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Classical harmonic oscillator (Read 2195 times)
Gerrit-Jan Linker
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Classical harmonic oscillator
05.03.09 at 13:43:57
 
Classical harmonic oscillator
 
Classical force on the oscillating bodies:
F = -k.x  
 
Equations of motion:
 
F = -k.x = m.a = m d2x/dt2
 
possible solution: x(t) = Acos(2πft+Φ)
 
Taking the derivatives:
d/dt x(t) = -A2πfsin(2πft+Φ)
d2/dt2 x(t) = -A(2πf)2cos(2πft+Φ)
 
Solving:
-kx = m d2x/dt2
-kAcos(2πft+Φ) = -mA(2πf)2cos(2πft+Φ)
k = m(2πf)2
k/m = (2πf)2
√(k/m) = 2πf
 
ω = √(k/m)
ω = angular frequency (2π.f)
k = force constant, spring stiffness coeff, spring constant
m = reduced mass, moment of inertia
 
Potential energy:
To calculate the potential energy, the energy stored in the spring, consider the spring at rest at x=0. We stretch it from x=0 to x to give it its initial potential energy U:
 
U = 0x Fdx = 0x kx dx = ½kx2
 
Total energy:
 
Utotal = Ukinetic + Upotential = ½mv2 + ½kx2
 
Quantum harmonic oscillator:
E(n) = (n+1/2).(h/2pi).√(k/m)
 
Reference:
http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator
 
see also:
Coherent States of the Simple Harmonic Oscillator
http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/CoherentStates.htm
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« Last Edit: 21.07.11 at 09:44:03 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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