For simple harmonic oscillator Hook's law and Newton's law can be obtained.
According to Hook's law the force of the SHO is directly proportional to the displacement (x) but in opposite direction.
F ∝ − x
F = − kx ........................(1)
According to Newton's law, the force of SHO is directly proportional to the mass(m) and acceleration (a)
F ∝ m
F ∝ a
F ∝ ma
F = ma .........................(2)
constant = 1
−kx = ma ....................(3)
The acceleration of SHO is obtained by differentiating the velocity which in term obtained by differentiating displacement with respect to time.
a = dx²
dt²
This displacement of SHO is mathematically defined as,
x = sin 2πυt
Differentiating displacement write time,
dx = cos 2πυt. 2πυ
dt
Differentiating the above write time again, we will get acceleration.
d²x = − sin 2πυt. 2πυ. 2πυ
dt²
(or)
a = − sin 2πυt. 4π²υ²
a = −4π²υ²x ..........................(4)
Substituting equation (4) in (3)
- kx = m(-4π²υ²x)
k = 4π²υ²m ...........................(5)
For a SHO, the potential energy is mathematically defined as,
V= 1/2 kx²
V = 1/2 (4π²υ²m) x²
V = 2π²υ²x²m ...................(6)
The fundamental schrodinger equation is,
∇²ψ + 8π²m (E-V) ψ = 0
h²
∂²ψ + 8π²m ( E−2πυxm ) ψ = 0....(7)
∂x² h²
This is the Schrodinger equation for SHO,
solving the above equation we will get, the wave function ψ and the eigen value E for SHO. The values are,
Eigen function :-
ψ = Nn. e⁻ᑫ²/² Hn (q)
Here,
ψ = Eigen function for SHO
Nn= Normalization constant
q = x√b
b = υm
h²
Hn (q) Hermite polynomial.
Eigen value :- (E)
Eₙ = ( n+1/2) hʋ
Here,
n = 0,1,2,3.........
h − plank's constant
υ − Frequency of SHO
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