← Wave Mechanics

Heisenberg Uncertainty Relationships

Thursday, February 10, 2022

Uncertainty with de Broglie Waves

Since uncertainty principles for classical waves apply to all waves, they can also be applied to de Broglie waves. We can use the fact that p=h/λp=h/\lambda and the differential dp=(h/λ2)dλdp=\left(-h/\lambda^2\right)d\lambda to determine that

Δp=hλ2Δλ\Delta p = \frac{h}{\lambda^2}\Delta\lambda

Combining this with the classical uncertainty principle, we get

ΔxΔpϵh\Delta x \Delta p \sim \epsilon h

This means the smaller the wave packet, the larger the uncertainty about its momentum. Without going into the quantum mechanics, it turns out the smallest possible value for ΔxΔp\Delta x \Delta p is h/4πh/4\pi, meaning

ΔxΔpx12\Delta x \Delta p_x \geq \frac{1}{2}\hbar

This is the first of the Heisenberg uncertainty relationships. Since this is the limit of how much can be known about a wave packet, it is reasonable to say that

ΔxΔpx\Delta x \Delta p_x \sim \hbar

Single-slit experiment

Upon approaching the single split, we know the momentum of the electrons exactly, meaning their x-position is completely unknowable. Once they pass through the slit, we know a range of their positions, and so less about their momenta. After the slit, we are not certain about their the electron's momentum or position, explaining the diffraction pattern seen.

Energy-Frequency Uncertainty Principle

Using the equation for energy for light waves, E=hfE=hf,

ΔEΔtϵh\Delta E \Delta t \sim \epsilon h

Again, since the minimum is ϵ=1/4π\epsilon = 1/4\pi,

ΔEΔt12\Delta E \Delta t \geq \frac{1}{2}\hbar

This is the second of the Heisenberg uncertainty relationships: the more precisely we try to measure the time coordinate of a particle, the less we know about the energy of that particle. And just like the first uncertainty relationship,

ΔEΔt\Delta E \Delta t \sim \hbar

Heisenberg Uncertainty Relationships

  1. It is not possible to make a simultaneous determination of the position and the momentum of a particle with unlimited precision.
  2. It is not possible to make a simultaneous determination of the energy and the time coordinate of a particle with unlimited precision.

Unsettling nature of the Heisenberg uncertainty relationships

It is not simply uncertainty that these relationships imply; it is an inability to determine the values of position and momentum simultaneously. In other words, it is not a limit of our current technology; it is a limit of the universe.

Statistical Interpretation of Uncertainty

In a single-slit experiment, detectors can be placed on the screen to determine the momentum of the particles after passing through the slit. This resembles a statistical distribution centered around px=0p_x=0 with a width of Δpx\Delta p_x. The definition of the standard deviation of a quantity AA, σA\sigma_A centered about AavA_{av}:


Similarly, the formal definition of the uncertainty of momentum is

Δpx=(px2)av(px,av)2\Delta p_x=\sqrt{\left(p_x^2\right)_{av}-\left(p_{x,av}\right)^2}

But since px,av=0p_{x, av}=0,

Δpx=(px2)av\Delta p_x=\sqrt{\left(p_x^2\right)_{av}}