Uncertainty principle

The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.

For other uses, see Uncertainty principle (disambiguation).

More formally, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the product of the accuracy of certain related pairs of measurements on a quantum system, such as position, x, and momentum, p.^[1] Such paired-variables are known as complementary variables or canonically conjugate variables.

First introduced in 1927 by German physicist Werner Heisenberg,^[2]^[3]^[4]^[5] 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^[6] later that year and by Hermann Weyl^[7] in 1928:

where $\hbar ={\frac {h}{2\pi }}$ is the reduced Planck constant.

The quintessentially quantum mechanical uncertainty principle comes in many forms other than position–momentum. The energy–time relationship is widely used to relate quantum state lifetime to measured energy widths but its formal derivation is fraught with confusing issues about the nature of time. The basic principle has been extended in numerous directions; it must be considered in many kinds of fundamental physical measurements.

Energy–time uncertainty principle[edit]

Energy spectrum line-width vs lifetime[edit]

An energy–time uncertainty relation like $\Delta E\Delta t\gtrsim \hbar /2,$ has a long, controversial history; the meaning of $\Delta t$ and $\Delta E$ varies and different formulations have different arenas of validity.^[12] However, one well-known application is both well established^[13]^[14] and experimentally verified:^[15]^[16] the connection between the life-time of a resonance state, $\tau _{\sqrt {1/2}}$ and its energy width $\Delta E$ : $\tau _{\sqrt {1/2}}\Delta E=\pi \hbar /4.$ In particle-physics, widths from experimental fits to the Breit–Wigner energy distribution are used to characterize the lifetime of quasi-stable or decaying states.^[17]

An informal, heuristic meaning of the principle is the following:^[18]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.^[19] 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).

Time in quantum mechanics[edit]

The concept of "time" in quantum mechanics offers many challenges.^[20] There is no quantum theory of time measurement; relativity is both fundamental to time and difficult to include in quantum mechanics.^[12] While position and momentum are associated with a single particle, time is a system property: it has no operator needed for the Robertson–Schrödinger relation.^[1] The mathematical treatment of stable and unstable quantum systems differ.^[21] These factors combine to make energy–time uncertainty principles controversial.

Three notions of "time" can be distinguished:^[12] external, intrinsic, and observable. External or laboratory time is seen by the experimenter; intrinsic time is inferred by changes in dynamic variables, like the hands of a clock or the motion of a free particle; observable time concerns time as an observable, the measurement of time-separated events.

An external-time energy–time uncertainty principle might say that measuring the energy of a quantum system to an accuracy $\Delta E$ requires a time interval $\Delta t>h/\Delta E$ .^[14] However, Yakir Aharonov and David Bohm^[22]^[12] have shown that, in some quantum systems, energy can be measured accurately within an arbitrarily short time: external-time uncertainty principles are not universal.

Intrinsic time is the basis for several formulations of energy–time uncertainty relations, including the Mandelstam–Tamm relation discussed in the next section. A physical system with an intrinsic time closely matching the external laboratory time is called a "clock".^[20]^: 31

Observable time, measuring time between two events, remains a challenge for quantum theories; some progress has been made using positive operator-valued measure concepts.^[12]

Mandelstam–Tamm[edit]

In 1945, Leonid Mandelstam and Igor Tamm derived a non-relativistic time–energy uncertainty relation as follows.^[23]^[12] From Heisenberg mechanics, the generalized Ehrenfest theorem for an observable B without explicit time dependence, represented by a self-adjoint operator ${\hat {B}}$ relates time dependence of the average value of ${\hat {B}}$ to the average of its commutator with the Hamiltonian:

${\frac {d\langle {\hat {B}}\rangle }{dt}}={\frac {i}{\hbar }}\langle [{\hat {H}},{\hat {B}}]\rangle .$

