# Heisenberg's Uncertainty Principle

Simply put, Heisenberg's Uncertainty Principle states that it is impossible to know both the exact position and the exact velocity of an object at the same time. However, the effect is tiny and so is only noticeable on a subatomic scale.

### History

Werner Heisenberg (1901-1976) was a German physicist who helped to formulate quantum mechanics at the beginning of the 20th century. He first presented the Heisenberg Uncertainty Principle in February 1927 in a letter to Wolfgang Pauli, then published it later that year.

### The Principle

Light can be considered as being made up of packets of energy called photons. To measure the position and velocity of any particle, you would first shine a light on it, then detect the reflection. On a macroscopic scale, the effect of photons on an object is insignificant. Unfortunately, on subatomic scales, the photons that hit the subatomic particle will cause it to move significantly, so although the position has been measured accurately, the velocity of the particle will have been altered. By learning the position, you have rendered any information you previously had on the velocity useless. In other words, the observer affects the observed.

### Detailed Explanation

To observe any particle, smaller particles must first be reflected off it. To find both the position and the momentum1 of, say, an electron, at least one photon (the smallest possible quantity of light) must be utilised. The photon must bounce off the electron then reflect back to the measuring device. For larger particles, such as sand grains or buses, the percentage uncertainty in the measurements of position and momentum is insignificant. For the far smaller subatomic particles, the percentage uncertainty in these measurements is larger, and finding the location and momentum becomes more of a problem.

As momentum = mass x velocity, an electron has momentum p = mv, where 'p' is momentum, 'm' is mass and 'v' is velocity.

'Mass' can be considered as the amount of 'stuff', or fundamental building blocks, an object is made of. 'Velocity' is the component of the particle's speed acting in a specific, defined direction.

The measurement is made with photons of light that have a wavelength2 'λ' (the Greek letter 'lambda'). Wavelength is the distance light travels in one complete wave cycle; frequency is the number of cycles per second (hertz, Hz). Visible light has a frequency of about 1014 hertz.

The equation velocity = frequency x wavelength can be used to determine the wavelength of a wave of known velocity and frequency:

• Velocity of a photon in a vacuum = c ≈ 3 x 108 ms-1 ≈ 108 ms-1

• Frequency of visible light ≈ 1014 Hz

• λ = wavelength = (c / frequency) ≈ (108 ms-1)/(1014 Hz ) ≈ 10-6 m

This is one millionth of a metre, one thousandth of a millimetre.

As the speed of light in a vacuum is constant, to reduce the wavelength of a photon, its frequency of the light must be increased. Using Energy = hf, where h is Planck's constant and f is the photon's frequency, the greater the photon's frequency, the greater the energy it carries, as h is always constant.

The wavelength of a particle is described by the DeBroglie formula as λ = (Planck's constant)/(momentum) = h/mc. Therefore, mc = h/λ

Photons have no rest mass but, when moving, have an apparent mass due to their kinetic energy (energy of motion). Using E=mc2, their apparent mass is E/c2

When one of these photons bounces off the electron, the electron's momentum will, therefore, be changed. This is similar to the way billiard balls change their trajectories when they collide.

The change in the electron's momentum (Δmv) is uncertain and will be of the same order and magnitude as the photon's momentum: Δmv ≈ (h/λ)

Since c in f = c / λ is constant, an increase in the photon's wavelength leads to a decrease in its frequency and, therefore, a decrease in the energy it carries. Therefore, the uncertainty in its momentum is smaller.

The photon of light cannot measure the electron's position (x) with perfect accuracy. Considering the photon of light as a wave, to increase the accuracy in position measurement, a photon with a smaller wavelength must be used. However, the smaller the wavelength of light, the higher its frequency and the higher its energy. Therefore, on collision with the electron, it will change its momentum a relatively large amount, meaning the momentum is less accurately determined when the position is more accurately determined. The opposite is also true - if the momentum is more accurately determined, the position must be less accurately determined.

An estimate of the position of the particle is at least one photon wavelength from the original position.

The uncertainty in position, Δx, ≥ one photon wavelength. Therefore, Δx ≥  λ

The smaller the wavelength, the greater the accuracy in the measurement of the position, but, as the wavelength of the photon gets shorter, its frequency increases. According to the equation E = hf, the greater its frequency, the greater its energy and the greater the change in momentum when it collides with a particle. That is, the momentum of the electron has greater uncertainty when using photons with smaller wavelengths.

The opposite is also true - the longer the wavelength, the lower the energy and the lower the change in momentum; however, the uncertainty in the position of the electron will be greater.

Crucially, we cannot know both the exact position and momentum of a particle - we can only find an approximation to each. The better the approximation to one, the worse the approximation to the other becomes.

### The equations

Since Δmv ≈ h /λ and Δx ≥  λ the two can be combined:

ΔxΔmv ≥  hλ/λ. Therefore, ΔxΔp ≥ h

1'Position' means the location of the particle and momentum is its mass multiplied by its velocity.2It should be noted that here photons are considered as waves, rather than particles. The wave-particle duality is described in this Entry