The Mathematics of OurUniverse”Classical similarities in theQuantum world”Student ID – 26628961Faculty of Social andMathematical Sciences – University of Southampton2018 AbstractInthis report, we start by defining key aspects of Classical Lagrangian mechanicsincluding the principle of least action and how one can use this to derive theEuler-Lagrange equation.

Symmetries andConservation laws shall also be introduced, deriving relations betweenposition, momenta and the Lagrangian of our system. Following this, we develop our study ofClassical mechanics further using Legendre transforms on the Euler-Lagrangeequation and our conservation laws to define Hamiltonian mechanics. In our new notation, we use Poisson brackets whenevaluating the rate of change of a classical observable. Next, we cross to Quantum mechanics, givingsome definitions which shall be used in later discussion. We then state and prove the Ehrenfesttheorem, from which we draw our first correspondence between Classical andQuantum mechanics, most notably between the Poisson bracket and the Commutator.

Furthermore, the Ehrenfest theorem applied tooperators of position and momentum shows a further correspondence withClassical results. Finally, we take anexample of the Simple Harmonic Oscillator, using both Classical and Quantummethods to solve for this system and comment on the similarities anddifferences between the results. 1. Introduction 2.

LagrangianMechanicsWebegin by exploring a re-formulation of Newtonian mechanics developed byJoseph-Louis Lagrange called Lagrangian Mechanics. For a given physical system we requireequations of motion which contain variables as functions of time, in order topinpoint the location of an object or particle at any given time. The majority of physical systems are notfree, and motion is restricted by properties of the system. These systems are called constrained systems.2 Definition 2.

1- Aconstrained system is a system that is subject to either 3:Geometric constraints: factors which impose some limit to theposition of an object. 2Kinematical constraints: factors whichdescribe how the velocity of a particle behaves. 2 Definition 2.2- A function for which the integral can becomputed is said to be integrable. 19 Definition 2.3- A system is said tobe Holonomic if it has only Geometricalor Integrable Kinematical Constraints. 2 Sincethe Classical Newtonian equations using Cartesian coordinates do not have theseconstraints we must find a new coordinate system to work with. Definitions 2.

3- Let S bea system and be a set of independent variables. If the position of every particle in S can be written as a function of these variableswe say that are a set of generalised coordinates for S. The time derivatives of these generalised coordinates are calledthe generalised velocities of S. 23 Definition 2.4- Let Sbe a holonomic system. The number of degrees of freedom of S is the number of generalisedcoordinates required to describe the configuration of S.

Thenumber of degrees of freedom of a system is equal to the number of equations ofmotion needed to find the motion of the system. 2 Definition 2.5- Let S be a holonomic system with generalisedcoordinates. Then the Lagrangian function is, Hereour Lagrangian function is dependent on the set of generalized coordinates , the generalised velocities , and time . 2 3. Calculus ofVariationsThemethod of calculus of variations is used to find the stationary values on apath, curve, surface, etc.

of a given function with fixed end points by usingan integral. Definition 3.1- Let be a real valued function, which we call an action of function for . We can write this in the form of an integral, Definition 3.

2- The correct path ofmotion of a mechanical system with holonomic constraints and conservativeexternal forces, from time to , is the stationarysolution of the action. The correct pathsatisfies Lagrange’s equations of motion, this is called the principle of least action. 4 Lemma 3.3-(Euler-Lagrange Lemma) 5 If is a continuous function on , and for all continuously differentiable functions which satisfy , then, Proof. A proof of theEuler-Lagrange Lemma can be found in 5 pg.

189. Example of F=-dV/dt? Theorem 3.4- Suppose the function minimises the action , then it mustsatisfy the following equation on This is called the Euler-Lagrangeequation. 2 Proof. Following similar derivations as in 5 and 9, we start with an action , where is a given function of and . Let be a twicedifferentiable function, with fixed at end points, Leavingthe following, We want to find the extremum points of the action inorder to find the value of such that is the required minimum.

