# The and motion is restricted by properties of

The Mathematics of Our

Universe

“Classical similarities in the

Quantum world”

Student ID – 26628961

Faculty of Social and

Mathematical Sciences – University of Southampton

2018

Abstract

In

this report, we start by defining key aspects of Classical Lagrangian mechanics

including the principle of least action and how one can use this to derive the

Euler-Lagrange equation. Symmetries and

Conservation laws shall also be introduced, deriving relations between

position, momenta and the Lagrangian of our system. Following this, we develop our study of

Classical mechanics further using Legendre transforms on the Euler-Lagrange

equation and our conservation laws to define Hamiltonian mechanics. In our new notation, we use Poisson brackets when

evaluating the rate of change of a classical observable. Next, we cross to Quantum mechanics, giving

some definitions which shall be used in later discussion. We then state and prove the Ehrenfest

theorem, from which we draw our first correspondence between Classical and

Quantum mechanics, most notably between the Poisson bracket and the Commutator. Furthermore, the Ehrenfest theorem applied to

operators of position and momentum shows a further correspondence with

Classical results. Finally, we take an

example of the Simple Harmonic Oscillator, using both Classical and Quantum

methods to solve for this system and comment on the similarities and

differences between the results.

1. Introduction

2. Lagrangian

Mechanics

We

begin by exploring a re-formulation of Newtonian mechanics developed by

Joseph-Louis Lagrange called Lagrangian Mechanics. For a given physical system we require

equations of motion which contain variables as functions of time, in order to

pinpoint the location of an object or particle at any given time. The majority of physical systems are not

free, and motion is restricted by properties of the system. These systems are called constrained systems.

2

Definition 2.1- A

constrained system is a system that is subject to either 3:

Geometric constraints: factors which impose some limit to the

position of an object. 2

Kinematical constraints: factors which

describe how the velocity of a particle behaves. 2

Definition 2.2- A function for which the integral can be

computed is said to be integrable. 19

Definition 2.3- A system is said to

be Holonomic if it has only Geometrical

or Integrable Kinematical Constraints. 2

Since

the Classical Newtonian equations using Cartesian coordinates do not have these

constraints we must find a new coordinate system to work with.

Definitions 2.3- Let S be

a system and be a set of independent variables. If the position of every particle in S can be written as a function of these variables

we say that are a set of generalised coordinates for S.

The time derivatives of these generalised coordinates are called

the generalised velocities of S. 23

Definition 2.4- Let S

be a holonomic system. The number of degrees of freedom of S is the number of generalised

coordinates required to describe the configuration of S. The

number of degrees of freedom of a system is equal to the number of equations of

motion needed to find the motion of the system. 2

Definition 2.5- Let S be a holonomic system with generalised

coordinates. Then the Lagrangian function is,

Here

our Lagrangian function is dependent on the set of generalized coordinates , the generalised velocities , and time . 2

3. Calculus of

Variations

The

method of calculus of variations is used to find the stationary values on a

path, curve, surface, etc. of a given function with fixed end points by using

an 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 of

motion of a mechanical system with holonomic constraints and conservative

external forces, from time to , is the stationary

solution of the action. The correct path

satisfies 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 the

Euler-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 must

satisfy the following equation on

This is called the Euler-Lagrange

equation. 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 twice

differentiable function, with fixed at end points, Leaving

the following,

We want to find the extremum points of the action in

order to find the value of such that is the required minimum.

We begin by assuming that is the function that minimises our action and

that satisfies the required boundary conditions on . Now, we introduce a continuous twice differentiable

function 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 stationary

function for , and for all we require,

Differentiating with respect to parameter ,

By a property of Calculus, we bring

the into the integral giving,

and using the chain rule to evaluate the integrand,

Applying our definition of , it’s clear to see

that and similarly that , hence,

Integrating the term containing the using the integration by parts formula, we

name 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 with

the integral,

Finally, applying Lemma 3.3 we see our

required result,

This is the Euler-Lagrange equation for It can be used to solve our problems involving

the least action principle. The reversal

of 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 generalised

coordinates and Lagrangian . Then the equations of motion of the system

can be written in the following form, 2

The Lagrangian approach to mechanics is to find the extrema minimum

value of an integral in order to derive the equations of motion for that

system.

4. Symmetries and Conservation Laws

Let S be a holonomic system with a set of generalised

coordinates and the Euler-Lagrange equations of motion with n degrees of freedom. The Lagrangian for this system is clearly be

given by,

Definition 4.1- If a generalised coordinate of a mechanical

system S is not contained in the

Lagrangian L such that,

Then we

call an ignorable coordinate. 67

At an

ignorable coordinate the Euler-Lagrange

equation states,

Here, the term , because has no dependence, hence,

Definition 4.2- Consider a

holonomic system S with Lagrangian , such that we can define a ,

which we call the momentum of a free particle. Now say S

is a system described by generalised coordinates . One can define quantities as,

This

is called the generalised momenta for

coordinate . 4

This concept of generalised momenta is

useful, because it can be substituted into equation (4.3) giving, a further

simplified Euler-Lagrangian equation such that . Therefore, this

shows that the generalised momentum for the ignorable coordinate, , is constant.

