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American Journal of Physics and Applications
2017; 5(6): 80- http://www.sciencepublishinggroup.com/j/ajpa doi: 10.11648/j.ajpa.20170506. ISSN: 2330-4286 (Print); ISSN: 2330-4308 (Online)
A Method for Deriving Quantum Dynamic Equations from
Classical Mechanics
Hua Ma
The College of Science, Air Force University of Engineering, Xi’an, People’s Republic of China
Email address:
[email protected]
To cite this article:
Hua Ma. A Method for Deriving Quantum Dynamic Equations from Classical Mechanics. American Journal of Physics and Applications. Vol. 5, No. 6, 2017, pp. 80-83. doi: 10.11648/j.ajpa.20170506.
Received : August 14, 2017; Accepted : September 6, 2017; Published : October 11, 2017
Abstract: Based on the operator theories and Hamiltonian canonical equation, an operator based quantum dynamics equation
is established, which has the same effect as the Hamiltonian equation in describing the state evolution of quantized dynamical systems. As the reasonable verification of this equation, Schrodinger equation can be derived theoretically, and the variational principle properties of quantum mechanics are revealed. This work will help to promote the development of quantum theory and to perfect the axiomatic system of quantum mechanics.
Keywords: Classical Mechanics, Quantum Mechanics, Variational Principle, Hamiltonian Canonical Equation,
Schrodinger Equation, Operator Theory
1. Introduction
The variational principle is the general principle of scientific principles in calculus variation, and thus for the discovery of functional extremes, depending on the development of these functions [1-3]. All laws of physics can be mathematically described by self adjoint operator as a variational principle. In classical mechanics, these expressions are called Hamiltonian principles, which were first proposed by William Ron Hamilton in 1833. Hamiltonian mechanics is a theory developed as a new development of classical mechanics. It predicts the same result of non-classical mechanics. It employs a variety of mathematical formalisms that provide a more abstract understanding of the theory. It is pointed out that the dynamics of a physical system is a single function basis by a functional variational problem on Lagrange, which contains information about all of the physical system and the force acting on it [1, 4, 5]. Quantum mechanics is a branch of physics as the fundamental theory of nature at small scales and low energy levels of atoms and subatomic particles, which gradually arose from Max Planck's solution to the black-body radiation problem and Albert Einstein's interpretation of the photoelectric effect. Early quantum theory was profoundly reconceived in the mid-1920s. The reconceived theory is formulated in various specially developed mathematical
formalisms. The sign of perfection and trend of quantum mechanics is the introduction of wave function, Max Born's statistical interpretation, and the establishment of Schrodinger equation [6, 7]. Schrodinger equation is the mathematical equation that describes the way that the quantum wave function, which plays the same role for a quantum system as that Newton's laws of motion play in the mechanics of everyday objects. Schrodinger equation comes from Erwin Schrodinger, when he first derived these formulas and applied them to the behavior of an electron. As a mathematical formulation for studying quantum mechanical systems, it is considered a central result in the study of quantum system and its derivation was a significant landmark in developing the theory of quantum mechanics [8-10]. Today, Quantum mechanics has entered the axiomatization system, especially having perfect combinations with classical mechanics. In recent years, many scholars have studied and put forward a rich new ideas about the derivation, explanation, solution, and the expansion of applications focusing on the quantum dynamic equations [11-14]. However, how the Schrodinger equation is established is still a mystery, and still a suspense about constructing the axiomatic system of quantum mechanics. Historically, many physicists tried to give different derivations: Erwin Schrodinger established it by operating harmonic equation, Heisenberg explained its
81 Hua Ma: A Method for Deriving Quantum Dynamic Equations from Classical Mechanics
variational nature by matrix mechanics, Richard Feynman proved the equivalence between it and path integral mechanics, and later it was attempted to derive by group theory [6-8, 10]. All these works contain the components of conjecture and analogy, and there are still some deficiencies in the construction of axiomatic system of Quantum mechanics. It's really hard to theoretically deduce the formula, just as Richard Feynman says: It is not possible to derive it from anything you know. It came out of the mind of Schrödinger [15]. After years of thinking, I have finally found a scheme for deriving the Schrodinger equation, which is based on the quantization hypothesis and the Hamilton principle. In the derivations, it was found that the Hamiltonian canonical equation can be handled by operator, so as to obtain a generalized dynamics equation based on operator description, which has a broader physical connotation and can be used as the basis of the Schrodinger equation. In the follows: firstly, from Hamiltonian mechanics, establish the general quantum dynamics equation using operator methods and quantization hypothesis; then, derive the expression of momentum operator based on the quantum hypothesis; and finally, apply the momentum operator expression to the general quantum dynamic equation to derive Schrodinger equation.
