Entanglement

HEISENBERG’S UNCERTAINTY PRINCIPLE

At the heart of quantum mechanics is a principle that places a fundamental limit on how accurately two non-compatible observables, such as momentum and position or time and energy, can be measured simultaneously. This principle, called the Heisenberg uncertainty principle, was proposed in 1927 by the German physicist Werner Heisenberg.

Heisenberg invented the uncertainty principle while investigating the question of measurement in the framework of Dirac-Jordan transformation theory. Intrinsic uncertainties or imprecisions appeared to surface during the simultaneous measurement of the position (x) and linear momentum (p) of a moving particle (and also in the simultaneous measurement of energy and time). Heisenberg realized that these variables can never be measured simultaneously with absolute precision. Expressed mathematically, Heisenberg’s uncertainty principle takes the iconic form that even non-physicists recognize:

\displaystyle \Delta x \Delta p \geq \hbar/2

The result of multiplying the errors in the measurements of position and momentum (the errors are represented by the Greek letter “delta”) must be a number greater than or equal to half of the constant “h-bar”. This is equivalent to saying that the error in measurement of the position of a particle is inversely proportional to the error in measurement of its momentum, or, the smaller the error in measurement of position, the greater is the error in measurement of its momentum.

It turns out that the uncertainty principle is a consequence of the intimate connection between subatomic particles and waves in nature and has nothing to do with inadequacies in the measuring instruments, the technique, or the observer. It appeared to place a fundamental limit on how accurately we can simultaneously measure two non-compatible observables like momentum and position.

In terms of wave functions, it turns out that there are no wave functions that can simultaneously be focused on a single value of position and also on a single value of momentum. If the wave function is focused on a single value of position, the amplitudes for different momenta will be spread out as widely as possible over all the possibilities. And vice versa: If the wave function is concentrated on a single momentum, it is spread out widely over all possible positions. So if we observe the position of an object, we lose any knowledge of what its momentum is, and vice versa. (If we only measure the position approximately, rather than exactly, we can retain some knowledge of the momentum; this is what actually happens in real-world macroscopic measurements).

The Heisenberg uncertainty principle triggered an intense philosophical debate among scientists. If a quantity cannot be measured, what effect does it have on reality? Classical physics is deterministic: by knowing the present state of the universe, one could predict all future states. On the other hand, quantum mechanics allows one to calculate the probability of an event. It does not make a deterministic statement of whether an event will happen or not. Many physicists, such as David Bohm and Albert Einstein, therefore could not accept the idea that nature is ultimately in-deterministic. Einstein was strongly opposed to this element of randomness and chance in the universe, summarized by his famous quote “God does not play dice”. Is there something else hidden that ultimately leads to deterministic observable quantities? Many questions surrounding the Heisenberg uncertainty principle have no definite answer and heavily depend on the interpretations of quantum mechanics.

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