Illustrated Physics
physics, drawn
Depth

Series — page 5

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
A wavefunction crossing a barrier it has not the energy for. An electron of 2 electronvolts meeting a 3 electronvolt barrier 0.3 nanometres wide, with the four matching conditions solved rather than sketched. Left of the barrier the incident and reflected waves add to a standing pattern; inside it the amplitude decays exponentially; to the right a travelling wave continues with amplitude 0.3912 of the incident one, so 15.3 per cent of the electrons get through. Classically none of them do.

Tunnelling

  1. 1 The wall that is not quite a wall
  2. 2 A wall that a factor of two makes impassable
  3. 3 The last atom does all the seeing
  4. 4 Two walls that let more through than one
  5. 5 How long the crossing takes
  6. +3 more
8 essays · quantum Where the electron actually is, by radius. The radial probability density of the hydrogen 1s, 2s, 2p states — the chance of finding the electron in a thin shell at each radius, in units of the Bohr radius. 1s is most likely at 1.00 Bohr radii and averages 1.50; 2s is most likely at 5.24 Bohr radii and averages 6.00; 2p is most likely at 4.00 Bohr radii and averages 5.00. Each curve integrates to one, and each has n − l − 1 radial nodes where the electron is never found.

Atomic structure

  1. 1 Where the electron probably is
  2. 2 The order the shells fill
  3. 3 Why an atom is the size it is
  4. 4 The metal that is yellow because it is heavy
  5. 5 The ion that exists because its electrons keep apart
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7 essays · quantum Where the particle is likely to be found. A particle confined between two walls one unit apart. States 1, 4, 16 are drawn, each riding on a line at its own energy — 1E₁, 16E₁, 256E₁ — because the energies go as n². The curves are |ψ|², the probability of finding the particle at each position. The dashed line on each is the classical answer: a ball bouncing between the walls at constant speed is equally likely to be anywhere, and the quantum density oscillates about it and converges onto it as n rises.

Correspondence

  1. 1 Where the quantum picture hands back the old one
  2. 2 The average that obeys Newton
  3. 3 The state that swings like a pendulum
  4. 4 The return a classical cloud never makes
  5. 5 The probability that goes below zero
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7 essays · quantum The correlation, and the best a shared list of answers can do. The coincidence correlation between two polarisation analysers against the angle between them, over two full turns of the correlation — a polariser turned through 180° is the same polariser, so the picture repeats. The singlet gives −cos 2Δ, drawn through −1.00 at 0°, 1.00 at 90°, −1.00 at 180°, 1.00 at 270°. Beside it is the best correlation any shared list of pre-agreed answers can produce: straight lines between the same four extremes, with corners where the cosine is smooth. The two agree exactly at the multiples of 45° and nowhere else, and they are furthest apart — by 0.2105 — at 19.77° and 70.23°, which is ½ arcsin(2/π) from either end of the quarter turn. The difference is not a matter of degree: it is a curve against a shape with a corner in it, and no list can be bent into the curve.

Entanglement

  1. 1 The correlation no instructions can produce
  2. 2 The disagreement that one run settles
  3. 3 What two have they cannot give a third
  4. 4 A link between two that never met
  5. 5 The corner of Hilbert space that is ever visited
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7 essays · quantum Light crossing an accelerating box. A pulse crosses a box 6 m wide while the box accelerates at 9.81 m/s². The crossing takes 2·10⁻⁸ s, in which the far wall gains 1.96·10⁻⁷ m/s, so the pulse lands 1.96·10⁻¹⁵ m below the height it left at — and the path is a parabola. An observer sealed inside cannot tell that from a beam of light bending in a gravitational field, and the equivalence principle says there is nothing to tell. The sag is drawn 5.6·10¹⁴ times its true size.

