# Uncertainty principle

{{#invoke:Hatnote|hatnote}} {{#invoke: Sidebar | collapsible }} 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 σx and the standard deviation of momentum σp 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

{{#invoke:main|main}}

Click to see animation. The evolution of an initially very localized gaussian wave function of a free particle in two-dimensional space, with colour and intensity indicating phase and amplitude. The spreading of the wave function in all directions shows that the initial momentum has a spread of values, unmodified in time; while the spread in position increases in time: as a result, the uncertainty Δx Δp increases in time.
The superposition of several plane waves to form a wave packet. This wave packet becomes increasingly localized with the addition of many waves. The Fourier transform is a mathematical operation that separates a wave packet into its individual plane waves. Note that the waves shown here are real for illustrative purposes only, whereas in quantum mechanics the wave function is generally complex.

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

{{#invoke:Multiple image|render}} {{#invoke:main|main}} {{#invoke:main|main}} 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 ${\displaystyle \Psi (x,t)}$. The time-independent wave function of a single-moded plane wave of wavenumber k0 or momentum p0 is

${\displaystyle \psi (x)\propto e^{ik_{0}x}=e^{ip_{0}x/\hbar }~.}$

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

${\displaystyle \operatorname {P} [a\leq X\leq b]=\int _{a}^{b}|\psi (x)|^{2}\,\mathrm {d} x~.}$

In the case of the single-moded plane wave, ${\displaystyle |\psi (x)|^{2}}$ 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

${\displaystyle \psi (x)\propto \sum _{n}A_{n}e^{ip_{n}x/\hbar }~,}$

where An represents the relative contribution of the mode pn 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

${\displaystyle \psi (x)={\frac {1}{\sqrt {2\pi \hbar }}}\int _{-\infty }^{\infty }\phi (p)\cdot e^{ipx/\hbar }\,dp~,}$

with ${\displaystyle \phi (p)}$ representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that ${\displaystyle \phi (p)}$ is the Fourier transform of ${\displaystyle \psi (x)}$ 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 ${\displaystyle |\psi (x)|^{2}}$ is a probability density function for position, we calculate its standard deviation.

The precision of the position is improved, i.e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σp. Another way of stating this is that σx and σp 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.

### Matrix mechanics interpretation

{{#invoke:main|main}} 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

${\displaystyle [{\hat {A}},{\hat {B}}]={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}.}$

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

${\displaystyle [{\hat {x}},{\hat {p}}]=i\hbar .}$

The physical meaning of the non-commutativity can be understood by considering the effect of the commutator on position and momentum eigenstates. Let ${\displaystyle |\psi \rangle }$ be a right eigenstate of position with a constant eigenvalue x0. By definition, this means that ${\displaystyle {\hat {x}}|\psi \rangle =x_{0}|\psi \rangle .}$ Applying the commutator to ${\displaystyle |\psi \rangle }$ yields

${\displaystyle [{\hat {x}},{\hat {p}}]|\psi \rangle =({\hat {x}}{\hat {p}}-{\hat {p}}{\hat {x}})|\psi \rangle =({\hat {x}}-x_{0}{\hat {I}})\cdot {\hat {p}}\,|\psi \rangle =i\hbar |\psi \rangle ,}$

where Template:Mvar is the identity operator.

Suppose, for the sake of proof by contradiction, that ${\displaystyle |\psi \rangle }$ is also a right eigenstate of momentum, with constant eigenvalue Template:Mvar. If this were true, then one could write

${\displaystyle ({\hat {x}}-x_{0}{\hat {I}})\cdot {\hat {p}}\,|\psi \rangle =({\hat {x}}-x_{0}{\hat {I}})\cdot p_{0}\,|\psi \rangle =(x_{0}{\hat {I}}-x_{0}{\hat {I}})\cdot p_{0}\,|\psi \rangle =0.}$

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

${\displaystyle [{\hat {x}},{\hat {p}}]|\psi \rangle =i\hbar |\psi \rangle \neq 0.}$

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,

${\displaystyle \sigma _{x}={\sqrt {\langle {\hat {x}}^{2}\rangle -\langle {\hat {x}}\rangle ^{2}}}}$
${\displaystyle \sigma _{p}={\sqrt {\langle {\hat {p}}^{2}\rangle -\langle {\hat {p}}\rangle ^{2}}}.}$

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 ${\displaystyle {\hat {\mathcal {O}}}}$ we can associate a standard deviation

${\displaystyle \sigma _{\mathcal {O}}={\sqrt {\langle {\hat {\mathcal {O}}}^{2}\rangle -\langle {\hat {\mathcal {O}}}\rangle ^{2}}},}$

where the brackets ${\displaystyle \langle {\mathcal {O}}\rangle }$ indicate an expectation value. For a pair of operators Template:Mvar and Template:Mvar, we may define their commutator as

