Quantum Mechanics Lectures Note

7/27/2019 Quantum Mechanics Lectures Note

1/69

Course Title : Quantum Chemistry and Spectroscopy

Lecturer : Dr. Bakhtyar Kamal Aziz

Chemistry Department / College of Science

Contact Detail : Tel. 07701902821 Email:[email protected]

Course overview :

Classical mechanics was invented by Sir Isaac Newton to describe and predict the

motions of objects such as the planets as they move about the sun. Although

classical mechanics was a great success when applied to objects much larger than

atoms, it was a complete failure when applied to atoms and molecules. It was

superseded by quantum mechanics, which has enjoyed great success in explainingand predicting atomic and molecular properties. However, quantum mechanics was

built upon classical mechanics, and someone has said that if classical mechanics

had not been discovered prior to quantum mechanics, it would have had to be

invented in order to construct quantum mechanics.

Sir Isaac Newton, 16421727, was a great British mathematician and physicistwho was also one of the inventors of calculus.

There are some important differences between classical mechanics and quantum

mechanics. Classical mechanics, like thermodynamics, is based on experimentallygrounded laws, while quantum mechanics is based on postulates, which means

unproved assumptions that can be accepted only if their consequences agree with

experiments.

However, thermodynamics, classical mechanics, and quantum mechanics are all

mathematical theories. Galileo once wrote The book of nature is written in thelanguage ofmathematics. We will review some of the mathematics that we use aswe encounter it, and there are a few mathematics topics presented in the

appendixes. There are also several books that cover the application of mathematicsto physical chemistry.
mailto:[email protected]:[email protected]:[email protected]:[email protected]


2/69

Course Objectives:

The main objectives of the course are to understand these points:

1. The molecular nature of matter is studied through quantum mechanics.

2. Classical mechanics is the historical precursor of quantum mechanics.In this theory the state of a system is specified by giving values of coordinates and

velocities.

3. Classical mechanics accurately describes the motions of objects of large mass

that are moving at speeds much slower than the speed of light.

4. Classical mechanics ascribes exact trajectories to particles.

5. The classical wave equation describes the vibrations of strings and light waves.

6. The old quantum theory includes Plancks black-body radiation theory,

Einsteins theory of the photoelectric effect, and Bohrs theory of the hydrogenatom.

7. The old quantum theories contained quantization as hypotheses.

8. The old quantum theories have been superseded by quantum mechanics.

Course Reading list and References:

1- Physical Chemistry, 3rd edition , Robert G. Mortimer, 2008 Elsevier.2- Essentials Physical Chemistry, S. Chand3- Quantum Chemistry, 3rd edition, John P. Lewe and Kirk A. Petreson, 2006.4- Quantum mechanics in Chemistry, Melvin W.Hanna5- Quantum mechanics, Powell & Crasemann.


3/69

A/ Mathematical Preliminaries

1-Coordinate Systems.

2-Determinats.

3-Sammation & Products Notation.

4-Vectors.

5-Complex numbers.

6-Operators.

7-Eigen Value Equations.

B/Classical Mechanics

1-Conservative Systems.

2-Example of Newtonian Mechanics.

3-Lagrangian & Hamiltonian Forms of the equation of Motion.

4-Basic assumptions of Classical Mechanics.

C/Quantum Mechanics

The origins of quantum mechanics.

*The Failurs of classical Mechanics.

1-Black body Radiation.

3-Heat Capacities.

4-Atomic & Molecular Spectra

*WaveParticle Duality.

*WaveParticle Duality.


4/69

1-The Particle character of EMR.

2-The Wave character of particles.

D/The Dynamics of Microscopic Systems.

1-Schrodinger Equation.

2-Born interpretation of Wavefunction.

E/Quantum Mechanical principles.

1-The Information in a Wavefunction.

2-The uncertainty principle.

F/Applications of Quantum Theory.


5/69


6/69

ii /Spherical Polar Coordinates:-

A point P(r , , ) is represented by one distance (r) and two angles ( , )

r :-is the length of the line between the origin and the point (P).

