we have
Similarly, for any other Hermitian operator in the same state The product of the two deviations can thus be expressed as In order to relate the two vectors and , we use the Cauchy–Schwarz inequality and thus Eq. (Template:EquationNote) can be written as Since is in general a complex number, we use the fact that the modulus squared of any complex number is defined as where is the complex conjugate of . The modulus squared can also be expressed as we let and and substitute these into the equation above to get The inner product is written out explicitly as and using the fact that and are Hermitian operators, we find Similarly it can be shown that Thus we have and We now substitute the above two equations above back into Eq. (Template:EquationNote) and get Substituting the above into Eq. (Template:EquationNote) we get the Schrödinger uncertainty relation This proof has an issue Note that in the general form of the Robertson–Schrödinger uncertainty relation, there is no need to assume that the operators and are self-adjoint operators. It suffices to assume that they are merely symmetric operators. (The distinction between these two notions is generally glossed over in the physics literature, where the term |

Since the Robertson and Schrödinger relations are for general operators, the relations can be applied to any two observables to obtain specific uncertainty relations. A few of the most common relations found in the literature are given below.

- For position and linear momentum, the canonical commutation relation implies the Kennard inequality from above:

- For two orthogonal components of the total angular momentum operator of an object:

- where
*i*,*j*,*k*are distinct and*J*_{i}denotes angular momentum along the*x*_{i}axis. This relation implies that unless all three components vanish together, only a single component of a system's angular momentum can be defined with arbitrary precision, normally the component parallel to an external (magnetic or electric) field. Moreover, for , a choice , in angular momentum multiplets,*ψ*= |*j*,*m*〉, bounds the Casimir invariant (angular momentum squared, ) from below and thus yields useful constraints such as*j*(*j*+ 1) ≥*m*(*m*+ 1), and hence*j*≥*m*, among others.

- In non-relativistic mechanics, time is privileged as an independent variable. Nevertheless, in 1945, L. I. Mandelshtam and I. E. Tamm derived a non-relativistic
, as follows.**time-energy uncertainty relation**^{[23]}For a quantum system in a non-stationary state Template:Mvar and an observable*B*represented by a self-adjoint operator , the following formula holds:

- where σ
_{E}is the standard deviation of the energy operator (Hamiltonian) in the state Template:Mvar, σ_{B}stands for the standard deviation of*B*. Although the second factor in the left-hand side has dimension of time, it is different from the time parameter that enters the Schrödinger equation. It is a*lifetime*of the state Template:Mvar with respect to the observable*B*: In other words, this is the*time interval*(*Δt*) after which the expectation value changes appreciably. - An informal, heuristic meaning of the principle is the following: A state that only exists for a short time cannot have a definite energy. To have a definite energy, the frequency of the state must be defined accurately, and this requires the state to hang around for many cycles, the reciprocal of the required accuracy. For example, in spectroscopy, excited states have a finite lifetime. By the time-energy uncertainty principle, they do not have a definite energy, and, each time they decay, the energy they release is slightly different. The average energy of the outgoing photon has a peak at the theoretical energy of the state, but the distribution has a finite width called the
*natural linewidth*. Fast-decaying states have a broad linewidth, while slow decaying states have a narrow linewidth.^{[24]} - The same linewidth effect also makes it difficult to specify the rest mass of unstable, fast-decaying particles in particle physics. The faster the particle decays (the shorter its lifetime), the less certain is its mass (the larger the particle's width).

- For the number of electrons in a superconductor and the phase of its Ginzburg–Landau order parameter
^{[25]}^{[26]}

## Examples

### Quantum harmonic oscillator stationary states

{{#invoke:main|main}} Consider a one-dimensional quantum harmonic oscillator (QHO). It is possible to express the position and momentum operators in terms of the creation and annihilation operators:

Using the standard rules for creation and annihilation operators on the eigenstates of the QHO,

the variances may be computed directly,

The product of these standard deviations is then

In particular, the above Kennard bound^{[2]} is saturated for the ground state *n*=0, for which the probability density is just the normal distribution.