${\displaystyle [{\hat {A}},{\hat {B}}]={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}},}$

In this notation, the Robertson uncertainty relation is given by

${\displaystyle \sigma _{A}\sigma _{B}\geq \left|{\frac {1}{2i}}\langle [{\hat {A}},{\hat {B}}]\rangle \right|={\frac {1}{2}}\left|\langle [{\hat {A}},{\hat {B}}]\rangle \right|.}$

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

where we have introduced the anticommutator,

${\displaystyle \{{\hat {A}},{\hat {B}}\}={\hat {A}}{\hat {B}}+{\hat {B}}{\hat {A}}.}$

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.

${\displaystyle \sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}}$
${\displaystyle \sigma _{J_{i}}\sigma _{J_{j}}\geq {\tfrac {\hbar }{2}}\left|\left\langle J_{k}\right\rangle \right|~,}$
where i, j, k are distinct and Ji denotes angular momentum along the xi 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 ${\displaystyle [{J_{x}},{J_{y}}]=i\hbar \epsilon _{xyz}{J_{z}}}$, a choice ${\displaystyle {\hat {A}}=J_{x},~~{\hat {B}}=J_{y}}$, in angular momentum multiplets, ψ = |j, m 〉, bounds the Casimir invariant (angular momentum squared, ${\displaystyle \langle J_{x}^{2}+J_{y}^{2}+J_{z}^{2}\rangle }$) from below and thus yields useful constraints such as j (j + 1) ≥ m (m + 1), and hence jm, among others.
${\displaystyle \sigma _{E}~{\frac {\sigma _{B}}{\left|{\frac {\mathrm {d} \langle {\hat {B}}\rangle }{\mathrm {d} t}}\right|}}\geq {\frac {\hbar }{2}},}$
where σE is the standard deviation of the energy operator (Hamiltonian) in the state Template:Mvar, σB 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 ${\displaystyle \langle {\hat {B}}\rangle }$ 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).
${\displaystyle \Delta N\Delta \phi \geq 1~.}$

## 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:

${\displaystyle {\hat {x}}={\sqrt {\frac {\hbar }{2m\omega }}}(a+a^{\dagger })}$
${\displaystyle {\hat {p}}=i{\sqrt {\frac {m\omega \hbar }{2}}}(a^{\dagger }-a).}$

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

${\displaystyle a^{\dagger }|n\rangle ={\sqrt {n+1}}|n+1\rangle }$
${\displaystyle a|n\rangle ={\sqrt {n}}|n-1\rangle ,}$

the variances may be computed directly,

${\displaystyle \sigma _{x}^{2}={\frac {\hbar }{m\omega }}\left(n+{\frac {1}{2}}\right)}$
${\displaystyle \sigma _{p}^{2}=\hbar m\omega \left(n+{\frac {1}{2}}\right)\,.}$

The product of these standard deviations is then

${\displaystyle \sigma _{x}\sigma _{p}=\hbar \left(n+{\frac {1}{2}}\right)\geq {\frac {\hbar }{2}}~.}$

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

{{#invoke:Multiple image|render}}

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 x0 as

${\displaystyle \psi (x)=\left({\frac {m\Omega }{\pi \hbar }}\right)^{1/4}\exp {\left(-{\frac {m\Omega (x-x_{0})^{2}}{2\hbar }}\right)},}$

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

${\displaystyle |\Psi (x,t)|^{2}\sim {\mathcal {N}}\left(x_{0}\cos {(\omega t)},{\frac {\hbar }{2m\Omega }}\left(\cos ^{2}{(\omega t)}+{\frac {\Omega ^{2}}{\omega ^{2}}}\sin ^{2}{(\omega t)}\right)\right)}$
${\displaystyle |\Phi (p,t)|^{2}\sim {\mathcal {N}}\left(-mx_{0}\omega \sin {(\omega t)},{\frac {\hbar m\Omega }{2}}\left(\cos ^{2}{(\omega t)}+{\frac {\omega ^{2}}{\Omega ^{2}}}\sin ^{2}{(\omega t)}\right)\right),}$

where we have used the notation ${\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}$ to denote a normal distribution of mean μ and variance σ2. Copying the variances above and applying trigonometric identities, we can write the product of the standard deviations as

{\displaystyle {\begin{aligned}\sigma _{x}\sigma _{p}&={\frac {\hbar }{2}}{\sqrt {\left(\cos ^{2}{(\omega t)}+{\frac {\Omega ^{2}}{\omega ^{2}}}\sin ^{2}{(\omega t)}\right)\left(\cos ^{2}{(\omega t)}+{\frac {\omega ^{2}}{\Omega ^{2}}}\sin ^{2}{(\omega t)}\right)}}\\&={\frac {\hbar }{4}}{\sqrt {3+{\frac {1}{2}}\left({\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\right)-\left({\frac {1}{2}}\left({\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\right)-1\right)\cos {(4\omega t)}}}\end{aligned}}}

