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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