The de Broglie wave equation as derived by de Broglie. It has no physical basis except that of the photon. Derivation by analogy.

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WAVE EQUATION
de Broglie’s derivation of his:

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Determining the Matter
Wave Equation
To determine the wavelength
of the wavy electron, de
Broglie made use of the
relations between the energy
𝐸𝐸, the velocity of light 𝑐𝑐, the
momentum 𝑝𝑝 and the
frequency 𝑓𝑓 of a photon or
particle established by Planck
and Einstein at the time.
To start with, de Broglie first
employed Einstein’s relativistic
energy equation.
Light 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 𝑐𝑐
Light f𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑓𝑓
𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = 𝑚𝑚
𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑜𝑜𝑜𝑜 𝑎𝑎 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

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Classical Momentum
In classical mechanics, the momentum 𝑝𝑝𝑝𝑝 of a particle is equal to the product of its
mass 𝑚𝑚𝑝𝑝 and velocity 𝑣𝑣𝑝𝑝, or 𝑝𝑝𝑝𝑝 = 𝑚𝑚𝑝𝑝 𝑣𝑣𝑝𝑝. If the speed is so high as close to the
speed of light 𝑐𝑐 (relativistic speed), its momentum will be governed by Einstein’s
relativistic equation.
𝑣𝑣𝑝𝑝 ≪ 𝑐𝑐 𝑣𝑣𝑝𝑝 ≈ 𝑐𝑐
Classical Newtonian Einsteinan
Your need to use
my equations
Velocity of particle Velocity of light

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Einstein’s Energy Equation
Einstein’s equation for the energy 𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟
of a particle at high speed is written as:
𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟
2
= 𝑝𝑝2
𝑐𝑐2
+ (𝑚𝑚𝑜𝑜 𝑐𝑐2
)2
Taking the square roots on both sides,
we have:
𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑝𝑝2 𝑐𝑐2 + (𝑚𝑚𝑜𝑜 𝑐𝑐2)2
At the same time, Einstein's theory of
relativity pointed out that for a particle
like a photon of zero rest mass 𝑚𝑚𝑜𝑜 = 0.
So we can neglect the (𝑚𝑚𝑜𝑜 𝑐𝑐2
)2
term
and the relativistic energy becomes:
𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑝𝑝2 𝑐𝑐2 + (𝑚𝑚𝑜𝑜 𝑐𝑐2)2
= 𝑝𝑝2 𝑐𝑐2 = 𝑝𝑝𝑝𝑝

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Planck’s Equation
On the other hand, according to Planck,
the energy 𝐸𝐸γ of a photon is related to its
frequency 𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 and Planck’s constant ℎ
by the famous Planck’s equation:
𝐸𝐸γ = ℎ𝑓𝑓γ
where ℎ is Planck's constant; 𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 is the
frequency of the radiation or photon.
𝑓𝑓γ
Photon
frequency
gamma - symbol
for photon h – Planck’s constant

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Speed & Wavelength
In radiation (light), the
frequency 𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 of a photon
is related to its velocity 𝑐𝑐 and
wave length 𝜆𝜆 by:
𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 =
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤
=
𝑐𝑐
λ
So in terms of λ, the Planck’s
energy relationship can be
written as:
𝐸𝐸𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = ℎ𝑓𝑓 = ℎ 𝑐𝑐/λ
Or:
λ𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑐𝑐/𝑓𝑓
𝐸𝐸𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = ℎ𝑐𝑐/λ
λ
c
λ𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑐𝑐/𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

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Planck + Einstein
Linking up Planck’s formulae with
Einstein’s energy equation, de
Broglie had:
𝐸𝐸 = ℎ𝑓𝑓 = 𝑝𝑝𝑝𝑝
ℎ𝑓𝑓 = 𝑝𝑝𝑝𝑝
or:
𝑝𝑝𝑝𝑝 = ℎ𝑓𝑓
That is: Planck’s frequency energy
= Einstein’s relativistic energy
Kinetic energy
of photon
Frequency
energy of photon

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Wavelength and
Momentum
By manipulating the equation a
little bit in moving the terms on
both sides, we have a new
equation which finally becomes:
𝜆𝜆 = ℎ/𝑝𝑝
As seen in previous page 𝑐𝑐/𝑓𝑓 = 𝜆𝜆.
𝑝𝑝 𝑐𝑐 = ℎ𝑓𝑓
𝑐𝑐/𝑓𝑓 = ℎ/𝑝𝑝
𝜆𝜆 = ℎ/𝑝𝑝
Swap side
Swap side

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De Broglie Hypothesis
At this point, de Broglie made an ingenious
intuitive guess that if the electron is also a
wave particle, its formulae should also be
like that of a photon wave. That is, the same
formula works also for the electron:
𝜆𝜆𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 =
ℎ
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝜆𝜆𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =
ℎ
𝑝𝑝𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
Photon
wave
Electron
wave

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de Broglie equation
This relation between the wavelength
and the momentum of the electron
later became known as the famous
de Broglie equation. 𝜆𝜆𝑒𝑒 is called the
de Broglie wavelength of the
electron:
𝜆𝜆𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =
ℎ
𝑝𝑝𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
So the particle bursts open and
becomes a wave-particle. It is an
assumption that if an electron is free,
it would behave like a photon.

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DERIVATION BY SK
To be continued on:
ABCC
The physical origin of de
Broglie’s particle-wave equation