:-The angle (polar angle) is the angle between Z -axis and the line (OP)

:-Is called the Azimuthal angle and it is angle between X-axis and projection ofline (OP) in the XY plane.


7/69

Z

X

O

P

r

Cartesian coordinates are related to the spherical polar coordinates by the

relations:-

X= r sin cos

Y= r sin sin

Z= r cos


8/69

ii i/Cyli nder ical Coordinates :-

The location of a point (P) is given by two distances and one angle.

The two distances are (Z) and the length of the projection of line (OP) in the plane

XY , (p).

The angle is the azimuthal angle between X-axis and the projection of OP in theplane XY.

Z

X

YO

P (p, , z)

p

The relations are given by :-

X= cos

Y= sin

Z=z


9/69


10/69

In problems of quantum mechanics, one will require to evaluate integrals over all

space. To do this, the differential volume element, called ( d ) , must be known foreach kind of coordinate system. These volume elements for the various coordinate

systems, and the limits of integration that include all space, are :-

Cartesian

d = dx dy dz

- x + - y + - z +

Spherical polar

d = r2 Sin dr d d 0 r +

0

0 2

Cylinderical

0 p + 0 2

- z +

Elliptical

- 1 -1 +10 2


11/69

2\ Determinants

There are many physical problems of interest that are most conveniently

described by writing down array of mathematical quantities.

Two types of arrays are used in quantum chemistry: Determinants and matrices.

A determinant is any arrangement of N2 quantities in to a square array with N rows

and N columns. The number N of rows and columns is called the order of the

determinant.

Thus the arraysXE B 8 5 3

3 5 8B X-E 5 3 8

Are determinants order = 2 order = 3

Each element in the array will have two subscripts. The first define the row the

second define the column in which the element appears.

So the quantity aij is the element from i th row and j th column ofThe most convenient way to evaluate a determinant is to make use of the method

of ( signed minors ) or ( cofactors ) .

The minor of an element Aij is the ( N1 ) th order determinant remaining whenthe row i and column j of the original determinant are struck out .

To form the cofactor, the minor is given a sing according to the position of the

element Aij in the original determinant .

The sing is ( - 1 ) i+j

A determinant is evaluated by picking either a row or a column, forming the

product of each element in the row (or column) with its cofactor, and summing the

products, (+ve, -ve, +ve, -ve etc).


12/69

Example :

Evaluate the determinant below by the method of cofactors.

8 5 3 5 8 3 8 3 5

3 5 8 = 8 - 5 + 35 3 8 3 8 5 8 5 3

= 8 ( 40-24)5(24-40) + 3 ( 925)= 160

Properties of determinants:

1- The value of the determinant changes sign when two rows or two columnsare interchanged.

2- If two rows or two columns are identical, the determinant is equal to zero .Example 2 :Evaluate the determinant below by the method of cofactors .

4 1 2 3

1 2 3 42 3 4 1

3 4 1 2

2 3 4 1 3 4 1 2 4 1 2 3=4 3 4 1 - 1 2 4 1 + 2 2 3 1 -- 3 2 3 4

4 1 2 3 1 2 3 4 2 3 4 1

= 4 X ( 1664 )1 X ( 848 ) + 2 X ( 636 )3 X ( 327 )

= - 140


13/69

3/ Summation and Product Notation

In physical chemistry, we often see equations of the of the form :-

y = a1 + a2 + a3ai + .. an.. (1)

Z = a1a2 ai . am ...(2)

To simplify , is used with limits of summation placed below and above the .

A similar is used for multiplication.

Equation (1) can written

And equation (2) as

Example :- Let ai be the series of even integers starting with a1=2. Evaluate

Y= (2+4+6+8) = 20

Z = (2 X 4 X 6 X 8 ) =384


14/69

4/ Vectors

A vector is used to represent a physical quantity which has both magnitude and

direction ( force , magnetic field or acceleration ).

