In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions. Such variable pairs are known as complementary variables or canonically conjugate variables, and, depending on interpretation, the uncertainty principle limits to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value. The uncertainty principle implies that it is in general not possible to predict the value of a quantity with arbitrary certainty, even if all initial conditions are specified.
Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. The formal inequality relating the standard deviation of position σx and the standard deviation of momentum σp was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928:
where is the reduced Planck constant, ).
Historically, the uncertainty principle has been confused with a related effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the system, that is, without changing something in a system. Heisenberg utilized such an observer effect at the quantum level as a physical "explanation" of quantum uncertainty. It has since become clearer, however, that the uncertainty principle is inherent in the properties of all wave-like systems, 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. 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.
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 or quantum optics systems. Applications dependent on the uncertainty principle for their operation include extremely low-noise technology such as that required in gravitational wave interferometers.
IntroductionThe uncertainty principle is not readily apparent on the macroscopic scales of everyday experience. So it is helpful to demonstrate how it applies to more easily understood physical situations. Two alternative frameworks for quantum physics offer different explanations for the uncertainty principle. The wave mechanics picture of the uncertainty principle is more visually intuitive, but the more abstract matrix mechanics picture formulates it in a way that generalizes more easily.
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. 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, where 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. For example, if a measurement of an observable is performed, then the system is in a particular eigenstate of that observable. However, the particular eigenstate of the observable need not be an eigenstate of another observable : If so, then it does not have a unique associated measurement for it, as the system is not in an eigenstate of that observable.
Wave mechanics interpretationAccording to the de Broglie hypothesis, every object in the universe is a wave, i.e., 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 k0 or momentum p0 is
The Born rule states that this should be interpreted as a probability density amplitude 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.
On the other hand, consider a wave function that is a sum of many waves, which we may write this as
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
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 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.
|Proof of the Kennard inequality using wave mechanics|
|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. 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 u and v in this vector space:
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 z can be expressed as
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 physics involved in this proof was that and are wave functions for position and momentum, which are Fourier transforms of each other. A similar result would hold for any pair of conjugate variables.
Matrix mechanics interpretationIn 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 and, 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. By definition, this means that Applying the commutator to yields
where is the identity operator.
Suppose, for the sake of proof by contradiction, that is also a right eigenstate of momentum, with constant eigenvalue. 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 relationsThe most common general form of the uncertainty principle is the Robertson uncertainty relation.
For an arbitrary Hermitian operator we can associate a standard deviation
where the brackets indicate an expectation value. For a pair of operators and, 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,
where we have introduced the anticommutator,
|Proof of the Schrödinger uncertainty relation|
|The derivation shown here incorporates and builds off of those shown in Robertson, Schrödinger and standard textbooks such as Griffiths. For any Hermitian operator, based upon the definition of variance, we have|
we let and thus
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 which is defined as
and thus Eq. 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
We now substitute the above two equations above back into Eq. and get
Substituting the above into Eq. we get the Schrödinger uncertainty relation
This proof has an issue related to the domains of the operators involved. For the proof to make sense, the vector has to be in the domain of the unbounded operator, which is not always the case. In fact, the Robertson uncertainty relation is false if is an angle variable and is the derivative with respect to this variable. In this example, the commutator is a nonzero constant—just as in the Heisenberg uncertainty relation—and yet there are states where the product of the uncertainties is zero. This issue can be overcome by using a variational method for the proof., or by working with an exponentiated version of the canonical commutation relations.
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.
Mixed statesThe Robertson–Schrödinger uncertainty relation may be generalized in a straightforward way to describe mixed states.
The Maccone–Pati uncertainty relationsThe Robertson–Schrödinger uncertainty relation can be trivial if the state of the system is chosen to be eigenstate of one of the observable. The stronger uncertainty relations proved by Maccone and Pati give non-trivial bounds on the sum of the variances for two incompatible observables. For two non-commuting observables and the first stronger uncertainty relation is given by
where,, is a normalized vector that is orthogonal to the state of the system and one should choose the sign of to make this real quantity a positive number.
The second stronger uncertainty relation is given by
where is a state orthogonal to.
The form of implies that the right-hand side of the new uncertainty relation
is nonzero unless is an eigenstate of. One may note that can be an eigenstate of without being an eigenstate of either
or. However, when is an eigenstate of one of the two observables the Heisenberg–Schrödinger uncertainty relation becomes trivial. But the lower bound in the new relation is nonzero
unless is an eigenstate of both.
Phase spaceIn 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:
Choosing, we arrive at
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,
ExamplesSince 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:
- 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 time–energy uncertainty relation, as follows. For a quantum system in a non-stationary state and an observable B represented by a self-adjoint operator, the following formula holds:
- For the number of electrons in a superconductor and the phase of its Ginzburg–Landau order parameter
where we impose periodic boundary conditions on. The definition of depends on our choice to have range from 0 to. These operators satisfy the usual commutation relations for position and momentum operators,.