We begin by assuming that is the function that minimises our action andthat satisfies the required boundary conditions on . Now, we introduce a continuous twice differentiablefunction defined on , which satisfies . Define, where is an arbitrarily small real parameter. We set, We want to find the extremum of at , this means that is a stationaryfunction for , and for all we require, Differentiating with respect to parameter , By a property of Calculus, we bringthe into the integral giving, and using the chain rule to evaluate the integrand, Applying our definition of , it’s clear to seethat and similarly that , hence, Integrating the term containing the using the integration by parts formula, wename and. and our equation (3.12) becomes, Evaluate the first term of (3.

14) using, Substituting into equation (3.14) leaves, By taking , we arrive at and by factoring out a (-1) we are left withthe integral, Finally, applying Lemma 3.3 we see ourrequired result, This is the Euler-Lagrange equation for It can be used to solve our problems involvingthe least action principle. The reversalof the argument also shows that if satisfies (3.18) then is an extremum of . Hence, Definition 3.5- (Lagrange’s Equations of Motion) If S is a holonomic system with generalisedcoordinates and Lagrangian . Then the equations of motion of the systemcan be written in the following form, 2 The Lagrangian approach to mechanics is to find the extrema minimumvalue of an integral in order to derive the equations of motion for thatsystem.

4. Symmetries and Conservation LawsLet S be a holonomic system with a set of generalisedcoordinates and the Euler-Lagrange equations of motion with n degrees of freedom. The Lagrangian for this system is clearly begiven by, Definition 4.1- If a generalised coordinate of a mechanicalsystem S is not contained in theLagrangian L such that, Then wecall an ignorable coordinate. 67 At anignorable coordinate the Euler-Lagrangeequation states, Here, the term , because has no dependence, hence, Definition 4.2- Consider aholonomic system S with Lagrangian , such that we can define a , which we call the momentum of a free particle. Now say Sis a system described by generalised coordinates .

One can define quantities as, Thisis called the generalised momenta forcoordinate . 4 This concept of generalised momenta isuseful, because it can be substituted into equation (4.3) giving, a furthersimplified Euler-Lagrangian equation such that . Therefore, thisshows that the generalised momentum for the ignorable coordinate, , is constant.

We can also find the time derivative of thisgeneralised momenta simply using (4.7) in the Euler-Lagrange equation (3.20). Then using common notation one can see the result, Theorem 4.3- For all ignorable coordinates, , the generalised momenta are not time dependent; this iscalled conserved momentum. 8 The conservation laws in Lagrangian mechanicsare more general than in Newtonian mechanics. Therefore, the Lagrangian can also be used to prove the conservationlaws that were proved previously in Newtonian mechanics. 5.

Hamiltonian MechanicsWe shall now introduce Hamiltonianmechanics and see how they can be derived from the Lagrangian mechanics that wehave already seen. The Hamiltonianformulation adds no new physics to what we have already learnt, however it doesprovide us with a pathway to the Hamilton-Jacobi equations and branches ofstatistical mechanics. Definition 5.1- An activevariable is the one that is transformed by a transformation between twofunctions.

The two functions may alsohave dependence on other variables that are not part of the transformation,these are called passive variables.2 Definition 5.2- We have the variables which are functions of the active variables and passive variables Suppose can be defined by thefollowing formula, whereF is a given function of . With inverse, Thefunction G is related to F by the formula, where is the standardvector dot product (. Moreover, the derivatives of F and G with respect to the passive variables are related by, Therelationship between the two functions F andG is symmetric and is said to be the Legendre Transform of the other. 2 Let be a Lagrangiansystem with degrees of freedomand generalised coordinates . Then the Euler-Lagrange equations of motionfor are, where is the Lagrangian of the system. We now want to convert this set of second order ODE’s into Hamiltonian form interms of unknowns , where {are the generalisedmomenta of (4.

