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koustav
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- In the series solution of simple harmonic oscillator,why do we have a factor of 1/2 in the trial solution?
koustav said:In the series solution of simple harmonic oscillator,why do we have a factor of 1/2 in the trial solution?
But still we are left with the term e^-q^2/2drvrm said:the factor is chosen from the asymptotic behavior of the solution of differential equation ...in the above situation as the 2nd order partial derivative should yield square of the variable times the function the factor 1/2 is chosen...
see for details
<http://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/lecture-notes/MIT8_04S13_Lec08.pdf>
however the trial functions are always chosen by anticipating the behaviour of the solution as r going to zero and infinity as it is easy to estimate. for bound states at both these extremities the wave function should go to zero.
But if we take the 2nd case still our assumption is correct for large value of qPeroK said:You take ##\psi = e^{-q^2 /2}## because that's the solution. A better question is how do you get to that solution?
If you try ##\psi = e^{-q}## then ##\psi'' = \psi## so that doesn't work.
If you try ##\psi = e^{-q^2}## then ##\psi'' = (4q^2 - 2)\psi## so that doesn't quite work, but it should give you the idea:
If you try ##\psi = e^{-q^2 /2}## then ##\psi'' = (q^2 - 1)\psi## and, if you ignore the ##-1## for large ##q##, then you have a solution.
koustav said:But if we take the 2nd case still our assumption is correct for large value of q
Ok got the point.thanks for simplifying the point!PeroK said:Really? Like ##4q^2 - 2 \approx q^2## for large ##q##?
A simple harmonic oscillator is a system that exhibits periodic motion, where the restoring force is directly proportional to the displacement from the equilibrium position. This type of oscillator can be found in various physical systems such as mass-spring systems, pendulums, and vibrating strings.
The equation of motion for a simple harmonic oscillator is given by F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from the equilibrium position. This equation is based on Hooke's Law, which states that the force exerted by a spring is directly proportional to its extension or compression.
The period of a simple harmonic oscillator is the time it takes for one complete cycle of motion, while the frequency is the number of cycles per unit time. The relationship between the two is given by T = 1/f, where T is the period and f is the frequency. This means that as the frequency increases, the period decreases, and vice versa.
The amplitude of a simple harmonic oscillator is the maximum displacement from the equilibrium position. It is a measure of the size of the oscillations and is determined by the initial conditions of the system. The amplitude remains constant in an ideal simple harmonic oscillator, as long as there is no external force acting on it.
In a simple harmonic oscillator, the total energy is conserved and is constantly changing between potential energy and kinetic energy. When the oscillator is at its equilibrium position, all of its energy is in the form of potential energy. As the oscillator moves away from the equilibrium position, the potential energy decreases and the kinetic energy increases. However, the total energy remains constant throughout the motion.