De Broglie wavelength of electron

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SUMMARY

The de Broglie wavelength of an electron with kinetic energy E (in eV) is expressed as λ = 12.3 × 10-8 / E1/2. This relationship is derived using the equations λ = h/p and E = p2 / (2m), where h is the reduced Planck's constant (ℏ = h / (2π)). The numerical factor of 12.3 arises from unit conversions and the specific properties of electrons.

PREREQUISITES
  • Understanding of quantum mechanics concepts, specifically wave-particle duality.
  • Familiarity with the de Broglie wavelength equation.
  • Knowledge of kinetic energy and momentum relationships in physics.
  • Basic proficiency in unit conversions and dimensional analysis.
NEXT STEPS
  • Study the derivation of the de Broglie wavelength in more detail.
  • Learn about the implications of wave-particle duality in quantum mechanics.
  • Explore the concept of reduced Planck's constant (ℏ) and its applications.
  • Investigate the relationship between kinetic energy and momentum in various contexts.
USEFUL FOR

Students and educators in physics, particularly those focusing on quantum mechanics and wave-particle duality, as well as anyone seeking to understand the behavior of electrons at a fundamental level.

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Homework Statement



Show that the de Broglie wavelength of an electron of kinetic energy E (eV) is

[tex]\lambda = \frac{12.3*10^{-8}}{E^{1/2}}[/tex]

Homework Equations



[tex]\lambda = \frac{h}{p}[/tex]
[tex]E = \frac{p^2}{2m}[/tex]

The Attempt at a Solution


I've played around with substituting and things like that, but I can't seem to find that 12.3 number anywhere. I feel like I must be missing something simple, but I'm not sure what.
 
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First convert the second equation to the right units and solve it for p, then plug it into the first equation and work out all the things you have numerical values for. Note that h is actually [itex]\hbar = h / (2\pi)[/itex].

I then get [itex]12.26 \times 10^{-9}[/itex].
 

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