We can also find the time derivative of this

generalised 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 is

called conserved momentum. 8

The conservation laws in Lagrangian mechanics

are more general than in Newtonian mechanics.

Therefore, the Lagrangian can also be used to prove the conservation

laws that were proved previously in Newtonian mechanics.

5. Hamiltonian Mechanics

We shall now introduce Hamiltonian

mechanics and see how they can be derived from the Lagrangian mechanics that we

have already seen. The Hamiltonian

formulation adds no new physics to what we have already learnt, however it does

provide us with a pathway to the Hamilton-Jacobi equations and branches of

statistical mechanics.

Definition 5.1- An active

variable is the one that is transformed by a transformation between two

functions. The two functions may also

have 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 the

following formula,

where

F is a given function of . With inverse,

The

function G is related to F by the formula,

where

is the standard

vector dot product (. Moreover, the derivatives of F and G with respect to the passive variables are related by,

The

relationship between the two functions F and

G is symmetric and is said to be the Legendre Transform of the other. 2

Let be a Lagrangian

system with degrees of freedom

and generalised coordinates . Then the Euler-Lagrange equations of motion

for are,

where

is the Lagrangian of the system. We now want to convert this set of second order ODE’s into Hamiltonian form in

terms of unknowns , where {are the generalised

momenta of (4.7).

These can be written in vector form,

We

want to eliminate the velocities from the Lagrangian. To do this we use the Legendre

transforms. Leaving us with,

This

leads us to the definition of the Hamiltonian function.

Definition 5.3- The function , which is the

Legendre transform of the Lagrangian function must obey the following equation, where is called the Hamiltonian function of .

We

can now use (5.4) to form a relation between with respect to the passive variable . 2

Using

this relation, we can transform the Lagrange equations into Hamilton’s

equations. Take (4.9) which has

equivalent vector form,

Which

can be transformed into Hamiltonian notation by using (5.9) giving,

Hence

this 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 Classical

observables. We define Poisson Bracket as, 2

Let be a system with degrees of freedom

and generalised coordinates . In the system,

we have an observable looking at its

time derivative we have,

Using the Hamilton’s equations in (5.12) we

can replace and leaving us with,

Now applying the definition of the Poisson

bracket, we can concisely write the first term,

We shall refer to this result when looking at

the Ehrenfest theorem. 18

Comparison between

Lagrangian and Hamiltonian mechanics?

6. Classical Limit and Correspondence Principle 17,

18

Quantum

Mechanics is built upon an analogy with the Hamiltonian Classical

Mechanics. Here we find a clear link

between the coordinates of position and momentum with the Quantum

observables. Statistical interpretation…

The theory of Quantum

Mechanics is built upon a set of postulates.

9 In brief summary, they state that:

– The state of a

particle can be represented by a vector | in the Hilbert

space.

– The independent

variables from classical

interpretations become hermitian operators . In general, observables from classical

mechanics become operators in quantum

mechanics.

– If we study a

particle in state |, a measurement of

observable will give an eigenvalue and a probability of yielding this state .

– The state vector | obeys the Schrodinger

equation:

where is the Quantum Hamiltonian Operator, equal to the sum of kinetic and potential energies.

9

Definition 6.1- The expectation value of a given observable,

represented by operator is the average value of the observable over

the ensemble. 12 Say every particle is in the state then,

Definition 6.2- Let be a Quantum operator representing a physical

observable. We say is a Hermitian

Operator if,

Where is the adjoint

of the operator (definition can be found in 12 pg. 22). An example of a Hermitian operator is the

Hamiltonian operator. 12

Definition 6.3- The commutator

of two Quantum operators is defined as,

If

then we say the operators commute. It is also noted that the order of the operators can

change the result, and that in general, . 14

Theorem 6.4- WORDS + HATS

Proof. First, we apply the definition of the commutator (6.4),

Two

commutation relations which we shall use in later discussion are,

The

proofs for these can be found in 12.

Theorem 6.5- (The Ehrenfest Theorem) WORDS

The

generalized Ehrenfest theorem for the

time derivative of the expectation value of a Quantum operator is,

where

is the Hamiltonian operator. …

Proof. We

start by applying the definition of the expectation value of a general operator

(6.14),

Taking

the derivative into the expectation value gives,

We

can now simply evaluate the time derivatives of in the bras and kets by rearranging the

Schrödinger equation (6.1).

and similarly using the fact is Hermitian.

Using

results (6.14) and (6.15) in (6.13) we have,

We

can now combine the first and third term in (6.16) using the commutation

relation (6.4).

Finally,

we apply the definition of expectation value (6.2) on both terms in (6.17) and we

are left with the Ehrenfest Theorem for a general Quantum operator (6.11).

