De Droglie particle-wave equation - Derivation by de Broglie

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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de Broglie’s
WAVE EQUATION

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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
????????????, the velocity of light ???????????? , the
momentum ????????????and the
frequency ????????????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 ????????????????????????????????????????????????????????????????????????????????????????????????=????????????
Light f????????????????????????????????????????????????????????????????????????????????????????????????=????????????
????????????????????????????????????????????????=????????????
??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

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

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

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

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Speed & Wavelength
In radiation (light), the
frequency ????????????
??????????????????????????????????????????????????????????????????of a photon
is related to its velocity ????????????and
wave length ????????????by:
????????????
??????????????????????????????????????????????????????????????????=
????????????????????????????????????????????????????????????
??????????????????????????????????????????????????????????????????????????????????????????????????????????????????
=
????????????
λ
So in terms of λ, the Planck’s
energy relationship can be
written as:
????????????
??????????????????????????????????????????????????????????????????=ℎ????????????=ℎ????????????/λ
Or:
λ
??????????????????????????????????????????????????????????????????=????????????/????????????
????????????
??????????????????????????????????????????????????????????????????=ℎ????????????/λ
λ
c
λ
??????????????????????????????????????????????????????????????????=????????????/????????????
??????????????????????????????????????????????????????????????????

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Planck + Einstein
Linking up Planck’s formulae with
Einstein’s energy equation, de
Broglie had:
????????????=ℎ????????????=????????????????????????
ℎ????????????=????????????????????????
or:
????????????????????????=ℎ????????????
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:
????????????=ℎ/????????????
As seen in previous page ????????????/????????????=????????????.
????????????????????????=ℎ????????????
????????????/????????????=ℎ/????????????
????????????=ℎ/????????????
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:
????????????
??????????????????????????????????????????????????????????????????=
ℎ
????????????
??????????????????????????????????????????????????????????????????
????????????
????????????????????????????????????????????????????????????????????????????????????????????????=
ℎ
????????????
????????????????????????????????????????????????????????????????????????????????????????????????
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. ????????????
????????????is called the
de Broglie wavelength of the
electron:
????????????
????????????????????????????????????????????????????????????????????????????????????????????????=
ℎ
????????????
????????????????????????????????????????????????????????????????????????????????????????????????
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.