Equivalence principle

  1. 1 The floor that cannot be told from gravity
  2. 2 The term free fall cannot remove
  3. 3 The fall that does not depend on what is falling
  4. 4 The binding energy that has to fall too
  5. 5 The radiation a difference in falling would make
  6. +2 more
7 essays · astrophysics The energy ladder of a box. The first 6 energy levels of a box, drawn to scale in E₁, at 1.0, 4.0, 9.0, 16.0, 25.0, 36.0. The levels spread apart as the square of n, so a box's spectrum has no top. The arrow marks a transition: 4 to 3 releases 7.000 E₁.

Exclusion

  1. 1 No two in the same state, and why matter has volume
  2. 2 The pressure that is not a temperature
  3. 3 The force with no force in it
  4. 4 The collisions the exclusion principle forbids
  5. 5 The label three identical quarks forced
  6. +2 more
7 essays · quantum The energy of a capacitor, booked as a density. The energy stored by a parallel-plate capacitor of 200 square centimetres — 0.0200 square metres — against the separation of its plates, drawn twice. Held at 15 nC the energy rises in proportion to the separation; held at 169 V it falls as the inverse. Both curves are obtained by integrating the energy density ½ε₀E² over the volume between the plates, and each agrees with ½QV to better than a part in 10¹². The two describe the same capacitor at 2.00 mm, where they cross at 1.27 µJ, and there their slopes are equal and opposite: the attraction between the plates is 635 µN, or 6.353·10⁻⁴ N, whichever quantity is held fixed. That force is Q²/2ε₀A — a property of the field in the gap and of the area it crosses, with no reference to the plates at all.

Field energy

  1. 1 Where the energy of a field actually is
  2. 2 The angular momentum that is in nothing at all
  3. 3 The momentum of something that is not moving
  4. 4 The force read off a surface that touches nothing
  5. 5 The momentum light carries into glass
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7 essays · electromagnetism An electron's wavelength against the voltage that accelerated it. The de Broglie wavelength of an electron after falling through a potential difference, in picometres. At 100 volts it is 122.6 picometres, at 400 volts it is 61.3 picometres, at 900 volts it is 40.9 picometres. The wavelength goes as the inverse square root of the voltage, so quadrupling the voltage halves it. The calculation is non-relativistic; at a kilovolt that costs a tenth of a per cent. The dashed line is the 215 picometre spacing between atomic planes in nickel, which is what makes an electron beam diffract off a crystal at all.

Matter waves

  1. 1 Everything has a wavelength, and almost nothing shows it
  2. 2 One arrival at a time, and the pattern still appears
  3. 3 The phase a magnet leaves on a path it never touched
  4. 4 The fall that leaves the mass in the phase
  5. 5 The grating made of light
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7 essays · quantum The same loop, spanned two ways. A 100 cm² capacitor with a 2 mm gap, charged at 1.0 million volts per second. A loop drawn round the wire can be spanned by a flat surface, which the current of 44.271 µA passes through, or by a bag-shaped surface that passes between the plates, which no charge crosses at all. Ampère's law as it stood gave two different answers for one circulation. The rate of change of electric flux between the plates, multiplied by ε₀, is 44.271 µA — the same number to every figure, because C = ε₀A/d is the same ε₀A/d either way. That is the term, and it is not an approximation or a correction: it is what makes the law consistent at all.

Maxwell equations

  1. 1 The term that made light
  2. 2 The potentials that are not unique
  3. 3 The two equations that are not laws of motion
  4. 4 The symmetry one missing charge would complete
  5. 5 The handedness a mirror cannot keep
  6. +2 more
7 essays · electromagnetism A beam through a chain of analysers. An unpolarised beam of intensity 1 entering 3 analysers oriented at 0°, 90°, 0°, selecting the + then + output each time. The intensities are 0.500 and 0.500 out of analyser 1; 0.250 and 0.250 out of analyser 2; 0.125 and 0.125 out of analyser 3. The last analyser is oriented the same way as the first, and it produces a "−" beam of 0.125 — a beam of exactly the kind the first analyser removed entirely. The middle analyser did not filter the beam; it replaced what the beam had an answer to.