A vector can be represented by a symbol with an arrow above or below it ( or ).It can be represented by boldface type letter also.

Z

Y

X

O

r

r : is a radius vector

r : is the magnitude

op : is the direction .

Any vector can be written in terms of its components ( projections )along the three

axis.

To do this , three mutually perpendicular unit vectors called i , j , k are defined

that point along the x , y , z axes respectively . Then any vector can be written in

term of its components .

r = xi + yj + zk


15/69

The combination of vectors follow some rules :-1-Addition :- addition of vectors can be done either graphically or analytically.

A+B = C

Vector addition is commutative , X+Y = Y+X

And associative , (X + Y) + Z = X + (Y + Z ) = X + Y + Z

2- Subtraction :- vectors are subtracted by adding the negative of the

appropriate vector.

C

+B

A

Addition

D

Sustraction

-B

A

A - B = DA + B = C

3-Magnitude

By elementary trigonometry.

Z

Y

X

X2 + Y2 + Z2 = r2

r = (X2 + Y2 + Z 2 )1/2


16/69

4-Multiplication :-

A.B is called dot or scalar product.

AxB is called cross or vector product.A.B = AB cos ( 180 )

If two vector are perpendicular

cos = cos 90 = 0

So A.B = 0

The two vector are said to be orthogonal.

A.B = AxBx + AyBy + A2B2

The cross product AxB is defined as :-

AxB = n AB sin n :- is a unit vector perpendicular to both A and B .

If A and B are parallel , sin = sin 0 = 0

AxB = 0 .

The right-hand rule must be used in evaluating the cross product.

To use this rule , place the bottom edge of the right palm along A and curl the

fingers toward B . The thumb will then point in the direction of ( n ) .

As a result of the this rule :-

AxB BxA ( Not cumulative ).

In fact AxB = - BxA


17/69

The geometrical interpretation of the cross product AxB is that of a vector

perpendicular to both A and B whose length is equal to the area of the

parallelogram defined by A and B .

AxB is most conveniently written in the form of a determinant.

i j kAxB = Ax Ay Az

Bx By Bz

= i(AyBz - AzBy) j(AxBz AzBx) + k(AxBy AyBx)

B sin

nB

A

Division of vectors is not defined.

Differentiation of vectors is done simply by differentiating its components.


18/69

r = xi + yj + zk

5/ Complex Numbers

A complex number is one which contain (

) ( i ) as it is usually symbolized .

A + iB is a complex number

If C = A + iB

then the complex conjugate of C , called C* is formed by replacing ( i) wherever

it appears by ( -i)

C* = AiBThe magnitude or absolute value of a complex number is defined as :-

(CC*)1/2 = (A2 + B2)1/2The magnitude of a complex number is always real.

Addition and subtraction follow the same rules as for vectors, that is the real and

imaginary parts are added independently.

Thus if Z1 =X1 + iy1

Z2 = X2 + iy2

Then Z1+Z2 = (X1 + X2) + i(y1 + y2 ) .

Eulers, formula


19/69

eix = cos x + i sin x

Examples:-

Quantities in the form A +iB

a) (1 + i)3 b) (x+iy) (4 + iv) c) ei/2

(1+i)3 = (1+i) (1+i)2 = (1+i) (1+2i-1)

=2i+2i2 = 2i-2

6-Operators

An operator is a symbol that tell us to do something to what follows the symbol .

For , is an operator telling us to take the square root of what follows (2).d/dx (x2+5x+1)

d/dx is an operator .

General operator are indicated by a symbol with a caret over it , P or Q .

If P = (

Q =(


20/69

Then PQ =

Operators are not necessarily commutative, because operations are like vector

multiplications so, when dealing with operators, one must be careful of the order of

operation sequence.

PQ QP= since

An operator that will be used frequently in quantum mechanics is :- This operator is related to the kinetic energy.

In Cartesian coordinates it can be seen

The , wher (f) is scalar function , is called the gradient of (f) .Ex:- if f = x2 + y2 + z2 then

f = 2xi + 2yi + 2zkIn Quantum mechanics , only linear operators are used .