### Quantum harmonic oscillator with Gaussian initial condition

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In a quantum harmonic oscillator of characteristic angular frequency ω, place a state that is offset from the bottom of the potential by some displacement *x*_{0} as

where Ω describes the width of the initial state but need not be the same as ω. Through integration over the propagator, we can solve for the Template:Not a typo-dependent solution. After many cancelations, the probability densities reduce to

where we have used the notation to denote a normal distribution of mean μ and variance σ^{2}. Copying the variances above and applying trigonometric identities, we can write the product of the standard deviations as

From the relations

we can conclude

### Coherent states

{{#invoke:main|main}} A coherent state is a right eigenstate of the annihilation operator,

which may be represented in terms of Fock states as

In the picture where the coherent state is a massive particle in a QHO, the position and momentum operators may be expressed in terms of the annihilation operators in the same formulas above and used to calculate the variances,

Therefore every coherent state saturates the Kennard bound

with position and momentum each contributing an amount in a "balanced" way. Moreover every squeezed coherent state also saturates the Kennard bound although the individual contributions of position and momentum need not be balanced in general.

### Particle in a box

{{#invoke:main|main}} Consider a particle in a one-dimensional box of length . The eigenfunctions in position and momentum space are

and

where and we have used the de Broglie relation . The variances of and can be calculated explicitly:

The product of the standard deviations is therefore

For all , the quantity is greater than 1, so the uncertainty principle is never violated. For numerical concreteness, the smallest value occurs when , in which case

### Constant momentum

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Assume a particle initially has a momentum space wave function described by a normal distribution around some constant momentum *p*_{0} according to

where we have introduced a reference scale , with describing the width of the distribution−−cf. nondimensionalization. If the state is allowed to evolve in free space, then the time-dependent momentum and position space wave functions are

Since and this can be interpreted as a particle moving along with constant momentum at arbitrarily high precision. On the other hand, the standard deviation of the position is

such that the uncertainty product can only increase with time as

## Additional uncertainty relations

### Mixed states

The Robertson–Schrödinger uncertainty relation may be generalized in a straightforward way to describe mixed states.^{[27]}

### Phase space

In the phase space formulation of quantum mechanics, the Robertson–Schrödinger relation follows from a positivity condition on a real star-square function. Given a Wigner function with star product ★ and a function *f*, the following is generally true:^{[28]}

Since this positivity condition is true for *all* *a*, *b*, and *c*, it follows that all the eigenvalues of the matrix are positive. The positive eigenvalues then imply a corresponding positivity condition on the determinant:

or, explicitly, after algebraic manipulation,

### Systematic error

The inequalities above focus on the statistical imprecision of observables as quantified by the standard deviation. Heisenberg's original version, however, was interested in *systematic error*, incurred by a disturbance of a quantum system by the measuring apparatus, i.e., an observer effect. If we let represent the error (i.e., accuracy) of a measurement of an observable and represent its disturbance by the measurement process, then the following inequality holds:^{[5]}

In fact, Heisenberg's uncertainty principle as originally described in the 1927 formulation mentions only the first term. Applying the notation above to Heisenberg's position-momentum relation, Heisenberg's argument could be rewritten as

Such a formulation is both mathematically incorrect and experimentally refuted.^{[29]} It is also possible to derive a similar uncertainty relation combining both the statistical and systematic error components.^{[30]}

### Entropic uncertainty principle

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For many distributions, the standard deviation is not a particularly natural way of quantifying the structure. For example, uncertainty relations in which one of the observables is an angle has little physical meaning for fluctuations larger than one period.^{[21]}^{[31]}^{[32]}^{[33]} Other examples include highly bimodal distributions, or unimodal distributions with divergent variance.

A solution that overcomes these issues is an uncertainty based on entropic uncertainty instead of the product of variances. While formulating the many-worlds interpretation of quantum mechanics in 1957, Hugh Everett III conjectured a stronger extension of the uncertainty principle based on entropic certainty.^{[34]} This conjecture, also studied by Hirschman^{[35]} and proven in 1975 by Beckner^{[36]} and by Iwo Bialynicki-Birula and Jerzy Mycielski^{[37]} is