Measurement

  1. 1 The answer that was not there before
  2. 2 The measurement that never touched it
  3. 3 Where the interference goes
  4. 4 The state that cannot be copied
  5. 5 The value a gentle measurement returns
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7 essays · quantum The two lengths every mass has. The Compton wavelength and the Schwarzschild radius of the same mass, against mass, on logarithmic axes. One falls and the other rises, so they cross exactly once — here at 1.539·10⁻⁸ kg and 2.286·10⁻³⁵ m, found by bisecting the difference rather than by writing down √(ħG/c³). The conventional Planck values are 2.176·10⁻⁸ kg and 1.616·10⁻³⁵ m; the crossing sits a factor of 1.414 away from them, which is exactly √2 and is the factor of two in the Schwarzschild radius coming through a square root. That is the whole precision this argument has, and it is worth saying, because a number written to four figures invites a reader to believe the definition is doing more work than it is. Nothing in physics is known at that length.

Planck scale

  1. 1 Where every model runs out at once
  2. 2 The length no experiment can resolve
  3. 3 The estimate that misses by a hundred and twenty
  4. 4 The scale that may not be where it looks
  5. 5 The photons that raced for seven billion years
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7 essays · astrophysics The shell of news, and the kink inside it. A charge that was moving at 0.6c to the right, stopped over a short interval, and has been at rest ever since. Outside the sphere of radius ct nothing has heard: the field there still points at the position the charge would have reached had it carried on, marked ahead of it. Inside, the field is that of a charge at rest. In between is a shell one deceleration-time thick, and across it the field line has to bend, because a field line cannot simply stop in empty space — the two ends are joined here to 8.5e-14 pixels. That bend is transverse to the radius, and it is the radiation. It is not an extra thing the charge emitted; it is the join between two static fields that do not line up, and it travels outward at c because that is where the news front is.

Retardation

  1. 1 The field that points where the charge is now
  2. 2 The distance where a field changes its mind
  3. 3 When the source is not heard all at once
  4. 4 The solution that is thrown away
  5. 5 The attraction that weakens when light is too slow
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7 essays · electromagnetism A spacetime diagram at β = 0.5. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light.

Spacetime diagram

  1. 1 Two axes, one speed, and a diagram that does the arguing
  2. 2 The quantity nobody argues about
  3. 3 The diagram a ruler cannot read
  4. 4 The transformation that never mentions light
  5. 5 What the light cones alone can decide
  6. +2 more
7 essays · relativity Two transitions, one shape and one not. The condensate fraction of an ideal Bose gas, 1 − (T/Tc)^3/2, drawn against the superfluid fraction of liquid helium-4, which goes as roughly 1 − (T/Tλ)^5.6. Both reach one at absolute zero and zero at their transition, and in between they disagree everywhere. The ideal calculation says why a transition has to exist; it does not describe the one that does, because its atoms do not interact and helium's do.

Superfluidity

  1. 1 The liquid that will not slow down
  2. 2 The whirlpool that comes in one size
  3. 3 The fountain a lamp can drive
  4. 4 The heat that arrives as a wave
  5. 5 The speed below which nothing can be made
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7 essays · quantum The field of a dipole. Field lines traced from a positive charge toward a negative one. Every line does eventually close on the negative charge, but the outer ones loop far outside any frame, so this picture is a crop rather than the whole field.

The field concept

  1. 1 Field lines are a choice, not a discovery
  2. 2 The field before the lines were drawn on it
  3. 3 The attraction that needs no charge
  4. 4 The shape that takes five numbers
  5. 5 The zeros a handful of charges can make
  6. +2 more
7 essays · electromagnetism A packet, and the wavenumbers it is made of. Above: a wave packet built by adding a continuum of plane waves centred on wavenumber 12 with a spread of 1.6. Below: the weight given to each wavenumber. The packet's width, measured as the standard deviation of its probability, is 0.442; the spread of wavenumbers is 1.131; their product is 0.500, which is a half and cannot be less. Narrowing one bracket widens the other by exactly as much. Nothing quantum has been used to draw either panel.