An operator is linear if it is true that (F+g) = f + gPPP

Or P Paf = a f


21/69

e.g :

7-Eigenvalue Equations

An equation of type :-

P

P(qi) G(qi) = p G(qi)

(qi) is an operator

G(qi) is a function

p is a constant called eigenvalue equation

both involving variable qi

P and p is eigenvalue.


22/69

In Quantum mechanics , P is generally a differential operator , therefore the

eigenvalue equation is a differential equation .

The principal mathematical problem of quantum mechanics is to find the solution

G and the eigenvalues p to these eigenvalue equations.

Ex:-Let us choose

G(x) = sin ax

If we differentiate twice , it result in the same original function :-

G (x) = d2

/dx G(x) = d/dx ( d/dx sin ax )

= d/dx ( a cos ax )

= -a2 sin ax

If a :- constant

Then the function G(x) = sin ax is an eigenfuction of the operator d/dx2 and the

eigenvalue is (-a2) .

G(x) =A e-ax

G(x) = d/dx (d/dx A e ax )= d/dx (- A a e- ax)

= A.a2 e- ax

a2 = eigenvalue .

show that eax2 is not eigenfunction of d/dx

=d/dx = eax2 = d/dx eax2 = 2axeax2


23/69

=2ax

Not eigenfunction because x is variable.


24/69

Classical Mechanics

Mechanics deal with the motion of bodies and with forces affect these motions.

Before 1900, it was a powerful example of mathematical theories to predict,

correlate and interpret observations on the nature of the physical world.Classical mechanics was first based on Newtons laws of motion. More powerfulformulations were made by Lagrange and Hamilton.

Quantum mechanics needs to have some knowledge about classical mechanics, the

problem is to solve the differential equations resulting from Newtons 2nd law,were:-

: the force acting on (i)th particle in the systemai : its acceleration

The construction of operators

Operators for other observables of interest can be constructed from the operators

for position and momentum. For example, the kinetic energy operator T can be

constructed by noting that kinetic energy is related to linear momentum by

T=p2/2m where m is the mass of the particle. It follows that in one dimension and

in the position representation

In three dimensions the operator in the position representation is


25/69

The operator2, which is read 'del squared' and called the laplacian, is the sum ofthe three second derivatives.

The potential energy of a particle in one dimension, V(x), becomes multiplicationby the function V{x) in the position representation. The same is true of the

potential energy operator in three dimensions. For example, in the positionrepresentation the operator for the Coulomb potential energy of an electron in the

field of a nucleus of atomic number Z is the multiplicative operator

where r is the distance from the nucleus to the electron. It is usual to omit themultiplication sign from multiplicative operators, but it should not be forgotten that

such expressions are multiplications.The operator for the total energy of a system is called the hamiltonian operator and

is denoted H:

H = T + V

The name commemorates W.R. Hamilton's contribution to the formulation ofclassical mechanics. To write the explicit form of this operator we simply

substitute the appropriate expressions for the kinetic and potential energy operators

in the chosen representation. For example, the hamiltonian for a particle of mass m

able to move in one dimension is

where V(x) is the operator for the potential energy. Similarly, the Hamiltonianoperator for an electron of mass me in a hydrogen atom is


26/69

The general prescription for constructing operators in the position representation

should be clear from these examples. In short:1. Write the classical expression for the observable in terms of position coordinates

and the linear momentum.

2. Replace x by multiplication by x, and replace px

by (

/i)

/

x (and likewise for

the other coordinates).

Conservative systems

Is a system in which the sum of the kinetic and potential energy remains constant

with time ( isolated system), and it can not have internal dissipative forces such asfriction.

