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First Quantization, Schrodinger Equation

It was postulated that the momentum p and position q are no longer mere numbers but are operators satisfying the commutative relation:

(qp - pq) = i.

This is called the canonical quantization. The procedure that the particle is endowed with wave property (via the wave function ) is referred to as first quantization. Actually only p is treated as an operator - a differential operator acting on (q), q remains to be a c-number. The commutative relation can be viewed as a mathematical statement for the Uncertainty Principle. Since it implies that p cannot be a function of q (it would mean p and q can be determined precisely at the same time), p can be related to q only in the form of an operator such as - i d/dq, where i = . The commutative relation is the direct result of such interpretation.

Application of this rule for p and q to the equation relating the total energy E to the kinetic energy p2/2m and the potential energy V(q), e.g., E = p2/2m + V(q), yields the time-independent Schrodinger Equation for one particle with mass m and interacting potential V (i.e., it describes the stationary states with well-defined energy E corresponding to E 0, and t in the uncertainty relation t E > ) :
Schrodinger Equation
For a given interacting potential V the problem is to find the Energy E and the corresponding wave function . The wave function is the probability amplitude of finding the particle at certain position (again do not confuse this amplitude with the state vector amplitude in superposition). The absolute square of the amplitude is the probability density. It turns out that the imaginary factory i in the quantization of p is crucial in describing a waveform for the case of E > V; otherwise, will be just an exponential function not compatible with observation. This is the origin of complex number in quantum theory. See for example : Mathematical Schrodinger's Cat (QM Basic).

This form of Schrodinger Equation can be used to find out the structure of the atom or molecule. The basic concepts of superposition, perturbation, transition, and quantum measurement etc. can be explored in a system with an infinite square well potential (see again "Mathematical Schrodinger's Cat (QM Basic)" for details).

The generalization to system of many particles and more than one source of interaction becomes rapidly un-manageable and is solvable only through methods of approximation. The Schrodinger Equation for a poly-atomic molecule with N atomic nucleus and n electrons can be written down as :

{(2/2)[(/m) + (/Mk)] + E - (e2)[(1/2)(1/|ri - rj|) - (Zk/|ri - Rk|) + (1/2)(ZkZl/|Rk - Rl|)]} = 0

where m is the mass of electron, Mk is the mass of the kth atomic nucleus with number of positive charge Zk, E is the total energy of the system, and . Since the atomic nucleus move much slower than the electrons, the Born-Oppenheimer approximation can be adopted to separate the wave function into = electrons nuclei and the resulting equations are solved separately - the part for the electrons yields the electronic configuration while the vibrational and rotational states (of the nuclei) are obtained from the nuclear part. This method makes the equation more or less solvable but the task is still formidable with increasing number of particles. The Molecular Orbital Theory is used further to alleviate the computational difficulty in calculating the molecular structure (the nuclear part).

See "The Different Perspectives of the 1st and 2nd Quantizations".

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