Heisenberg Uncertainty Principle: Definition, Equation & How to Use It

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Quantum mechanics obeys very different laws than classical physics. Many influential scientists have worked in this field, including Albert Einstein, Erwin Schrodinger, Werner Heisenberg, Niels Bohr, Louis De Broglie, David Bohm and Wolfgang Pauli.

The standard Copenhagen interpretation of quantum physics states that all that can be known is given by the wave function. In other words, we can't know certain properties of quantum particles in any absolute terms. Many have found this notion unsettling and proposed all sorts of thought experiments and alternative interpretations, but the mathematics consistent with the original interpretation still bears out.

Wavelength and Position

Think of shaking a rope repeatedly up and down, creating a wave traveling down it. It makes sense to ask what the wavelength is – this is easy enough to measure – but less sense to ask where the wave is, because the wave is really a continuous phenomenon all along the rope.

In contrast, if a single wave pulse is sent down the rope, identifying where it is becomes straightforward, but determining its wavelength no longer makes sense because it is not a wave.

You can also imagine everything in between: sending a wave packet down the rope, for example, the position is somewhat defined, and the wavelength as well, but not both completely. This difference is at the heart of Heisenberg's Uncertainty Principle.

Wave-Particle Duality

You will hear people use the words photon and electromagnetic radiation interchangeably, even though it seems like they are different things. When speaking of photons, they are typically talking about the particle properties of this phenomenon, whereas when they are talking about electromagnetic waves or radiation, they are speaking to the wavelike properties.

Photons or electromagnetic radiation exhibit what is called particle-wave duality. In certain situations and in certain experiments, photons exhibit particle-like behavior. One example of this is in the photoelectric effect, where light hitting a surface causes the release of electrons. The specifics of this effect can only be understood if light is treated as discrete packets that the electrons must absorb in order to be emitted.

In other situations and experiments, they act more like waves. A prime example of this is the interference patterns observed in single- or multiple-slit experiments. In these experiments, light is passed through narrow, closely spaced slits, and as a result, it produces an interference pattern consistent with what you would see in a wave.

Even stranger, photons are not the only thing that exhibit this duality. Indeed, all fundamental particles, even electrons and protons, seem to behave in this way! The larger the particle, the shorter its wavelength, so the less this duality appears. This is why we don’t notice anything like this at all on our everyday macroscopic scale.

Interpreting Quantum Mechanics

Unlike the clear-cut behavior of Newton’s laws, quantum particles exhibit a sort of fuzziness. You cannot say exactly what they are doing, but only give probabilities of what measurement results might yield. And if your instinct is to assume this is because of an inability to measure things accurately, you would be incorrect, at least in terms of the standard interpretations of the theory.

The so-called Copenhagen interpretation of quantum theory states that all that can be known about a particle is contained within the wave function that describes it. There are no additional hidden variables or things we simply haven’t discovered that would give more detail. It is fundamentally fuzzy, so to speak. The Heisenberg Uncertainty Principle is just another development that solidifies this fuzziness.

Heisenberg Uncertainty Principle

The uncertainty principle was first proposed by its namesake, German physicist Werner Heisenberg, in 1927 while he was working at Neils Bohr’s institute in Copenhagen. He published his findings in a paper titled “On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics.”

The principle states that the position of a particle and the momentum of a particle (or the energy and time of a particle) cannot both be known simultaneously with absolute certainty. That is, the more precisely you know the position, the less precisely you know the momentum (which is directly related to wavelength), and vice versa.

Applications of the uncertainty principle are numerous and include particle confinement (determining the energy required to contain a particle within a given volume), signal processing, electron microscopes, understanding quantum fluctuations and zero-point energy.

Uncertainty Relations

The primary uncertainty relationship is expressed as the following inequality:

\sigma_x\sigma_p\geq\frac{\hbar}{2}

where ℏ is the reduced Planck's constant and ​σx​ and ​σp​ are the standard deviation of position and momentum, respectively. Note that the smaller one of the standard deviations becomes, the larger the other one must become in order to compensate. As a result, the more precisely you know one value, the less precisely you know the other.

Additional uncertainty relationships include uncertainty in orthogonal components of angular momentum, uncertainty in time and frequency in signal processing, uncertainty in energy and time, and so on.

The Source of Uncertainty

One common way to explain the origins of uncertainty is to describe it in terms of measurement. Consider that, to measure the position of an electron, for example, requires interacting with it in some way – typically hitting it with a photon or other particle.

However, the act of hitting it with the photon causes its momentum to change. Not only that, there is a certain amount of inaccuracy in the measurement with the photon associated with the photon’s wavelength. A more accurate position measurement can be achieved with a shorter wavelength photon, but such photons carry more energy and hence can cause a greater change in the electron’s momentum, making it impossible to measure both position and momentum with perfect accuracy.

While the measurement method certainly makes it difficult to obtain the values of both simultaneously as described, the actual problem is more fundamental than that. It isn’t just an issue of our measurement capabilities; it’s a fundamental property of these particles that they don’t have both a well-defined position and momentum simultaneously. The reasons lie in the "wave on a string" analogy made previously.

Uncertainty Principle Applied to Macroscopic Measurements

One common question people ask with regards to the strangeness of quantum mechanical phenomena is how come they don’t see this weirdness on the scale of everyday objects?

It turns out that it isn’t that quantum mechanics simply doesn’t apply to larger objects, but that the strange effects it as are negligible at large scales. Particle-wave duality, for instance, isn’t noticed on the large scale because the wavelength of matter waves becomes vanishingly small, hence the particle-like behavior that dominates.

As for the uncertainty principle, consider the how big the number on the right-hand side of the inequality is. ℏ/2 = 5.272859 × 10-35 kgm2/s. So the uncertainty in position (in meters) times the uncertainty in momentum (in kgm/s) must be greater than or equal to this. On the macroscopic scale, getting near this limit this implies impossible levels of accuracy. For example, a 1-kg object can be measured as having a momentum of 1.00000000000000000 ±10-17 kgm/s while at a position of 1.00000000000000000 ±10-17 m and still more than satisfy the inequality.

Macroscopically, the right side of the uncertainty inequality is relatively so small as to be negligible, but the value is not negligible in quantum systems. In other words: the principle still applies to macroscopic objects – it just becomes irrelevant due to their size!

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