Proof of de Broglie wavelength for electronAccording to de Broglie's wave-particle duality, the relation between electron's wavelength and momentum is λ = h / m vThe proof of this is given in my textbook as follows:1.De Broglie first used Einstein's famous equation relating matter and energy, E = m c 2 ,where E = energy, m = mass, c = speed of light.2.Using Planck's theory which states every quantum of a wave has a discrete amount of energy given by Planck's equation, E = h ν ,where E = energy, h = Plank's constant ( 6.62607 × 10 − 34 J s ), ν = frequency.3.Since de Broglie believes particles and wave have the same traits, the two energies would be the same: m c 2 = h ν . m c 2 = h ν .4.Because real particles do not travel at the speed of light, de Broglie substituted v, velocity, for c, the speed of light: m v 2 = h ν .I want a direct proof without substituting v for c. Is it possible to prove directly λ = h / m v without substituting v for c in the equation λ = h / m c?

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Proof of de Broglie wavelength for electronAccording to de Broglie&#039;s wave-particle duality, the relation between electron&#039;s wavelength and momentum is   λ  =  h      /    m  vThe proof of this is given in my textbook as follows:1.De Broglie first used Einstein&#039;s famous equation relating matter and energy,  E  =  m      c    2    ,where   E  = energy,   m  = mass,   c  = speed of light.2.Using Planck&#039;s theory which states every quantum of a wave has a discrete amount of energy given by Planck&#039;s equation,  E  =  h  ν  ,where   E  = energy,   h  = Plank&#039;s constant (  6.62607  ×      10          −      34              J        s  ),   ν  = frequency.3.Since de Broglie believes particles and wave have the same traits, the two energies would be the same:  m      c    2    =  h  ν  .  m      c    2    =  h  ν  .4.Because real particles do not travel at the speed of light, de Broglie substituted   v, velocity, for   c, the speed of light:  m      v    2    =  h  ν  .I want a direct proof without substituting   v for   c. Is it possible to prove directly   λ  =  h      /    m  v without substituting   v for   c in the equation   λ  =  h      /    m  c?

Bentyrchjurvk · Accepted Answer

De Broglie proposed that the relation   p  =  h      /    λ would not only hold for photons but also for massive particles. This inspired Schrödinger to propose his famous equation.

Peia6tvsr · Accepted Answer

The   c in   m      c    2   is not the actual speed of the particle (unless we’re talking about light, but then m would be zero).   m      c    2   is simply the energy the particle has at rest. I don’t know exactly how De Broglie did it, but you can prove it like this:First you can prove that the momentum operator should be       ℏ    i        d          d      x       by finding the generator of the translation operator in the quantum mechanical way (which gives you something like       d          d      x       as the generator) and the classical way (which gives you the momentum as the generator), and simply state that both results should be equivalent. If this sounds unfamiliar, then I suggest you look into Noether’s theorem (it’s one of the coolest theorem’s in maths/physics, so I would suggest it anyway). But if you’re okay with just assuming that the momentum operator is equal to       ℏ    i        d          d      x      , then you can just start from there.Using   p  =      ℏ    i        d          d      x       and the assumption that particles have a wave-like nature, you can prove that   p  =      h          λ      . Since in general, we can write the wave function of a given state of a particle as:       Ψ    (  x  ,  t  )  =            Ψ        0        e          i      (      k      x      −              ω            t      )      , which gives us:   p      Ψ    =      ℏ    i              d              Ψ                    d      x        =  ℏ  k      Ψ  , so   p  =  ℏ  k  =  h      /        λ  You can do the same thing to prove Planck’s theorem by first finding the generator of time translation and prove that the operator of energy should be   i  ℏ      d          d      t      , and then letting this operator act on       Ψ   again.N.B.: In the most general case       Ψ    (  x  ,  t  ) should be a superposition of wave functions with different             Ψ        0  ,   k and/or       ω  , but then you can’t be sure what the momentum of your particle is anymore.

Proof of de Broglie wavelength for electron According to de Broglie's wave-particle duality, the re

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