T + V = E

Conservative system represent mathematically the ( -ve) gradient of some

potential function V , that is :-

= - ----------------------------------(1)

From Newtons 2nd law:-

-------------------------------------(2)

If eq. 1 holds : Substitution of eq. 3 into 2 gives:


27/69

E will represent total energy of a particle ( E ). So

T + V = E

An example on Newtonians mechanics ( Classical Oscillator )A simple realization of the harmonic oscillator in classical mechanics is a

particle which is acted upon by a restoring force proportional to its displacementfrom its equilibrium position. Considering motion in one dimension, this means

F = - k x ..(1)

Such a force might originate from a spring which obeys Hooke's law, as shown inFig. 1. According to Hooke's law, which applies to real springs for sufficiently

small displacements, the restoring force is proportional to the displacement eitherstretching or compression from the equilibrium position. The force constant k is a

measure of the stiffness of the spring.


28/69

The variable x is chosen equal to zero at the equilibrium position, positive for

stretching, negative for compression. The negative sign in (1) reflects the fact thatF is a restoring force, always in the opposite sense to the displacement x.

Applying Newton's second law to the force from Eq (1), we find

where m is the mass of the body attached to the spring, which is itself assumed

massless. This leads to a differential equation of familiar form, although withdifferent variables:

The dot notation (introduced by Newton himself) is used in place of primes when

the independent variable is time. The general solution to (3) is

which represents periodic motion with a sinusoidal time dependence. This isknown as simple harmonic motion and the corresponding system is known as a

harmonic oscillator. The oscillation occurs with a constant angular frequency

This is called the natural frequency of the oscillator. The corresponding circular

frequency in hertz (cycles per second) is

The general relation between force and potential energy in a conservative system

in one dimension is

Thus the potential energy of a harmonic oscillator is given by


29/69

which has the shape of a parabola, as drawn in Fig. 2. A simple computation showsthat the oscillator moves between positive and negative turning points xmax

where the total energy E equals the potential energy 1/2kx2max while the kineticenergy is momentarily zero. In contrast, when the oscillator moves past x = 0, the

kinetic energy reaches its maximum value while the potential energy equals zero.


30/69

Lagrang and Hamilton forms of the equations of motion

For problems of equations of motion in other coordinate system (except Cartesian),

it is difficult to write down the equations of motion.

So, it is more convenient to derive equations of motion in the form which isindependent of a coordinate system. Such equations were derived by two

Mathematician Lagrang and Hamilton.Hamiltonian form is more important because it is used in the transformation of

classical to quantum mechanics, it is necessary to introduce general coordinates,

velocities and momenta.

Suppose a conservative system containing three particles. Inorder to specifycompletely the state of the system at a given time (t), we have to specify the

position and velocities of the three particles. To do this, we have to specify nine

(x1, y1, z1, x3, y3, z3) and nine velocities (

.

In general, for a system with Nparticles, we have to specify 3N coordinates and3N velocities. Such system would have 6N degrees of freedom.

To formulate classical mechanics in a general way, we introduce for a systemcontaining N particles, 3N generalized coordinates (qi), and 3N generalized

velocities Then we derive the Lagrangian and Hamiltonian forms of the equations of motion

in terms of these generalized coordinates and velocities.

The generalized quantities are given a specific form.

The Lagrangian function L( is defined as :-

( T ) is the kinetic energy in terms of generalized velocities & coordinates and

time (t), V is the potential energy in terms of generalized coordinate and the time(t).

The Lagrangian fuction (L) (for conservative system) and potential energy (V) willnot explicitly depend on the time.


31/69

Basic assumption of classical mechanics

It was implied that an experimentalist can precisely measure the positionsand the velocities of all particles in a system at the same time (t) inorder to

describe the state of the system. When the initial state is defined, the laws of mechanics and knowledge of

the forces acting on the system enable the system to be characterized at anylater time.

This means, an experimentalist can measure the position, velocity, energy,

momentum,etc for any particle at any time and compare it with the theoreticalprediction.

The three statements summarize the assumption inherent in this view:-1- There is no limit to the accuracy with which one or more of the dynamical

variables of classical system can be simultaneously measured except the

limit imposed by the precision of the measuring instrument.2- There is no restriction on the number of dynamical variables that can be

accurately measured simultaneously.

