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 $p=h/\lambda$ and the differential $dp=\left(-h/\lambda^2\right)d\lambda$ to determine that

$\Delta p = \frac{h}{\lambda^2}\Delta\lambda$

Combining this with the classical uncertainty principle, we get

$\Delta x \Delta p \sim \epsilon h$

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 $\Delta x \Delta p$ is $h/4\pi$ , meaning

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

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

$\Delta x \Delta p_x \sim \hbar$

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, $E=hf$ ,

$\Delta E \Delta t \sim \epsilon h$

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

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

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,

$\Delta E \Delta t \sim \hbar$

Heisenberg Uncertainty Relationships

It is not possible to make a simultaneous determination of the position and the momentum of a particle with unlimited precision.
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 $p_x=0$ with a width of $\Delta p_x$ . The definition of the standard deviation of a quantity $A$ , $\sigma_A$ centered about $A_{av}$ :

$\sigma_A=\sqrt{\left(A^2\right)_{av}-\left(A_{av}\right)^2}$

Similarly, the formal definition of the uncertainty of momentum is

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

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

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