Let us suppose an electron moving with velocity of light, it undergoes wave motion.
So, if we want to describe the motion of the electron, we have to use quantum mechanics and quantum mechanics principle.
Here, electron moving like a wave there must be a vibration, frequency and amplitude there must be phase.
Let ψ = ψ₀ sin(ωt-kx). . . .(1)
it is simple harmonic wave equation.
where, ψ= wave function,
ω= angular velocity,
t = time,
k= wave vector and
x= displacement
Diff. eq(1) with respect to x
we get,
dψ/dx = ψ₀ cos(ωt-kx)(-k)
Diff. again with respect to x
we get,
d²ψ/dx²= -ψ₀ sin(ωt-kx)(-k)(-k)
= -k²ψ₀ sin(ωt-kx)
= -k²ψ [ψ = ψ₀ sin(ωt-kx). . . .(1)] d²ψ/dx² +k²ψ =0 . . . .(2)
we know k=2π/λ
putting the value of k in eq(2)
we get,
d²ψ/dx² + 4π²/(λ²ψ) =0 . . . .(3)
according to de-broglie hypothesis
λ=h/(mv)
similarly, λ²= h²/(m²v²)
putting the value of λ² in eq(3)
we get,
d²ψ/dx² + 4π²m²v²ψ/h²=0 . . . .(4)
E=K+V
E=1/2 (mv²) + v
(E-V) = 1/2 (mv²)
2(E-V)= mv²
multiply m both side,
we get,
2m(E-V) = m²v²
putting the value of m²v² in eq(4)
d²ψ/dx² + 4π²/h² 2m(E-V)ψ=0
according to bohr
ħ=h/2π or ħ²=h²/4π²
d²ψ/dx² +2m(E-V)ψ/ħ²=0
This equation represent time independent Schrodinger wave in one dimension.
In three dimension it may be written in the following way
d²ψ/dx² + d²ψ/dy² + d²ψ/dz² + 2m(E-V)ψ/ħ² =0
∆²ψ+ 2m(E-V)ψ/ħ²=0
where ∆=id/dx + jd/dy + kd/dz.
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