3- Since the expressions for velocity are continuously varying functions oftime, the velocity, and hence the kinetic energy, can vary continuously, thatis, there is no restrictions on the values that a dynamical variable can have.


32/69

Quantum Mechanics

Origins of Quantum MechanicsIt was thought that the motion of atoms and subatomic particles could be expressed

using the laws of classical mechanics which was introduced by Isaac Newton.These laws were very successful at explaining the motion of everyday objects and

planets.

Classical Mechanic principles allow:-

1- The prediction of precise trajectory for particles.2- Allow translational, Vibrational and rotational modes of motion to be exited

to any energy simply by controlling the applied energy.

The failures of Classical Mechanics1-Black-body radiation

In retrospectand as will become clearwe can now see that theoretical physicshovered on the edge of formulating a quantum mechanical description of matter asit was developed during the nineteenth century. However, it was a series of

experimental observations that motivated the revolution. Of these observations, themost important historically was the study of black-body radiation, the radiation in

thermal equilibrium with a body that absorbs and emits without favouringparticular frequencies. A pin-hole in an otherwise sealed container is a good

approximation (Fig. 0.1).

Two characteristics of the radiation had been identified by the end of the centuryand summarized in two laws. According to the Stefan-Boltzmann law, theexcitance, M, the power emitted divided by the area of the emitting region, is

proportional to the fourth power of the temperature:

The Stefan-Boltzmann constant, a, is independent of the material from which the

body is composed, and its modern value is 5.67x10-8 Wm-2K-4. So, a region of area

1 cm

2

of a black body at 1000 radiates about 6 W if all fre-frequencies are takeninto account.Not all frequencies (or wavelengths, with = c/v), though, are equally representedin the radiation, and the observed peak moves to shorter wave-lengths as thetemperature is raised. According to Wien's displacement law,

mT = Constant -------------(2)with the constant equal to 2.9 mm K.


33/69

One of the most challenging problems in physics at the end of the nineteenth

century was to explain these two laws. Lord Rayleigh, with minor help from JamesJeans,1 brought his formidable experience of classical physics to bear on the

problem, and formulated the theoretical Rayleigh-Jeans law for the energy density

(, the energy divided by the volume) in the wavelength range d:

where is Boltzmann's constant (k = 1.381 x 10-23JK-1). This formula sum-summarizes the failure of classical physics. It suggests that regardless of the

temperature, there should be an infinite energy density at very short wave-wavelengths. This absurd result was termed the ultraviolet catastrophe by

Ehrenfest.

At this point, Planck made his historic contribution. His suggestion was equivalentto proposing that an oscillation of the electromagnetic field of fre- frequency vcould be excited only in steps of energy of magnitude hv, where h is a new

fundamental constant of nature now known as the Planck constant.According to this quantization of energy, the oscillator can have the energies 0, hv,

2hv,... and no other energy. Classical physics allowed a continuous variation in

energy, so even a very high frequency oscillator could be excited with a very small

energy: that was the root of the ultraviolet catastrophe.Quantum theory is characterized by discreteness in energies (and, as we shall see,

of other properties), and the need for a minimum excitation energy effec-effectively switches off oscillators of very high frequency, and hence eliminates

the ultraviolet catastrophe.When Planck implemented his suggestion, he derived the following Planck

distribution for the energy density of a black-body radiator:

This expression, which is plotted in Fig. 0.2, avoids the ultraviolet catastrophe, and

fits the observed energy distribution extraordinarily well if we take h =6.626 x10 -34

Js. Just as the Rayleigh-Jeans law epitomizes the failure of classical physics, the

Planck distribution epitomizes the inception of quantum theory. It began the new

century as well as a new era, for it was pub-published in 1900.


34/69

Fig. 0.2 The Planck distribution.


35/69

2-Heat capacitiesIn 1819, science had a deceptive simplicity. Dulong and Petit, for example, were

able to propose their law that 'the atoms of all simple bodies have exactly the same

heat capacity'. In modern terms, we would phrase the law in terms of the molarisochoric (constant volume) heat capacity, CV,m, and write Cv,m 3R for a solidelement, where R is the gas constant (R = NAk, with NA the Avogadro constant).

