Foundations — Heisenberg uncertainty principle — Δx Δp ≥ ℏ - 2

Before you can read the parent note, you need to own every squiggle it throws at you. We build them one at a time, each on top of the last. Nothing is used before it is drawn.

1. Position $x$ — "where is it?"

Picture a ruler laid flat. The electron is a mark on that ruler. If the mark is at the "3 cm" line, then $x=3\text{cm}$. That's all $x$ means: a single point on a number line.

The topic needs $x$ because the whole question is "how well do we know where the particle is?" — and you can't talk about knowing a location without first naming the location.

2. The wavefunction $\psi$ and probability density $|\psi (x){|}^2$ — what the "hump" really is

Every "hump" we draw on this page is really $|\psi (x){|}^2$. Tall where the particle is likely; near zero where it almost never is. Because some value of $x$ must occur, the total area under the hump is exactly $1$ (100% probability). This is the machinery behind Atomic orbitals and probability density and comes straight out of the Schrödinger equation and wavefunction ψ.

The topic needs $|\psi {|}^2$ because $\Delta x$ and $\Delta p$ (defined next) are literally the widths of this curve — you cannot measure a spread until you have the curve whose spread you're measuring.

3. The spread $\Delta$ — "how blurry?"

Here is the single most important picture on this page.

Look at the two chalk humps — both are $|\psi (x){|}^2$. The narrow hump on the left means we know the position well: $\Delta x$ is small. The wide hump on the right means the particle is smeared over a big region: $\Delta x$ is large.

The topic needs $\Delta$ because uncertainty is the entire subject. Without a symbol for "blurriness", the principle can't even be written.

4. Average $⟨\cdot ⟩$ and the standard deviation — the exact rule for "width"

Before we can measure width, we need to say exactly what "the average position" means for a smeared-out particle.

Now the width itself:

Read the recipe inside-out, because that is the order you compute it:

Why bother with this exact rule instead of eyeballing width? Because "where does a hump end?" is vague, but $\sigma$ gives ONE definite number for any shape of curve. That is why the parent note insists $\Delta x,\Delta p$ are standard deviations, not errors — they describe the particle's own inherent smear, present before anyone measures. (We will apply the very same $\sigma$ recipe to momentum $p$ once it is defined in the next section.)

5. Momentum $p$ — "how much motion?"

Picture a rolling ball. A heavy ball rolling fast is hard to stop — big $p$. A light ball drifting slowly is easy to stop — small $p$. Momentum bundles how heavy and how fast into one number.

Momentum has a direction: a particle moving left has negative $p$, one moving right has positive $p$. Now that $p$ exists, we apply the exact same $\sigma$ recipe from Section 4 to it — a momentum spread $\Delta p={\sigma }_p=\sqrt{⟨(p-⟨p⟩{)}^2⟩},$ where $⟨p⟩$ is the average momentum. The centre $⟨p⟩$ might even be zero (a particle equally likely to go left or right), yet $\Delta p$ can be large — because the squaring in step 3 makes leftward ($-$) and rightward ($+$) momenta both contribute positively. So negative momenta never cancel the spread; they add to it.

Because $p=mv$, if we know the mass $m$ we can flip between momentum and velocity freely: $v=\frac{p}{m},\Delta v=\frac{\Delta p}{m}.$

The topic needs $p$ because the second thing we try to pin down about a particle is its motion — and momentum is the natural partner of position (they form what physicists call a conjugate pair).

6. Wavelength $\lambda$ and the wave nature

Look at the ripple. Peak to peak is one $\lambda$. A stretched-out wave (long $\lambda$) wiggles slowly; a squashed wave (short $\lambda$) wiggles fast.

Now the deep link: de Broglie discovered that a particle's momentum and its wavelength are locked together. See de Broglie wavelength — λ = h/p: $p=\frac{h}{\lambda }.$

Read it as a see-saw: short wavelength $\Rightarrow$ big momentum, long wavelength $\Rightarrow$ small momentum. This is why the parent note can swap "spread in wavelength" for "spread in momentum" — they are the same information. This whole idea rests on Wave-particle duality: the electron is both a particle (has $p$) and a wave (has $\lambda$).

7. Wave-number $k$ — counting wiggles instead of measuring wavelength

Why introduce $k$ at all when we already have $\lambda$? Because the Fourier machinery in the next section naturally speaks in "wiggles per metre", not "metres per wiggle". A spread in wavelengths becomes a spread in $k$, written ${\sigma }_k$ — computed with the same $\sigma$ recipe from Section 4, now applied to the list of $k$-values that make up the wave packet.

8. Why localizing a wave costs wavelength-sharpness

Top row: one lone wave — you know its $k$ exactly, but it is everywhere, so $\Delta x$ is huge. Bottom: adding several waves of slightly different $k$ makes them cancel out everywhere except one bump — now the particle is located ($\Delta x$ small), but we used a spread of wave-numbers ${\sigma }_k$, so $k$ (hence $\lambda$, hence momentum) is now fuzzy.

This is the engine of the whole principle: you can't have a narrow bump AND a single wave-number. Squeeze one, the other bulges. This links forward to the Schrödinger equation and wavefunction ψ, which describes exactly these wave packets.

9. Planck's constant $h$ and its cousin $\hslash$

We meet the two constants now, before using them in the bound, so no symbol runs ahead of its definition.

Why two symbols for basically one number? Because the $2\pi$ shows up constantly (waves love full circles = $2\pi$, exactly the $2\pi$ inside $k$), so physicists pre-divide once and write $\hslash$ to keep formulas tidy. The bar through the $h$ just reminds you "the $2\pi$ is already inside me".

The topic needs $h$ because it is the exact size of the "floor" under the uncertainty — the number below which the blur can never drop.

10. Where the $\frac{1}{2}$ comes from — outline of the bound

Now every symbol in the bound ($x$, $\Delta$, $\sigma$, $p$, $k$, $\hslash$) has already been built, so we can assemble it honestly.

11. Energy $E$, time $t$, and the second form

There is a second, similar-looking relation: $\Delta E\Delta t\ge \frac{\hslash }{2}.$ A state that only lives for a short time $\Delta t$ cannot have a perfectly sharp energy $\Delta E$ — which is why spectral lines have a natural width (re-met in Atomic orbitals and probability density).

12. The "$\ge$" sign — a floor, not a target

Picture a floor in a room: you can float at any height above it, never underneath. The floor is $\hslash /2$. Read the sign carefully — it does three things at once:

This "at least $\hslash /2$" is exactly what makes the principle a law of nature rather than a suggestion: no state, no apparatus, no cleverness can ever land below the floor.

How the pieces feed the principle

The map below shows the dependency chain — read the arrows as "is needed to build". Duality gives the wave; the wave $\psi$ gives the probability curve $|\psi {|}^2$; that curve's standard-deviation widths are $\Delta x$ and $\Delta p$; de Broglie converts wave-number-spread into momentum-spread; the Fourier theorem ties the two widths together; Planck's constant sets the exact floor. Everything meets at the parent principle the uncertainty principle.

Equipment checklist

Test yourself — cover the right side and see if you can answer before revealing.