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Uncertainty Principle

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In quantum physics, the Heisenberg Uncertainty Principle states that one cannot simultaneously know both the position and the momentum of a given object to arbitrary precision. It furthermore precisely quantifies the imprecision. It is one of the cornerstones of quantum mechanics and was discovered by Werner Heisenberg in 1927.

The uncertainty principle is sometimes erroneously explained by claiming that the measurement of position necessarily disturbs a particle's momentum. Heisenberg himself offered this explanation initially. Disturbance plays no part, however, since the principle even applies if position is measured in one copy of the system and momentum is measured in another, identical one. It is more accurate to say that the particle is a wave, not a point-like object, and does not have a well-defined simultaneous position and momentum.

Consider the following analogy: suppose you have a time-varying signal such as a sound wave, and you want to know the exact frequencies in your signal at an exact moment in time. This is impossible: in order to determine the frequencies accurately, you need to sample the signal for some time and you thereby lose time precision. (In other words, a sound cannot have both a precise time, as in a short pulse, and a precise frequency, as in a continuous pure tone.) The time and frequency of a wave in time are analogous to the position and momentum of a wave in space.

The uncertainty principle is frequently confused with another odd quantum mechanical phenomenon known as wavefunction collapse in which the act of observing a particle appears to change the equations describing the particle. These two phenomenon are separate but related. The uncertainty principle states that a particle does not have a fixed value for momentum and position, but when you observe a particle it seems to accquire a fixed and distinct value for the quantity you are measuring.

Definition The statement is as follows. If several identical copies of a system in a given state are prepared, measurements of position and momentum will vary according to known probability distributions; this is the fundamental postulate of quantum mechanics. We could measure the standard deviation Δx of the position measurements and the standard deviation Δp of the momentum measurements. Then we will find that

<math>\Delta x \Delta p \ge \frac{h}{4\pi} </math>

where h is Planck's constant and π is Archimedes' constant. (In some treatments, the "uncertainty" of a variable is taken to be the smallest width of a range which contains 50% of the values, which, in the case of normally distributed variables, leads to a lower bound of h/2π for the product of the uncertainties.) Note that this inequality allows for several possibilities: the state could be such that x can be measured with high precision, but then p will only approximately be known, or conversely p could be sharply defined while x cannot be precisely determined. In yet other states, both x and p can be measured with "reasonable" (but not arbitrary high) precision.

In everyday life, we don't observe these uncertainties because the value of h is extremely small.

Generalized Uncertainty Principle

The uncertainty principle does not just apply to position and momentum. In its general form, it applies to every pair of conjugate variables. Two variables are conjugate if the associated operators do not commute. An example of a pair of conjugate variables is the x-component of angular momentum (spin) vs. the y-component of angular momentum. In general, and unlike the case of position versus momentum discussed above, the lower bound for the product of the uncertainties of two conjugate variables depends on the state the system is in. The uncertainty principle becomes then a theorem in the theory of operators (see functional analysis). The uncertainty principle also applies to the pair of variables time and energy, but the mathematical treatment of this case differs somewhat from the operator approach mentioned above. The full Heisenberg uncertainty relation is the following:

<math>
\Delta A\,\Delta B \ge \frac{1}{2} \left|\left\langle\left[\hat{A},\hat{B}\right]\right\rangle_\psi\right| </math> where
A and B are two observables,
<math>\hat{A}</math> and <math>\hat{B}</math> their corresponding operators,
<math>[\hat{A}, \hat{B}]</math> denotes the commutator of <math>\hat{A}</math> and <math>\hat{B}</math>,
<math>\left\langle\,\right\rangle_\psi</math> denotes averaging for the state |ψ⟩, and
ΔX is the Standard Deviation of X: <math>\sqrt{\langle \hat{X}^2\rangle_\psi - \langle \hat{X}\rangle_\psi ^2}</math>.

This relation, which is readily obtained as a consequence of the Cauchy-Bunyakovski-Schwarz inequality, was first pointed out in 1930 by Howard Percy Robertson[?] and (independently) by Erwin Schrödinger. It is therefore also known as the Robertson-Schrödinger relation[?]. It may be evaluated not only for pairs of conjugate operators (e.g. those defining measurements of distance and of momentum, or of duration and of energy) but generally for any pair of Hermitian operators. There is also an uncertainty relation between the field strength and the number of particles which is responsible for the phenonomenon of virtual particles.

Interpretations Albert Einstein was not happy with the uncertainty principle, and he challenged Niels Bohr with a famous thought experiment: we fill a box with a radioactive material which randomly emits radiation. The box has a shutter, which is opened and immediately thereafter shut by a clock at a precise time, thereby allowing some radiation to escape. So the time is already known with precision. We still want to measure the conjugate variable energy precisely. No problem, says Einstein: just weigh the box before and after. The equivalence between mass and energy from special relativity will allow you to determine precisely how much energy left the box. Bohr countered as follows: should energy leave, then the now lighter box will rise slightly on the scale. That changes the position of the clock. Thus the clock deviates from our stationary reference frame, and again by special relativity, its measurement of time will be different from ours, leading to some unavoidable margin of error. In fact, a detailed analysis shows that the imprecision is correctly given by Heisenberg's relation.

Within the widely but not universally accepted Copenhagen interpretation of quantum mechanics, the uncertainty principle is taken to mean that on an elementary level, the physical universe does not exist in a deterministic form--but rather as a collection of probabilities, or potentials. For example, the pattern (probability distribution) produced by millions of photons passing through a diffraction slit can be calculated using quantum mechanics, but the exact path of each photon cannot be predicted by any known method. The Copenhagen interpretation holds that it cannot be predicted by any method.

It is this interpretation that Einstein was questioning when he said "I cannot believe that God would choose to play dice with the universe." Bohr who was one of the authors of the Copenhagen interpretation responded, "Einstein, don't tell God what to do".

Einstein was convinced that this interpretation was in error. His reasoning was that all previously known probability distributions arose from deterministic events. The distribution of a flipped coin or a rolled dice can be described with a probability distribution (50% heads, 50% tails). But this does not mean that their physical motions are unpredictable. Ordinary mechanics can be used to calculate exactly how each coin will land, if the forces acting on it are known. And the heads/tails distribution will still line up with the probability distribution (given random initial forces).

Einstein assumed that there are similar hidden variables in quantum mechanics which underly the observed probabilities.

Neither Einstein or anyone since has been able to construct a satisfying hidden variable theory, and the Bell inequality illustrates some very thorny issues in trying to do so. Although the behavior of an individual particle is random, they are also correlated with the behavior of other particles. Therefore, if the uncertainty principle is the result of some deterministic process, it must be the case that particles at great distances instantly transmit information to each other to ensure that the correlations in behavior between particles occur.

In some situations the Heisenberg uncertainty principle is called the Heisenberg indeterminacy principle. See: Quantum indeterminacy.



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