2. Deduction of the General Quantum
Dynamic Equation
2.1. Quantization Hypothesis
Quantization hypothesis, established by Max Planck, Albert Einstein, and De Broglie et al ., which can be summarized as: for a quantum system, energy, momentum and other quantities are restricted to discrete values, and objects have characteristics of wave-particle duality [6, 7]. In following derivations, some basic ideas were adopted as the foundations. First, the assumption of quantum states: Quantum systems behave some states, which are represented by wave functions. Second, the statistical explanation on wave functions: Wave functions have complex-valued probability amplitudes, from which the probabilities for the possible results of measurements made on the system can be calculated. Third, the mathematical approach based on the operator theory: Any observable physical quantities, i.e. position, momentum, energy, angular momentum etc., should be associated with a self adjoint linear operator [16]. The operators must yield real eigenvalues, since they are values that may come up as the result of the experiment. As the results, the physical operators must be Hermitian in Hilbert space. In accordance with the above assumptions and requirements, the wave function must be square integrable meaning in region R with N-dimensional coordinate vector q :
(^2) N * N ∫R ψ^ ( , ) q t^^ d^ q^ =^ ∫Rψ^ ( , ) q t^ ψ^ ( , ) q t d^ q < ∞ (1)
And the normalization condition results in:
- N ∫R ψ^ ( , ) q t^ ψ^ ( , ) q t d^^ q =^1
The ideas expressed by Eqs. (1) and (2) will be used as the basis for the following derivations.
2.2. The Time Dependent Differential Form of Hamiltonian Canonical Equation
In Hamiltonian mechanics, a classical physical system is described by a set of canonical coordinates ( q , p ) and Hamiltonian H , where q and p are default as vectors in full text. The time evolution of a dynamical system is uniquely defined by Hamiltonian canonical equation [1, 4, 5]:
∂ = − ∂
d p H dt q
d q H dt p
In actual physical processes, momentum and coordinates are functions of time:
p = p t ( ) , q = q t ( ) (4)
So, Hamiltonian H can be expressed as:
H = H q t ( ( ), p t ( )) (5)
And thus its differential is:
H H H dq H dp dH dq dp dt q p q dt p dt
Taking Hamiltonian canonical Eq. (3) into Eq. (6), it reads:
dp dq dq dp dH dt dt dt dt dt
2.3. Derivation of the Quantum Dynamical Equation in the Form of Operator
Introducing operators q → q ˆ and p → p ˆ into Eq. (7), and using ≅ denote the equivalence relation between two operators, which means that the two operators with different expressions have the same effect when they act on a wave function based on the quantum hypothesis. So, operator based Eq. (7) is:
ˆ ˆ^ ˆ^ ˆ^ ˆ ≅ −( + ) dp dq dq dp dH dt dt dt dt dt
The right term of Eq. (8) can be simplified as:
[ ]
2 2 2 2
( − + ) = ( ˆˆ^ − ˆ ˆ^ ) = ˆ^ , ˆ
dp dq dq dp d d dt qp pq dt q p dt dt dt dt dt dt dt
where [,] is commutation operator. Bringing Eq. (9) into Eq. (8), and performing integrate operation on both sides of Eq. (8):
[ ] [ ]
2 2
∫ ∫ ∫
d d d dH q p dt q p dt dt dt dt
83 Hua Ma: A Method for Deriving Quantum Dynamic Equations from Classical Mechanics
mechanics has more deep rules needed to be further explored. For instance, in some other quantum systems, the motion law of the quantum system is no longer described by the Schrodinger equation if the commutation relation between position and momentum is in other forms. By analyzing the momentum operator based on the quantization hypothesis, Schrodinger equation can be derived from the general dynamic equation, certifying the validity of the general quantum mechanical formula established here. This work reveals the nature of the variational principle of the dynamics of the quantum system, and thus promotes the perfection of the axiomatic system of quantum mechanics. The establishment of the general quantum dynamic equation from classical mechanics, as well as the application in deriving Schrodinger equation, are expected to open up a new dynamics system, using which to describe the evolution law of quantized and non-quantized dynamical systems in a unified way.
References
[1] https://en.wikipedia.org/wiki/Classical_mechanics.
[2] C Lanczos, The Variational Principles of Mechanics (University of Toronto Press, Toronto, 1964).
[3] Cassel, Kevin W., Variational Methods with Applications in Science and Engineering (Cambridge University Press, 2013).
[4] Landau LD and Lifshitz EM, Mechanics (Pergamon Press, 3rd ed., pp. 2–4, 1976).
[5] Hand L. N., Finch J. D., Analytical Mechanics (Cambridge University Press, 2008). [6] Kragh, Helge, Quantum Generations: A History of Physics in the Twentieth Century (Princeton University Press, p. 58, 2002). [7] Mehra J., Rechenberg H., The historical development of quantum theory (New York: Springer-Verlag, 1982). [8] Schrödinger E., Physical Review. 28 (6): 1049–1070 (1926). [9] Griffiths, David J., Introduction to Quantum Mechanics (Prentice Hall, 2nd ed., 2004). [10] Sakurai J. J., Modern Quantum Mechanics (Massachusetts: Addison-Wesley, p. 68, 1995). [11] J. Köppe, W. Grecksch, W. Paul, “Derivation and application of quantum Hamilton equations of motion”, Annalen Der Physik , 2016, 529 (3): 1600251. [12] VG Zelevinsky, “Microscopic derivation of a quantum hamiltonian for adiabatic collective motion”, Nuclear Physics A , 2016, 337 (1): 40-76. [13] J Zheng-Johansson, “PI Johansson, Inference of Schrödinger Equation from Classical-Mechanics Solution”, Physics , 2006, 17 (3): 240-244. [14] MA Ajaib, “A Fundamental Form of the Schrodinger Equation”, Foundations of Physics , 2015, 45 (12): 1586-1598. [15] T. Hey, P. Walters, The New Quantum Universe (Cambridge University Press, 2009). [16] D. McMahon, Quantum Mechanics Demystified (Mc Graw Hill (USA), 2006).