The Ehrenfest Theorem corresponds structurally to a

result in Classical Mechanics. If we

take a Classical observable which depends on set of generalised

coordinates and momenta , then calculate its rate

of change we see as shown for (5.16) that,

From

this we can see an immediate correspondence between the Classical Poisson

bracket (5.13) and the Quantum commutator (6.4),

what do we learn from this??

Now, we look at some key results from the Ehrenfest theorem

and how they can help us find further correspondence between Classical and

Quantum Mechanics.

Example 6.6- In this example we

shall look at a specific case of the Ehrenfest theorem where we set the position operator. 17 For a Hamiltonian,

We

begin by subbing into (6.11),

It

is clear to see that the second term in this equation disappears as has no time dependence. We now use our Hamiltonian to expand the

commutator.

Here

(Definition 6.3) so we are only left with the

commutator .

Applying

Theorem 6.4 setting , the commutator can

be expand leaving,

Utilizing

the commutator result ,

This

result can be compared with from Classical Mechanics. It is also possible to translate it into an

expression involving the Hamiltonian, only if it is legal to take the

derivative of the Hamiltonian operator with respect to another operator, namely

as shown,

This

clearly 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 Ehrenfest

theorem,

Again

has no time dependence so the second term

disappears. Using the same Hamiltonian (6.20)

Here

commutes with and so we are left with

By

utilizing the result from (6.10) for the commutator. Some trivial simplification leaves,

In

one dimension, we can see that the rate of change of the average momentum is

equal to the average derivative of the potential V. Again, the behavior of

the average Quantum variables corresponds with the Classical expressions for

these observables. In Classical terms

(6.32) reduces to .

Explanations

Again,

one sees resemblance between this Quantum result and the Classical Hamilton’s

equations (5.12),

Evaluation of above results in relation

to Classical Mechanics

The key differences between the Classical and

Quantum versions of (

The

main difference between the quantum and classical forms is that the quantum

version is a relation between mean values, while the classical version is

exact. We can make the correspondence exact provided that it’s legal to take

the averaging operation inside the derivative and apply it to each occurrence

of X and P. That is, is it

legal to say that,

CORRESPONDENCE PRINCIPLE pg. 253-255 Taylor

7. Simple Harmonic

Oscillator

Example 7.1- Lagrangian

Harmonic Oscillator 9

Consider a system containing the undamped Harmonic

Oscillator in 3-D, with displacement coordinate , which is a generalised coordinate. We first form a Lagrangian relation for this

system,

Now,

we consider the case of the 1-D Harmonic Oscillator (i.e. Constraining y and z

to both be zero, ). 4 Leaving us to

find the following equations,

Hence

our equations of motion for the system,

All

that is left is to rearrange this equation and to solve,

Definition 7.2- Scaled Quantum operators for position and

momentum and are defined as,

Hence

lowering and raising operators can be defined in the following way,

They

have commutation relation,

We

shall use the ladder operators or

more notably the raising operator when analyzing the Quantum Harmonic

Oscillator in Section 8. 12

Definition 6.7- Ground State Ket?

Remark 7…- The theory of Quantum Mechanics makes

predictions using probabilities for the result of a measurement of an

observable . The probabilities are found by obtaining the

real eigenvalues of and using the relation stated in the

postulates.

Example 8.2- Quantum

12

We start with our

scaled operators of position and momentum,

For

the Quantum Harmonic Oscillator, we need a Hamiltonian operator based on the

Classical Simple Harmonic Oscillator.

Replacing observables and with operators we have,

We

use raising and lowering operators defined in (6.12) in order to find the wave

function for the Simple Harmonic Oscillator.

We have scaled operators of position and momentum as in (6.11), so we

can write in terms of our ,

Lowering operator can act in our X-space on ground state ket |. Such that,

as

we cannot lower past the ground state.

Apply the definition of the expectation values,

Evaluating the two terms inside the bracket

we see,

So, we have equation (8.8) rewritten as,

Giving us solution the solution for our

ground state wave function,

Now

we have our ground state we can apply raising operator to | and using a similar approach to above,

By

repeating this process, at the end of the story we find a generalised form of

the normalised wave function,

where

are Hermite polynomials.

We can compare the probability density function of the

classical approach with the quantum ground state . It is clear to see that the classical

mechanics has a minimum at , where it has

maximum kinetic energy, whereas for quantum mechanics peaks at for the ground state. However, as increases the quantum wave functions begin to

represent a similar distribution to that of classical mechanics as shown in

figure 8.2. For a very large with macroscopic energies, the classical and

quantum curves are indistinguishable, due to limitations of experimental

resolution.

Chat

about measurements… 275 18

Conclusion

In this report, we have defined

Lagrangian, Hamiltonian and Quantum mechanics

In further study, one could…

10

Bohr’s correspondence

principle 16

Large values of n

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