Transcribing"…so the more precisely we know position, the less we can say about momentum…"
✦Quick check
As you measure an electron's position more and more sharply, its momentum becomes:
correct · B
Δx·Δp ≥ ℏ/2 — shrinking Δx forces Δp up without limit. The trade is in the wavefunction, not your instrument.
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Where does the ℏ/2 bound actually come from?
21:51
AI
It falls out of treating position and momentum as Fourier conjugates: a wave packet narrow in x is necessarily broad in k, and p = ℏk. The ℏ/2 is the tightest the product can get, hit only by a Gaussian packet.
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Topics — your synthesis, in your words
§1Light as quanta: the photoelectric effectUpcoming00:00–07:30
In one line — when you turn light brighter, what do you get more of?
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§2Electrons as waves: de BroglieUpcoming07:30–15:40
Why do we never notice the wavelength of a cricket ball?
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§3The uncertainty principleUpcoming15:40–24:10
In one line, what does this inequality forbid you from knowing at once?
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§4Wave packets and localisationUpcoming24:10–31:20
Δk · Δx ≈ 1⊞ AI placed
To pin the position down tighter, what do you have to give up?
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§5What 'position' means for an electronUpcoming31:20–38:50
In your words — does the electron have a definite position before you measure it?
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Equations (3)
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[1]Heisenberg uncertainty22:30
Δx · Δp ≥ ℏ/2
Stated as the product of the spreads in position and momentum has a hard floor.
[2]de Broglie wavelength12:05
λ = h/p
Every particle has a wavelength set by its momentum.
[3]Photon energy03:40
E = hf
Light arrives in quanta whose energy scales with frequency.
Cards (6)
Flash card
What does brightness add to a beam of light?
More photons — not more energetic ones. E = hf is fixed by frequency.
Flash card
What is the de Broglie wavelength of a particle?
λ = h/p — inversely proportional to momentum.
Flash card
State the uncertainty principle.
Δx · Δp ≥ ℏ/2 — you cannot sharpen both at once.
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Δx · Δp ≥ ℏ/2
22:30
λ = h/p
12:05
wave packet
25:40
E = hf
03:40
21:40So the question we keep coming back to is: how well can you simultaneously know where a particle is and how fast it's moving?
22:05And the answer, which is genuinely strange, is that there's a hard limit — Δx times Δp is bounded below by ℏ over 2.
22:30The more precisely we know position, the less we can say about momentum. This isn't a statement about clumsy instruments.
Simulations (2)
Interactive simulations tied to this lecture. Drag the sliders to see how the wavefunction responds.
Simulation
Wave packet — position vs momentum spread
Narrow the packet in x and watch Δp widen. The product Δx·Δp never drops below ℏ/2.
Simulation
Single-slit diffraction
Shrink the slit; the pattern spreads — sharper position buys a wider momentum spread.
Tangents (3)
Short detours and 'why does this matter' asides the lecturer touched on.
Tangent
Why ℏ and not h?
The reduced Planck constant ℏ = h/2π falls out naturally from the Fourier / angular forms.
Tangent
Heisenberg's microscope
A useful heuristic — but the bound is deeper than measurement disturbance.
Tangent
Does this apply to a cricket ball?
Yes — but ℏ is so small the spread is utterly unobservable at that scale.
Problem set
Problems generated from this lecture's objectives, graded with step-by-step feedback.
Problem · 1
An electron is localised to Δx = 1 nm. Estimate the minimum Δp and the corresponding velocity spread.
Problem · 2
Show that a Gaussian wave packet saturates the bound, i.e. Δx·Δp = ℏ/2.
Full notebook
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Notebook
Quantum Mechanics — The Uncertainty Principle
5 objectives · 2 done · 3 clips filed · last edited just now
Notebook
Quantum Mechanics — de Broglie & matter waves
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End Session
Quantum Mechanics — The Uncertainty Principle · 22 min