Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

11.3.2: Uncertainty in Quantum Mechanics

( \newcommand{\kernel}{\mathrm{null}\,}\)

In order to bring this into quantum mechanics, we already know how to calculate the average a, which we call the “expectation value”. If the state of the system is |ψ and the operator corresponding to the observable a is ˆA, then

a=ψ|ˆA|ψ

Similarly, now that we recognize that we can interpret ˆA2 as just applying the operator ˆA twice, we can calculate a2:

a2=ψ|ˆA2|ψ

For example, let’s consider the state |ψ=|+z and the observable spin-z. We expect the uncertainty here to be zero, because we know exactly what we’ll get if we measure spin-z. Let’s see if it works out that way:

sz=ψ|ˆSz|ψ=2[10][1001][10]=2[10][10]=2

As expected, the expectation value for spin-z is +/2. For the other part:

s2z=+z|ˆSzˆSz|z=24[10][1001][1001][10]=24[10][1001][10]=24[10][10]=24

If we take the difference sz2sz2, we get 2/42/4=0, as expected.

What if we want to know the uncertainty on Sx for this state?

sx=+z|ˆSx|+z=2[10][0110][10]=2[10][01]=0

If the system is in the state |+z, we know that we have a 50% chance each for finding spin-x to be +/2 or /2. Thus, it’s no surprise that the average value of spin-x is zero, even though zero isn’t a value we might measure. To figure out the variance:

s2x=+z|ˆSxˆSx|+z=24[10][0110][0110][10]=24[10][0110][01]=24[10][10]=24

Thus, in this case, the formal uncertainty Δsx on the x-spin is /2.


This page titled 11.3.2: Uncertainty in Quantum Mechanics is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pieter Kok via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?