Frank Rioux
No language which lends itself to visualizability
In spite of Max Born's warning, we are going to present a simple model for a quantum jump that is based on visualization.
This tutorial deals with the quantum mechanical particle (quon to use Nick Herbert's term) on a ring. Previously the behavior of the
http://www.users.csbsju.edu/~frioux/q-jump/benzene-por.pdf
http://www.users.csbsju.edu/~frioux/q-jump/PiElectronsBenzene.pdf
In the absence of perturbations, such as oscillating electromagnetic fields, quantum systems are in the stationary states obtained from the solution of the time-independent Schrodinger equation. When a quantum system is subject to a perturbation such as an oscillating electromagnetic field it is forced into a time-dependent superposition of its stationary states.
For the system to absorb energy from the electromagnetic field and undergo a transition from one allowed energy state to another two conditions must be met.
According to this model, when an electron on a ring is subjected to a electromagnetic perturbation, the electron responds
The model is illustrated below for several possible transitions for an electron on a ring and the selection rule determined. Previously this model was used for the particle in a box and the harmonic oscillator. Links to these tutorials are provided below.
http://www.users.csbsju.edu/~frioux/q-jump/njump.pdf
http://www.users.csbsju.edu/~frioux/q-jump/jumpsho.htm
http://www.users.csbsju.edu/~frioux/q-jump/p-jump.htm
Set up for graphics:
Form the time-dependent superposition of initial and final states:
Initial state:
Final state:
Select Animation and then Record from the Tools menu and follow instructions. Choose from 0 to 30 at 3 frames/sec for a reasonable animated display.
Before runing the animation note that this superposition shows dipole character in that the electron density is a minimum at the opposite side of the ring at which it is a maximum. This dipole rotates around the ring during the animation, providing a mechanism for coupling with the electromagnetic field illuminating the particle on a ring, if the Bohr frequency condition is satisfied. Thus the transition from the n = 0 state to the n = 1 state is allowed. It is easy to show in this way that any transiton for which
Now we look at a transition for which
Initial state:
Final state:
It is easy to show that while n = 1 to 2 and n = -1 to -2 transitions are allowed, the n = 1 to -2
Acknowledgement
The preparation of this tutorial and others on my page dealing with quantum jumps was stimulated by reading "Fluctuating Electric Dipoles and the Absorption of Light," by David R. McMillin [J. Chem. Ed. 55, 7 (1978)] .