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First Quantization, Schrodinger Equation (Revised in 2021)

    There were more experimental evidences (such as photoelectric effect, Compton scattering, etc) since the formulation of the Planck's Law in 1900. The new formula requires the quantization of energy in discrete amount. Theoretical development to reconcile the discrepancies with classical theory took about 25 years to something creditable. It happened that there are many routes to arrive at the solution, which basically is to treat the momentum p and energy E as operator (differentiation or otherwise), e.g., p i(d/dx) and E -i(d/dt). The followings are brief sketches to show the essence of the different methods.

  1. Probability Wave (1900 - 1925) -
  2. Matrix Mechanics (1925) - The formulation is based on the non-commutative relation between p (the linear momentum) and x (the spatial coordinate), i.e., xp - px = i. The scheme is implemented by matrices as shown in Figure 12-05a,b. The replacement of ordinary number by matrix can be interpreted as turning a definite entity into a blurry one, and that has exactly been shown by its relationship with the uncertainty principle px (see "Schwartz's Inequality").

  3. Canonical Quantization (1926) - This method tries to derive a quantum version from the classical theory with minimal deviation.
    The Hamiltonian formulation of classical mechanics treats the dynamic of the system in terms of energy (aka Hamiltonian H, see Figure 12-05a,c), instead of force (as F = ma in Newtonian mechanics). It involves a mathematical term called Poisson Bracket (PB). For the case of coordinate x and momentum p, the PB {xp - px} = 1. Quantization just replaces the PB with the commutation rule [xp - px] = i from which we obtain p -i(d/dx), and all occurrences of p in the Hamiltonian H is substituted by such operator. See the wordy justification in 321 pages by P.A.M. Dirac in his 1926 thesis "The Principles of Quantum Mechanics".

  4. Time Development Operator (2014) - A recent formulation does away all the baggages of the previous methodologies by starting with just a simple operator running in Hilbert space, i.e.,

    All quantum properties such as probability wave, commutation rules, uncertainty principle, and p = -i(d/dx) can be derived as shown in Figure 12-05a,d (with some hindsights ?). See "Short-cut to the Introduction of Quantum Theory" for more details, and the original book on "Quantum Mechanics" by L. Susskind, 2014.
Meanwhile, all the arguments about p, x are applicable for the energy E and time t, such that :
energy of black body radiation : E = h = , = 2,
commutation rule : [tE - Et] = -i,
uncertainty principle Et ,
and E i(d/dt).

The quantization rules can be applied to the kinetic energy E of a free particle (i.e., no force acting on it) :
The real part of the traveling wave and standing wave is shown in Figure 12-05b and 12-05c respectively.

Traveling Wave Standing Wave These examples do not exhibits qunatization of E. It occurs only when a potential (energy) V is added to the free particle equation and V has to be something like a well to confine the particle. Such general form is called Schrodinger Equation
Schrodinger Equation

Figure 12-05b Traveling Wave [view large image]

Figure 12-05c Standing Wave
[view large image]

which has the time dependent part separated out similar to the standing wave. For a given interacting potential V the problem is to find the Energy E and the corresponding wave function .
Energy Quantization This form of Schrodinger Equation can be used to find out the structure of the atom or molecule (in steady state and often with approximation). It was very important for finding out the intensity of spectral lines via the transition between energy levels. The basic concepts of superposition, perturbation, transition, and quantum measurement etc. can be explored in a system with an infinite square well potential, see "Mathematical Schrodinger's Cat (QM Basic)" for details. Another good example is the harmonic oscillator with V = kx2/2 (Figure 12-05d).

Figure 12-05d Energy Quantization

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 :

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 "Many-Body Problem (2022 Edition)".

Under the condition of a lot of particles, statistical method is used to describe the system (see "Bose-Einstein Distribution"). One of the application of this method is to derivate the Planck's Law for black body radiation. At this point, the progress of quantum theory has come a full circle to derivate the Planck's formula without quantizing the energy of the particles explicitly.

The formalism above is called "Quantum Mechanics". It is non-relativistic (not invariant under Lorentz transformation) and cannot deal with high energy phenomena such as particle creation, etc. See "Quantum Field and 2nd Quantization".

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