Predict matter-wave directions from accelerating energy
The de Broglie relation λ=h/p associates wavelength with momentum, not with the physical diameter of a particle. An electron accelerated from effectively zero kinetic energy through potential V gains kinetic energy eV. Within the nonrelativistic approximation p²/(2m)=eV, so λ is proportional to 1/sqrt(V). The supplied rounded electron coefficient is λ=1.226/sqrt(V) nm when V is in volts. This is not the photon relation E=pc.
Learning outcome: Compare first-order existence at 25 and100 V for spacing 0.2 nm, then change spacing while keeping energy fixed. Record the candidate sine and speed/c before reporting a direction. State why grating and crystal angle conventions differ.
Starting recap
Separate per-event energy, expectation rate and measurement indication. Check particle identities, conservation, threshold branches and calibration roles before making a source or detector claim.
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At 100 V the wavelength is 0.1226 nm. With spacing 0.4 nm, first-order sine is 0.3065 and angle is about 17.8484°. Quadrupling the voltage would halve wavelength within the same approximation, not quarter it.
Open the experimentPredict
Within the supplied nonrelativistic model, multiplying accelerating voltage by four makes wavelength:
Experiment
Compare first-order existence at 25 and100 V for spacing 0.2 nm, then change spacing while keeping energy fixed. Record the candidate sine and speed/c before reporting a direction. State why grating and crystal angle conventions differ.
Predict a matter-wave diffraction direction
Worked teaching model · synthetic values · Electrons start effectively at rest, acquire kinetic energy eV, and remain nonrelativistic on this bounded grid. Supplied rounded wavelength constant is 1.226 nm sqrt(V). A coherent ideal transmission grating gives d sin(theta)=lambda for first order; this is not a crystal Bragg-angle convention or apparatus plan.
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A square-root acceleration law changes the matter wavelength
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: nonrelativistic matter wavelength (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for A square-root acceleration law changes the matter wavelength
nonrelativistic matter wavelength
| accelerating potential (V) | wavelength (nm) |
|---|---|
| 25 | 0.2452 |
| 27.5 | 0.233789 |
| 30 | 0.223836 |
| 32.5 | 0.215055 |
| 35 | 0.207232 |
| 37.5 | 0.200205 |
| 40 | 0.193848 |
| 42.5 | 0.18806 |
| 45 | 0.182761 |
| 47.5 | 0.177887 |
| 50 | 0.173383 |
| 52.5 | 0.169204 |
| 55 | 0.165314 |
| 57.5 | 0.16168 |
| 60 | 0.158276 |
| 62.5 | 0.155078 |
| 65 | 0.152067 |
| 67.5 | 0.149224 |
| 70 | 0.146535 |
| 72.5 | 0.143986 |
| 75 | 0.141566 |
| 77.5 | 0.139264 |
| 80 | 0.137071 |
| 82.5 | 0.134978 |
| 85 | 0.132978 |
| 87.5 | 0.131065 |
| 90 | 0.129232 |
| 92.5 | 0.127473 |
| 95 | 0.125785 |
| 97.5 | 0.124162 |
| 100 | 0.1226 |
| 102.5 | 0.121096 |
| 105 | 0.119645 |
| 107.5 | 0.118246 |
| 110 | 0.116895 |
| 112.5 | 0.115588 |
| 115 | 0.114325 |
| 117.5 | 0.113102 |
| 120 | 0.111918 |
| 122.5 | 0.11077 |
| 125 | 0.109657 |
| 127.5 | 0.108576 |
| 130 | 0.107527 |
| 132.5 | 0.106508 |
| 135 | 0.105517 |
| 137.5 | 0.104554 |
| 140 | 0.103616 |
| 142.5 | 0.102703 |
| 145 | 0.101814 |
| 147.5 | 0.100947 |
| 150 | 0.100102 |
| 152.5 | 0.0992786 |
| 155 | 0.0984747 |
| 157.5 | 0.09769 |
| 160 | 0.0969238 |
| 162.5 | 0.0961754 |
| 165 | 0.095444 |
| 167.5 | 0.094729 |
| 170 | 0.0940299 |
| 172.5 | 0.093346 |
| 175 | 0.0926769 |
| 177.5 | 0.0920219 |
| 180 | 0.0913806 |
| 182.5 | 0.0907526 |
| 185 | 0.0901373 |
| 187.5 | 0.0895344 |
| 190 | 0.0889434 |
| 192.5 | 0.0883639 |
| 195 | 0.0877957 |
| 197.5 | 0.0872382 |
| 200 | 0.0866913 |
| 202.5 | 0.0861545 |
| 205 | 0.0856276 |
| 207.5 | 0.0851102 |
| 210 | 0.084602 |
| 212.5 | 0.0841029 |
| 215 | 0.0836125 |
| 217.5 | 0.0831306 |
| 220 | 0.0826569 |
| 222.5 | 0.0821912 |
| 225 | 0.0817333 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Electron kinetic energy | 100 | eV |
| de Broglie wavelength | 0.1226 | nm |
| Candidate first-order sine | 0.3065 | ratio |
| Approximate speed divided by c | 0.0197836 | ratio |
| First propagating order exists | 1 | flag |
| First-order angle | 17.8484 | degrees |
A matter-wave direction changes with momentum, not simply with particle count.
K=eV=p²/(2m) gives p proportional to sqrt(V), hence lambda=h/p proportional to 1/sqrt(V). The supplied rounded coefficient applies to nonrelativistic electrons, not arbitrary particles or high acceleration. The bounded speed/c values show why that approximation is appropriate here. First-order absence is not absence of particles: zero order and other scattering remain possible. A diffraction pattern is a probability distribution accumulated over many events, not a revealed path of each electron.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A matter-wave direction changes with momentum, not simply with particle count.
K=eV=p²/(2m) gives p proportional to sqrt(V), hence lambda=h/p proportional to 1/sqrt(V). The supplied rounded coefficient applies to nonrelativistic electrons, not arbitrary particles or high acceleration. The bounded speed/c values show why that approximation is appropriate here. First-order absence is not absence of particles: zero order and other scattering remain possible. A diffraction pattern is a probability distribution accumulated over many events, not a revealed path of each electron.
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