7). These can be written in vector form, Wewant to eliminate the velocities from the Lagrangian. To do this we use the Legendretransforms. Leaving us with, Thisleads us to the definition of the Hamiltonian function. Definition 5.3- The function , which is theLegendre transform of the Lagrangian function must obey the following equation, where is called the Hamiltonian function of . Wecan now use (5.

4) to form a relation between with respect to the passive variable . 2 Usingthis relation, we can transform the Lagrange equations into Hamilton’sequations. Take (4.

9) which hasequivalent vector form, Whichcan be transformed into Hamiltonian notation by using (5.9) giving, Hencethis leaves us with the two transformed Lagrange equations (5.7) and (5.11),these are known as Hamilton’s equations,which have expanded form, Definition 5.4- Let be two Classicalobservables.

We define Poisson Bracket as, 2 Let be a system with degrees of freedomand generalised coordinates . In the system,we have an observable looking at itstime derivative we have, Using the Hamilton’s equations in (5.12) wecan replace and leaving us with, Now applying the definition of the Poissonbracket, we can concisely write the first term, We shall refer to this result when looking atthe Ehrenfest theorem. 18 Comparison betweenLagrangian and Hamiltonian mechanics? 6.

Classical Limit and Correspondence Principle 17,18QuantumMechanics is built upon an analogy with the Hamiltonian ClassicalMechanics. Here we find a clear linkbetween the coordinates of position and momentum with the Quantumobservables. Statistical interpretation… The theory of QuantumMechanics is built upon a set of postulates.9 In brief summary, they state that: – The state of aparticle can be represented by a vector | in the Hilbertspace. – The independentvariables from classicalinterpretations become hermitian operators . In general, observables from classicalmechanics become operators in quantummechanics.

– If we study aparticle in state |, a measurement ofobservable will give an eigenvalue and a probability of yielding this state .- The state vector | obeys the Schrodingerequation: where is the Quantum Hamiltonian Operator, equal to the sum of kinetic and potential energies.9Definition 6.1- The expectation value of a given observable,represented by operator is the average value of the observable overthe ensemble. 12 Say every particle is in the state then, Definition 6.

2- Let be a Quantum operator representing a physicalobservable. We say is a HermitianOperator if, Where is the adjointof the operator (definition can be found in 12 pg. 22). An example of a Hermitian operator is theHamiltonian operator. 12Definition 6.

3- The commutatorof two Quantum operators is defined as, If then we say the operators commute. It is also noted that the order of the operators canchange the result, and that in general, . 14 Theorem 6.4- WORDS + HATS Proof.

First, we apply the definition of the commutator (6.4), Twocommutation relations which we shall use in later discussion are, Theproofs for these can be found in 12. Theorem 6.5- (The Ehrenfest Theorem) WORDSThegeneralized Ehrenfest theorem for thetime derivative of the expectation value of a Quantum operator is, where is the Hamiltonian operator. … Proof. Westart by applying the definition of the expectation value of a general operator(6.

14), Takingthe derivative into the expectation value gives, Wecan now simply evaluate the time derivatives of in the bras and kets by rearranging theSchrödinger equation (6.1). and similarly using the fact is Hermitian. Usingresults (6.14) and (6.15) in (6.13) we have, Wecan now combine the first and third term in (6.

16) using the commutationrelation (6.4). Finally,we apply the definition of expectation value (6.2) on both terms in (6.17) and weare left with the Ehrenfest Theorem for a general Quantum operator (6.11). The Ehrenfest Theorem corresponds structurally to aresult in Classical Mechanics. If wetake a Classical observable which depends on set of generalisedcoordinates and momenta , then calculate its rateof change we see as shown for (5.

16) that, Fromthis we can see an immediate correspondence between the Classical Poissonbracket (5.13) and the Quantum commutator (6.4), what do we learn from this?? Now, we look at some key results from the Ehrenfest theoremand how they can help us find further correspondence between Classical andQuantum Mechanics. Example 6.6- In this example weshall look at a specific case of the Ehrenfest theorem where we set the position operator.