From the relations

${\displaystyle {\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\geq 2,\,\,\,|\cos {(4\omega t)}|\leq 1,}$

we can conclude

${\displaystyle \sigma _{x}\sigma _{p}\geq {\frac {\hbar }{4}}{\sqrt {3+{\frac {1}{2}}\left({\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\right)-\left({\frac {1}{2}}\left({\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\right)-1\right)}}={\frac {\hbar }{2}}.}$

### Coherent states

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

${\displaystyle {\hat {a}}|\alpha \rangle =\alpha |\alpha \rangle ,}$,

which may be represented in terms of Fock states as

${\displaystyle |\alpha \rangle =e^{-{|\alpha |^{2} \over 2}}\sum _{n=0}^{\infty }{\alpha ^{n} \over {\sqrt {n!}}}|n\rangle }$

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,

${\displaystyle \sigma _{x}^{2}={\frac {\hbar }{2m\omega }}}$
${\displaystyle \sigma _{p}^{2}={\frac {\hbar m\omega }{2}}.}$

Therefore every coherent state saturates the Kennard bound

${\displaystyle \sigma _{x}\sigma _{p}={\sqrt {\frac {\hbar }{2m\omega }}}\,{\sqrt {\frac {\hbar m\omega }{2}}}={\frac {\hbar }{2}}.}$

with position and momentum each contributing an amount ${\displaystyle {\sqrt {\hbar /2}}}$ 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 ${\displaystyle L}$. The eigenfunctions in position and momentum space are

${\displaystyle \psi _{n}(x,t)={\begin{cases}A\sin(k_{n}x)\mathrm {e} ^{-\mathrm {i} \omega _{n}t},&0

and

${\displaystyle \phi _{n}(p,t)={\sqrt {\frac {\pi L}{\hbar }}}\,\,{\frac {n\left(1-(-1)^{n}e^{-ikL}\right)e^{-i\omega _{n}t}}{\pi ^{2}n^{2}-k^{2}L^{2}}},}$

where ${\displaystyle \omega _{n}={\frac {\pi ^{2}\hbar n^{2}}{8L^{2}m}}}$ and we have used the de Broglie relation ${\displaystyle p=\hbar k}$. The variances of ${\displaystyle x}$ and ${\displaystyle p}$ can be calculated explicitly:

${\displaystyle \sigma _{x}^{2}={\frac {L^{2}}{12}}\left(1-{\frac {6}{n^{2}\pi ^{2}}}\right)}$
${\displaystyle \sigma _{p}^{2}=\left({\frac {\hbar n\pi }{L}}\right)^{2}.}$

The product of the standard deviations is therefore

${\displaystyle \sigma _{x}\sigma _{p}={\frac {\hbar }{2}}{\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}.}$

For all ${\displaystyle n=1,\,2,\,3\,...}$, the quantity ${\displaystyle {\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}}$ is greater than 1, so the uncertainty principle is never violated. For numerical concreteness, the smallest value occurs when ${\displaystyle n=1}$, in which case

${\displaystyle \sigma _{x}\sigma _{p}={\frac {\hbar }{2}}{\sqrt {{\frac {\pi ^{2}}{3}}-2}}\approx 0.568\hbar >{\frac {\hbar }{2}}.}$

### Constant momentum

{{#invoke:main|main}}

File:Wave function of a Gaussian state moving at constant momentum.gif
Position space probability density of an initially Gaussian state moving at minimally uncertain, constant momentum in free space.

Assume a particle initially has a momentum space wave function described by a normal distribution around some constant momentum p0 according to

${\displaystyle \phi (p)=\left({\frac {x_{0}}{\hbar {\sqrt {\pi }}}}\right)^{1/2}\cdot \exp {\left({\frac {-x_{0}^{2}(p-p_{0})^{2}}{2\hbar ^{2}}}\right)},}$

where we have introduced a reference scale ${\displaystyle x_{0}={\sqrt {\hbar /m\omega _{0}}}}$, with ${\displaystyle \omega _{0}>0}$ 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

${\displaystyle \Phi (p,t)=\left({\frac {x_{0}}{\hbar {\sqrt {\pi }}}}\right)^{1/2}\cdot \exp {\left({\frac {-x_{0}^{2}(p-p_{0})^{2}}{2\hbar ^{2}}}-{\frac {ip^{2}t}{2m\hbar }}\right)},}$