Calculate de broglie wavelenght of an electron moving with a speed of 1/100of speed of light in vaccum
Answers
The De Broglie wave equation for wavelength of a moving particle or object is : w = h/mv, where w is the wavelength in meter, h is Planck’s universal constant, m is the mass of the moving particle or object, and v is its velocity. This follows from the momentum equation for light which is : mc = h/w, where m is the mass of the photon, c is its velocity, h as noted above, and w is the wavelength of light. The light’s momentum equation came from equating Einstein’s E = mc^2 to Planck’s equation for the energy of the quantum of light or photon which is E = hf or hc/w. That makes the expression mc^2 = hc/w, which when solved for mc gives you the momentum equation for light, mc = h/w, which on further rearranging gives w = h/mc for light. De Broglie’s genius was to think of wave theory of matter, as there was already the well established theory of wave-particle duality of light that came predominantly from Einstein and some initial aspects from Planck. De Broglie extended that concept to all matter having ‘wave-like property ‘ while in motion. You can see the similarity of the De Broglie equation for matter-wave’s wavelength, w = h/mv, to the equation above for light’s wavelength and its momentum. Greater the velocity of the moving particle or object, the shorter will be the wavelength corresponding to greater frequency of the ‘matter-wave’ form. Or lower the velocity of the moving particle or object, greater will be the wavelength, and therefore lower the frequency of the ‘matter-wave’. Now going back to your question the mathematical expression of a moving electron’s wavelength at 10% of c will be : w = h/m•0.1c. You have to plug in values for h( 6.6261•10^-34 m^2kg/s), electron’s mass(9.1094•10–31kg), and c(3.00•10^8 m/s) to get the wavelength which is an extremely tiny fraction of a meter. Once you plug these values in your answer will be 2.4246•10^-11 m. Now let me give you something to compare to: What will the wavelength of the waveform of a 70 kg man running at 1 m/s be like ?? When calculated with the De Broglie equation as above it works out to be an incredibly short wavelength of 9.47•10^-36 m !! This also shows you that greater the mass of the moving object, shorter the wavelength of its ‘wave-form’ will be which is also consistent with the wave equation, as mass appears in the denominator with velocity. This also tells you that with the extremely short wavelength as demonstrated above, of the running man, greater the frequency will be of the waveform of a moving macroscopic object, due to its greater kinetic energy. Kaiser T, MD.