Download
1 / 8
80 likes | 188 Views
The text to the right has been copied from a GSU webpage “ Hyperphysics ” – the whole thing can be viewed if you click on this link: Quantum Harmonic Oscillator. Why is the quantum simple harmonic oscillator important? For many reasons – for instance, any potential having a
E N D
The text to the right has been copied from a GSU webpage “Hyperphysics” – the whole thing can be viewed if you click on this link: Quantum Harmonic Oscillator
Why is the quantum simple harmonic oscillator important? For many reasons – for instance, any potential having a Local minimum can be approximated by a parabolic potential In the neighborhood of that minimum.
Important examples of potentials with minima are the potentials associated with the interactions between atoms in molecules. The interaction between two atoms in a molecule is pretty well descri- bed by the so-called “Morse Potential” which is definitely NOT a harmonic potential. However, in the vicinity of the minimum it can be approximated by a para- bolic harmonic potential, and such an approximation is known to work quite well If the oscillation energy is relatively small.
Let’s play for a moment with a classical simple harmonic oscillator… The picture displays a classical har- monic oscillator, at the equilibrium position (A), at the positions of maxi- mum displacement to the left (B) and to the right (C). Suppose that the angular frequency of this oscillator is ω . What is the equa- tion describing the displacement as a function of time? (suppose that at t=0 the displacement x is zero).
Now, we want to solve the following problem. Suppose that the oscillator is placed in a dark room. You cannot see it. But you have a camera with a flash. At some moment, you take a picture of the oscillator. What is the probability that on the picture the mass is positioned between x and x+dx ?
The probability of finding the mass between x and x+dx is proportional to (answer): ……. Yes! You are right! It is proportional to dt , i.e., to the time the mass spends between these two points. But how can we find dt ? Please answer:
Now, please compare the profile of the function we have found with the square of the wavefunction of the SHO quantum state with n=100 ! The striking similarity of the two functions, the classical one and the quantum one, is a beautiful illustra- tion of the Bohr’s Principle of Correspondence