Now let be any of the eigenstates of, which are given by. These states are normalizable, unlike the eigenstates of the momentum operator on the line. Also the operator is bounded, since ranges over a bounded interval. Thus, in the state, the uncertainty of is zero and the uncertainty of is finite, so that
Although this result appears to violate the Robertson uncertainty principle, the paradox is resolved when we note that is not in the domain of the operator, since multiplication by disrupts the periodic boundary conditions imposed on. Thus, the derivation of the Robertson relation, which requires and to be defined, does not apply.
For the usual position and momentum operators and on the real line, no such counterexamples can occur. As long as and are defined in the state, the Heisenberg uncertainty principle holds, even if fails to be in the domain of or of.
Quantum harmonic oscillator stationary statesConsider a one-dimensional quantum harmonic oscillator. 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 energy eigenstates,
the variances may be computed directly,
The product of these standard deviations is then
In particular, the above Kennard bound is saturated for the ground state, for which the probability density is just the normal distribution.
Quantum harmonic oscillators with Gaussian initial conditionIn 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
where Ω describes the width of the initial state but need not be the same as ω. Through integration over the, we can solve for the -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 σ2. Copying the variances above and applying trigonometric identities, we can write the product of the standard deviations as
From the relations
we can conclude the following: .
Coherent statesA 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 quantum harmonic oscillator, 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 boxConsider a particle in a one-dimensional box of length. The eigenfunctions in position and momentum space are
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 momentumAssume a particle initially has a momentum space wave function described by a normal distribution around some constant momentum p0 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
Systematic and statistical errorsThe inequalities above focus on the statistical imprecision of observables as quantified by the standard deviation. 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 represent the error of a measurement of an observable A and 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:
Heisenberg's uncertainty principle, as originally described in the 1927 formulation, mentions only the first term of Ozawa inequality, regarding the systematic error. Using the notation above to describe the error/disturbance effect of sequential measurements, it could be written as
The formal derivation of the Heisenberg relation is possible but far from intuitive. It was not proposed by Heisenberg, but formulated in a mathematically consistent way only in recent years.
Also, it must be stressed that the Heisenberg formulation is not taking into account the intrinsic statistical errors and. There is increasing experimental evidence that the total quantum uncertainty cannot be described by the Heisenberg term alone, but requires the presence of all the three terms of the Ozawa inequality.
Using the same formalism, it is also possible to introduce the other kind of physical situation, often confused with the previous one, namely the case of simultaneous measurements :
The two simultaneous measurements on A and B are necessarily unsharp or weak.
It is also possible to derive an uncertainty relation that, as the Ozawa's one, combines both the statistical and systematic error components, but keeps a form very close to the Heisenberg original inequality. By adding Robertson
and Ozawa relations we obtain
The four terms can be written as:
as the inaccuracy in the measured values of the variable A and
as the resulting fluctuation in the conjugate variable B,
an uncertainty relation similar to the Heisenberg original one, but valid both for systematic and statistical errors:
Quantum entropic uncertainty principleFor 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. 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. This conjecture, also studied by Hirschman and proven in 1975 by Beckner and by Iwo Bialynicki-Birula and Jerzy Mycielski is that, for two normalized, dimensionless Fourier transform pairs and where
the Shannon information entropies
are subject to the following constraint,
where the logarithms may be in any base.
The probability distribution functions associated with the position wave function and the momentum wave function have dimensions of inverse length and momentum respectively, but the entropies may be rendered dimensionless by
where and are some arbitrarily chosen length and momentum respectively, which render the arguments of the logarithms dimensionless. Note that the entropies will be functions of these chosen parameters. Due to the Fourier transform relation between the position wave function and the momentum wavefunction, the above constraint can be written for the corresponding entropies as
where is Planck's constant.
Depending on one's choice of the product, the expression may be written in many ways. If is chosen to be, then
If, instead, is chosen to be, then
If and are chosen to be unity in whatever system of units are being used, then
where is interpreted as a dimensionless number equal to the value of Planck's constant in the chosen system of units. Note that these inequalities can be extended to multimode quantum states, or wavefunctions in more than one spatial dimension.
The quantum entropic uncertainty principle is more restrictive than the Heisenberg uncertainty principle. From the inverse logarithmic Sobolev inequalities
, it readily follows that this entropic uncertainty principle is stronger than the one based on standard deviations, because
In other words, the Heisenberg uncertainty principle, is a consequence of the quantum entropic uncertainty principle, but not vice versa. A few remarks on these inequalities. First, the choice of base e is a matter of popular convention in physics. The logarithm can alternatively be in any base, provided that it be consistent on both sides of the inequality. Second, recall the Shannon entropy has been used, not the quantum von Neumann entropy. Finally, the normal distribution saturates the inequality, and it is the only distribution with this property, because it is the maximum entropy probability distribution among those with fixed variance.