Physics derive: OB Tsai Write & Edit: BMC


 The study of the motion is an ancient one, making Classical Mechanics one of the oldest and largest subjects in physics. It is also widely known as Newtonian Mechanics. The initial development of Classical Mechanics is often referred to as Newtonian Mechanics, with the mathematical methods invented by Isaac Newton. The most famous equation is known as Newton's Second Law of Motion.


Isaac Newton(1643-1727)

【Isaac Newton(1643-1727)】


 After Newton, more abstract and general methods were developed, leading to reformulations of Classical Mechanics known as Lagrangian Mechanics and Hamiltonian Mechanics. Joseph Lagrange and William Hamilton extend substantially beyond Newton's work, particularly through their use of analytical mechanics.


 Using mathematical analysis and a function called the Lagrangian, choosing the generalized coordinates, with Principle of Least Action, Lagrangian Mechanics is a reformulation of Classical Mechanics. It is widely used to solve mechanical problems when Newton's formulation of Classical Mechanics is not convenient.


Joseph Lagrange(1736-1813)

【Joseph Lagrange(1736-1813)】


 Hamiltonian Mechanics is also a theory developed from Classical Mechanics. Based on Lagrangian Mechanics, Hamilton uses a different mathematical formalism, providing a more abstract understanding of the theory. It was an important reformulation of Classical Mechanics, which later contributed to the formulation of Statistical Mechanics and Quantum Mechanics.


William Hamilton(1805-1865)

【William Hamilton(1805-1865)】


 Classical Mechanics and Quantum Mechanics are the two major sub-fields of mechanics. With Wave-particle Duality, Quantum Mechanics use a mathematical way, the wave function, to provide information about the probability amplitude of position, momentum, and other physical properties of a particle. Erwin Schrödinger developed the fundament of Quantum Mechanics and formulated his Schrödinger Equation that describes how the quantum state of a quantum system changes with time. In the Copenhagen Interpretation, the wave function is the most complete description that can be given of Quantum Mechanics' system. Solutions to Schrödinger's Equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems, possibly even the whole universe.


Erwin Schrödinger(1887-1961)

【Erwin Schrödinger(1887-1961)】


 With Pierre de Fermat's principle of light, Leonhard Euler's mathematical method, Christiaan Huygens' theory of wave source, and Richard Feynman's guess, now we will try to show how finally Schrödinger Equation can be derived from Newtonian Mechanics, during Lagrangian and Hamiltonian, after Wave-particle Duality of Quantum Physics.



Chapter 1-Lagrangian Mechanics


 From Newton's Second Law of Motion



 And Conservative Force is















 Define Lagrangian






 Principle of Least Action



 Also define






 Here using integration by parts






 We get Lagrange's Equations of the first kind




Chapter 2-Hamiltonian Mechanics


 From Newtonian Mechanics



 Partial differential of Lagrangian is






 With spatial translation invariance



 And time translation invariance












 Now define Hamiltonian






 We get Hamiltonian Energy




Chapter 3-Partial Differential between Lagrangian and Hamiltonian


 From Lagrangian






 With partial differential of Lagrangian



 Taking(3.1)、(3.2)、(3.3)into following formula






 Minus(3.4)and plus   



 Come to Hamiltonian






 Then come back to






 Because of
















Chapter 4-Hamiltonian Energy with Action


 Considering about












 Hamiltonian Energy is



 Taking into(4.1)、(4.2)we get




Chapter 5-Derivation of Time-independent Schrödinger Equation


 For light wave function



 Fermat Principle that the path taken between two points by a ray of light is the path that can be traversed in the least time. The shortest path can be corresponded to the shortest path of phase .


Pierre de Fermat(1601-1665)

【Pierre de Fermat(1601-1665)】


 Principle of Least Action of Newtonian Mechanics is



 For light wave of shortest path of phase, we can guess



 Where  is Planck Constant over 2π. Because  is non-dimensional, we divide S by  for dimensionless.



 So light wave function can become



 For 1-dimension space, returning to






 For Hamiltonian Energy






 To find the solutions of(5.1)which are the functions for which a given functional is stationary, let



 Taking into Euler-Lagrange Equation



Leonhard Euler(1707-1783)

【Leonhard Euler(1707-1783)】


 Here we get



 This is Time-independent Schrödinger Equation.



Chapter 6-Derivation of Time-dependent Schrödinger Equation


 Huygens Principle proposed that every point which a light wave disturbance reaches becomes a source of a spherical wave. The sum of these secondary waves determines the form of the wave at any subsequent time.


Christiaan Huygens(1629-1695)

【Christiaan Huygens(1629-1695)】


 For 1-dimension space, Feynman guesses



Richard Feynman(1918-1988)

【Richard Feynman(1918-1988)】


 Using integral by summation









 We get






 Taking into power series about



 Because  is very small and 






 Let , then  and 



 Here using Taylor expansion



 is very small, so(6.2)becomes



 According to Gaussian integral formula






 (6.3)goes to



 Taking into power series and  for









 For normalization constant, we can choose






 Here we get



 This is Time-dependent Schrödinger Equation.






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