The value of $\langle [{\hat {H}},{\hat {B}}]\rangle$ is then substituted in the Robertson uncertainty relation for the energy operator ${\hat {H}}$ and ${\hat {B}}$ : $\sigma _{H}\sigma _{B}\geq \left|{\frac {1}{2i}}\langle [{\hat {H}},{\hat {B}}]\rangle \right|,$ giving $\sigma _{H}{\frac {\sigma _{B}}{\left|{\frac {d\langle {\hat {B}}\rangle }{dt}}\right|}}\geq {\frac {\hbar }{2}}$ (whenever the denonminator is nonzero). While this is a universal result, it depends upon the observable chosen and that the deviations $\sigma _{H}$ and $\sigma _{B}$ are computed for a particular state. Identifying $\Delta E\equiv \sigma _{E}$ and the characteristic time $\tau _{B}\equiv {\frac {\sigma _{B}}{\left|{\frac {d\langle {\hat {B}}\rangle }{dt}}\right|}}$ gives an energy–time relationship $\Delta E\tau _{B}\geq {\frac {\hbar }{2}}.$ Although $\tau _{B}$ has the dimension of time, it is different from the time parameter t that enters the Schrödinger equation. This $\tau _{B}$ can be interpreted as time for which the expectation value of the observable, $\langle {\hat {B}}\rangle ,$ changes by an amount equal to one standard deviation.^[24] Examples:

Intrinsic quantum uncertainty[edit]

Historically, the uncertainty principle has been confused^[26]^[27] with a related effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the system,^[28]^[29] that is, without changing something in a system. Heisenberg used such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty.^[30] It has since become clearer, however, that the uncertainty principle is inherent in the properties of all wave-like systems,^[31] and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects.^[32] Thus, the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology.^[33]

Position–linear momentum uncertainty relation: for the position and linear momentum operators, the canonical commutation relation $[{\hat {x}},{\hat {p}}]=i\hbar$ implies the Kennard inequality from above: $\sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}.$

Angular momentum uncertainty relation: For two orthogonal components of the operator of an object: $\sigma _{J_{i}}\sigma _{J_{j}}\geq {\frac {\hbar }{2}}{\big |}\langle J_{k}\rangle {\big |},$ 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 $[J_{x},J_{y}]=i\hbar \varepsilon _{xyz}J_{z}$ , a choice ${\hat {A}}=J_{x}$ , ${\hat {B}}=J_{y}$ , in angular momentum multiplets, ψ = |j, m⟩, bounds the Casimir invariant (angular momentum squared, $\langle J_{x}^{2}+J_{y}^{2}+J_{z}^{2}\rangle$ ) from below and thus yields useful constraints such as j(j + 1) ≥ m(m + 1), and hence j ≥ m, among others.

total angular momentum

Additional uncertainty relations[edit]

Heisenberg limit[edit]

In quantum metrology, and especially interferometry, the Heisenberg limit is the optimal rate at which the accuracy of a measurement can scale with the energy used in the measurement. Typically, this is the measurement of a phase (applied to one arm of a beam-splitter) and the energy is given by the number of photons used in an interferometer. Although some claim to have broken the Heisenberg limit, this reflects disagreement on the definition of the scaling resource.^[55] Suitably defined, the Heisenberg limit is a consequence of the basic principles of quantum mechanics and cannot be beaten, although the weak Heisenberg limit can be beaten.^[56]

Systematic and statistical errors[edit]

The inequalities above focus on the statistical imprecision of observables as quantified by the standard deviation $\sigma$ . Heisenberg's original version, however, was dealing with the systematic error, a disturbance of the quantum system produced by the measuring apparatus, i.e., an observer effect.

If we let $\varepsilon _{A}$ represent the error (i.e., inaccuracy) of a measurement of an observable A and $\eta _{B}$ the disturbance produced on a subsequent measurement of the conjugate variable B by the former measurement of A, then the inequality proposed by Ozawa−encompassing both systematic and statistical errors—holds:^[27]

Problem 1 – If the photon has a short , and therefore, a large momentum, the position can be measured accurately. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. If the photon has a long wavelength and low momentum, the collision does not disturb the electron's momentum very much, but the scattering will reveal its position only vaguely.

wavelength

Problem 2 – If a large is used for the microscope, the electron's location can be well resolved (see Rayleigh criterion); but by the principle of conservation of momentum, the transverse momentum of the incoming photon affects the electron's beamline momentum and hence, the new momentum of the electron resolves poorly. If a small aperture is used, the accuracy of both resolutions is the other way around.

aperture

Applications[edit]

Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. All forms of spectroscopy, including particle physics use the relationship to relate measured energy line-width to the lifetime of quantum states. 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^[116] or quantum optics^[117] systems. Applications dependent on the uncertainty principle for their operation include extremely low-noise technology such as that required in gravitational wave interferometers.^[118]

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]