阿衡的探秘地圖

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）】

　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）】

　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）】

　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）】

　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）】

　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）】

　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）】

　For 1-dimension space, Feynman guesses

【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.