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

 


 

 Let(1.1)=(1.2)

 


 

 LHS

 


 

 RHS

 


 

 LHS=RHS

 



 

 Define Lagrangian

 


 

 Action

 


 

 Principle of Least Action

 


 

 Also define

 


 

 And

 


 

 Here using integration by parts

 


 

 So(1.3)becomes

 


 

 We get Lagrange's Equations of the first kind

 


 


 

Chapter 2-Hamiltonian Mechanics

 

 From Newtonian Mechanics

 


 

 Partial differential of Lagrangian is

 


 

 Where

 


 

 With spatial translation invariance

 


 

 And time translation invariance

 


 

 Here

 


 

 And

 


 

 So(2.1)becomes

 


 

 Now define Hamiltonian

 


 

 Where

 


 

 We get Hamiltonian Energy

 


 


 

Chapter 3-Partial Differential between Lagrangian and Hamiltonian

 

 From Lagrangian

 


 

 Here

 


 

 With partial differential of Lagrangian

 


 

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

 


 

 Where

 


 

 Minus(3.4)and plus   

 


 

 Come to Hamiltonian

 


 

 Where

 


 

 Then come back to

 


 

 Where

 


 

 Because of

 


 

 And

 


 

 Here

 


 

 And

 


 

 So

 


 


 

Chapter 4-Hamiltonian Energy with Action

 

 Considering about

 


 

 Action

 


 

 And

 


 

 So

 


 

 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

 


 

 With

 


 

 For Hamiltonian Energy

 


 

 So

 


 

 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

 


 

 If 

 


 

 Where

 


 

 We get

 


 

 So

 


 

 Taking into power series about

 


 

 Because  is very small and 

 


 

 (6.1)becomes

 


 

 Let , then  and 

 


 

 Here using Taylor expansion

 


 

 is very small, so(6.2)becomes

 


 

 According to Gaussian integral formula

 


 

 With

 


 

 (6.3)goes to

 


 

 Taking into power series and  for

 


 

 So

 


 

 And 

 


 

 For normalization constant, we can choose

 


 

 So(6.4)becomes

 


 

 Here we get

 


 

 This is Time-dependent Schrödinger Equation.

 

 

 

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