17 For a Hamiltonian, Webegin by subbing into (6.11), Itis clear to see that the second term in this equation disappears as has no time dependence. We now use our Hamiltonian to expand thecommutator. Here (Definition 6.3) so we are only left with thecommutator . ApplyingTheorem 6.4 setting , the commutator canbe expand leaving, Utilizingthe commutator result , Thisresult can be compared with from Classical Mechanics. It is also possible to translate it into anexpression involving the Hamiltonian, only if it is legal to take thederivative of the Hamiltonian operator with respect to another operator, namelyas shown, Thisclearly shows a correspondence with one of Hamilton’s equations seen in (5.

12), Evaluation Example 6.7- We now follow a similar route as in 17 using the operator for momentum in the Ehrenfesttheorem, Again has no time dependence so the second termdisappears. Using the same Hamiltonian (6.20) Here commutes with and so we are left with Byutilizing the result from (6.10) for the commutator.

Some trivial simplification leaves, Inone dimension, we can see that the rate of change of the average momentum isequal to the average derivative of the potential V. Again, the behavior ofthe average Quantum variables corresponds with the Classical expressions forthese observables. In Classical terms(6.32) reduces to .Explanations Again,one sees resemblance between this Quantum result and the Classical Hamilton’sequations (5.

12), Evaluation of above results in relationto Classical MechanicsThe key differences between the Classical andQuantum versions of (Themain difference between the quantum and classical forms is that the quantumversion is a relation between mean values, while the classical version isexact. We can make the correspondence exact provided that it’s legal to takethe averaging operation inside the derivative and apply it to each occurrenceof X and P. That is, is itlegal to say that,CORRESPONDENCE PRINCIPLE pg. 253-255 Taylor 7.

Simple HarmonicOscillatorExample 7.1- LagrangianHarmonic Oscillator 9Consider a system containing the undamped HarmonicOscillator in 3-D, with displacement coordinate , which is a generalised coordinate. We first form a Lagrangian relation for thissystem, Now,we consider the case of the 1-D Harmonic Oscillator (i.e.

Constraining y and zto both be zero, ). 4 Leaving us tofind the following equations, Henceour equations of motion for the system, Allthat is left is to rearrange this equation and to solve, Definition 7.2- Scaled Quantum operators for position andmomentum and are defined as, Hencelowering and raising operators can be defined in the following way, Theyhave commutation relation, Weshall use the ladder operators ormore notably the raising operator when analyzing the Quantum HarmonicOscillator in Section 8. 12 Definition 6.

7- Ground State Ket? Remark 7…- The theory of Quantum Mechanics makespredictions using probabilities for the result of a measurement of anobservable . The probabilities are found by obtaining thereal eigenvalues of and using the relation stated in thepostulates. Example 8.

2- Quantum12 We start with ourscaled operators of position and momentum, Forthe Quantum Harmonic Oscillator, we need a Hamiltonian operator based on theClassical Simple Harmonic Oscillator. Replacing observables and with operators we have, Weuse raising and lowering operators defined in (6.12) in order to find the wavefunction for the Simple Harmonic Oscillator.

We have scaled operators of position and momentum as in (6.11), so wecan write in terms of our , Lowering operator can act in our X-space on ground state ket |. Such that, aswe cannot lower past the ground state. Apply the definition of the expectation values, Evaluating the two terms inside the bracketwe see, So, we have equation (8.

8) rewritten as, Giving us solution the solution for ourground state wave function, Nowwe have our ground state we can apply raising operator to | and using a similar approach to above, Byrepeating this process, at the end of the story we find a generalised form ofthe normalised wave function, where are Hermite polynomials. We can compare the probability density function of theclassical approach with the quantum ground state . It is clear to see that the classicalmechanics has a minimum at , where it hasmaximum kinetic energy, whereas for quantum mechanics peaks at for the ground state.

However, as increases the quantum wave functions begin torepresent a similar distribution to that of classical mechanics as shown infigure 8.2. For a very large with macroscopic energies, the classical andquantum curves are indistinguishable, due to limitations of experimentalresolution. Chatabout measurements.

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