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Historical backgroundPrior to 1900 all physical phenomena were believed to be explicable BUT there were three critical experiments in the pre-quantum era which could NOT be explained by a straightforward application of classical physics:
Today the combined work of these three men is known as the Old Quantum Theory. The old quantum theory, resulting in Bohr's orbital model of the atom could point to certain real successes: Derivation of the Balmer formula, quantum numbers and selection rules for energy states in an atom, explanation of the periodic table and the Pauli exclusion principle. The old quantum theory relied heavily on the Newtonian mechanics, but sought to supplement it with supplementary conditions. But what about the particle / wave character of the electron ? (de Broglie, 1923) In the early 1920 's it was clear that the quantum theory as it then existed was unsatisfactory. In the mid 1920 's two distinct and seemingly independent versions of a new quantum theory were presented:
Soon after their discovery these two formulations where shown to be equivalent, forming the basis of present-day quantum theory.
ConceptsThe theory of quantum mechanics asserts that with every possibility for an event in nature to take place, there is a quantity called amplitude associated with each alternative. Furthermore, the amplitude associated with the overall event is obtained by adding the amplitudes of each of the alternatives. The probability that the event will happen is equal to the square of the absolute value of the overall amplitude. Thus, if f1 and f2 are the amplitudes of the two possibilities for a particular event to take place, the amplitude for the total event is f = f1 + f2 and the probability for the event to occur is given by P = | f1 + f2 |2 In the macroscopic world the total probability for an event to take place is given by P = P1 + P2 the sum of the probabilities of each alternative. In quantum mechanics P = | f1 |2 + | f2 |2 + f1f2* +f1*f2 = P1 + P2 + f1f2* + f1*f2 showing that the law of computing probabilities is not that of classical physics. The two additional terms are due to the interference of alternatives. If the event is interrupted before its conclusion, for example by determining if the event takes place through alternative 1, the amplitudes of all other alternatives can no longer be added to the total amplitude. The fact that the total probability follows from the knowledge of the amplitudes of all interfering alternatives forms the basis of what is called the Heisenberg uncertainty principle. The uncertainty principle asserts that there is a natural limit to the accuracy of any measurement. For instance the momentum of a particle cannot be precisely specified without loosing all information about its position, and vice versa. The uncertainty principle demonstrates that there are fundamental limitations to the use of concepts based on every-day experience.
Schrödinger equationThe time evolution of a quantum system follows from the solution of the TDSE,
the Time-Dependent Schrödinger Equation. For simplicity consider the
TDSE describing a system that can be in no more than two states. For a quantum
system that has only two possible states f1
and f2 the TDSE reads
where
is the Hamiltonian describing the system. The solution of this equation gives a complete description of the time evolution of the quantum system. For instance, the probability to find the system in state 1 at time T is given by P1( T ) = | f1( T ) |2 The Hamiltonian for a particle in an electromagnetic potential is given by
where m is the mass and e is the charge of the particle. The quantum state of the particle is characterized by the amplitude Y ( r , t ) for any point in space and time. This amplitude is also called the wave function of the particle. As before, the TDSE governs the time evolution of the wave function. The probability to find the particle at the position R at time T is given by P ( R , T ) = | Y ( R , T ) |2 The wave function contains all the information about the quantum system. Once it is known for all points in space and time, any physical quantity can be calculated.
Simulation methodThe simulation method employs a sophisticated numerical algorithm to solve the TDSE. Essentially, the algorithm repeatedly performs the sum over all alternatives. For all practical purposes the wave function generated by this algorithm is exact. All technical details about the algorithm and the simulation technique can be found in the paper: Computer Simulation of Quantum Phenomena in Nano-Scale Devices, by Hans De Raedt, Annual Reviews of Computational Physics IV (ed. D. Stauffer, World Scientific, 1996). [download *.pdf file] Computer animation is used as a tool to present the simulation results. The animation shows the time evolution of the probability as a function of the position in space. Unless stated explicitly, the animations show the probability distribution (the probability for each point in space) of a single particle.
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in terms of what we now call classical physics. Newtonian mechanics (1687)
explained the motion of mechanical objects on both celestial and terrestrial
scales. Application of this theory to the motion of molecules resulted in the
kinetic theory of gases. J.J. Thomson's discovery of the electron (1897)
involved demonstrating that its behavior is described by Newton's equations of
motion. The wave nature of light had been suggested by the diffraction
experiments of Young (1803). It became more obviously so by Maxwell's discovery
in 1864 of the connection between optical, magnetic and electrical phenomena.
Taking all this into account it is not surprising that at the end of the
nineteenth century many physicists thought that all interesting questions had
been asked and that finding the right answers would be merely a matter of
time. 