Uncertainty

  1. 1 Sharpness has to be paid for
  2. 2 The motion that cannot be stopped
  3. 3 The questions that can be asked together
  4. 4 The fastest a state can stop being itself
  5. 5 The angle the uncertainty principle cannot be written for
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7 essays · quantum A horizon 0.97 light years behind, made by nothing but the motion. Position across and time up, in units where light travels at 45°, for a rocket holding a constant proper acceleration of 1 gravity. The worldline is the hyperbola x² − c²t² = (c²/a)², asymptotic to the light line it never crosses. Three light signals are drawn: one released at x = 0.55 catches up at t = 0.63, one released at x = 0 never arrives, one released at x = -0.6 never arrives. The dividing line is the asymptote itself. Everything at or behind it is permanently out of reach, and for one gravity that boundary sits 0.97 light years behind the rocket's starting point. Nothing is there — no mass, no field, no surface. The horizon is a consequence of never stopping.

Accelerated frames

  1. 1 The wall of silence behind a rocket that never stops
  2. 2 The ship that never arrives at c
  3. 3 The temperature of an acceleration
  4. 4 The clock that does not feel the turn
  5. 5 The heat hidden in a Doppler shift
  6. +1 more
6 essays · relativity The term that abolishes the inner orbits. The effective potential of the Schwarzschild geometry per unit mass, in units of c², at 4 angular momenta, against radius in Schwarzschild radii. Newton's version has a minimum — a stable circular orbit — at L̃²/GM for every angular momentum there is, however small, and it is the dashed curve at the same L̃, here with its minima at 10.13, 7.61, 6.00, 5.12 rs. General relativity adds one term, −GM L̃²/c²r³, and that minimum stops existing below a definite angular momentum. At L̃ = 4.50 GM/c the barrier and the well are still separate, at 1.83 and 8.29 rs. The two merge at L̃ = √12 GM/c, at 3 rs = 6GM/c² — the innermost stable circular orbit, where the curve drawn here has neither a maximum nor a minimum but a single inflection. Below that momentum — the curve at L̃ = 3.20 GM/c — there is no stationary point anywhere outside the horizon, so no circular orbit exists at all, at any angular momentum whatever. None of that is a property of matter; it is a statement about the geometry, and it fixes the energy of the innermost orbit at √(8/9) = 0.94281 of mc², so 5.719% of the rest mass has been radiated by anything that reached it. Every well drawn here was located on the emitted curve by golden section and checked against the closed form.

Orbit stability

  1. 1 The orbit that cannot be made smaller
  2. 2 The orbit that does not come back to itself
  3. 3 The arrow that says which way the orbit points
  4. 4 The orbit special relativity cannot close
  5. 5 The hilltop that holds the Trojans
  6. +1 more
6 essays · astrophysics Photocurrent against applied voltage, at two intensities. Photocurrent against retarding voltage for caesium lit at 12×10¹⁴ hertz, at relative intensities of 1 and 2. Both curves reach zero at the same stopping voltage of 2.82 volts, because the brighter light delivers more photons and not more energetic ones. The saturation currents are in the ratio of the intensities. The slope between the stopping voltage and zero assumes the emitted electrons' energies are spread uniformly below the maximum, which they are not; the crossing point does not depend on that assumption.

Photon

  1. 1 Light arrives in lumps, and brightness only changes how many
  2. 2 A photon with a momentum, and a collision that proves it
  3. 3 The experiment a wave cannot pass
  4. 4 The outcomes identical photons refuse
  5. 5 The noise pushed below the floor
  6. +1 more
6 essays · quantum

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