Dulong and Petit's rather primitive observations, though, were done at room

temperature, and it was unfortunate for them and for classical physics when

measurements were extended to lower temperatures.

It was found that all elements had heat capacities lower than predicted by Dulong

and Petit's law, and the values tended toward zero as T

0.

Dulong and Petit's law was easy to explain in terms of classical physics. All it was

necessary to do was to suppose that each atom acted as an oscillator in three

dimensions, and then to use classical physics to calculate the corresponding heat

capacity. That the heat capacities were smaller than predicted was a serious

embarrassment. Einstein recognized the similarity between this problem and black-

body radiation, for if each atomic oscillator required a certain minimum energy

before it would actively oscillate and hence contribute to the heat capacity, then at

low temperatures some would be inactive and the heat capacity would be smaller

than expected. He applied Planck's suggestion for electromagnetic oscillators to the

material, atomic oscillators of the solid,

and deduced the following expression:

where the Einstein temperature,

E, is related to the frequency of atomic

oscillators by E = /k. This function is plotted in Fig. 0.3, and closelyreproduces the experimental curve. In fact, the fit is not particularly good at verylow temperatures, but that can be traced to Einstein's assumption that all the atoms

oscillated with the same frequency. When this restriction was removed by Debye,

he obtained


36/69

where the Debye temperature,

D, is related to the maximum frequency of the

oscillations that can be supported by the solid. This expression gives a very good

fit with observation.

The importance of Einstein's contribution is that it complemented Planck's.

Planck had shown that the energy of radiation is quantized; Einstein showed that

matter is quantized too. Quantization appears to be universal. Neither was able to

justify the form that quantization took (with oscillators excitable in steps of ),but that is a problem we shall solve later in the text.


37/69

3-The photoelectric and Compton effectsVisible or ultraviolet radiation impinging on clean metal surfaces can cause

electrons to be ejected from the metal.Such an effect is not inconsistent with classical theory since electromagnetic waves

are known to carry energy and momentum. But the detailed behavior as a functionof radiation frequency and intensity can not be explained classically.

The energy required to eject an electron from a metal is determined by its work

function . For example, sodium has = 1.82 eV. The electron-volt is aconvenient unit of energy on the atomic scale: 1 eV = 1.602x10-19J.

This corresponds to the energy which an electron picks up when accelerated across

a potential difference of 1 volt. The classical expectation would be that radiationof sufficient intensity should cause ejection of electrons from a metal surface, with

their kinetic energies increasing with the radiation intensity. Moreover, a time

delay would be expected between the absorption of radiation and the ejection ofelectrons. The experimental facts are quite different. It is found that no electronsare ejected, no matter how high the radiation intensity, unless the radiation

frequency exceeds some threshold value o for each metal. For sodiumo=4.39x10

14Hz (corresponding to a wavelength of 683 nm), as shown in Fig. 6.

For frequencies () above the threshold, the ejected electrons acquire a kineticenergy given by


38/69

Evidently, the work function can be identified with ho, equal to 3.65x10-19J

=1.82 eV for sodium. The kinetic energy increases linearly with frequency above

the threshold but is independent of the radiation intensity.Increased intensity does, however, increase the number of photoelectrons.

Einstein's explanation of the photoelectric effect in 1905 appears trivially simple

once stated. He accepted Planck's hypothesis that a quantum of radiation carries an

energy h. Thus, if an electron is bound in a metal with an energy , a quantum of

energy ho = will be sufficient to dislodge it. And any excess energy h( - o)

will appear as kinetic energy of the ejected electron. Einstein believed that the

radiation field actually did consist of quantized particles, which he named photons.

Although Planck himself never believed that quanta were real, Einstein's success

with the photoelectric effect greatly advanced the concept of energy quantization.

Threshold value


39/69

The puzzling features of the effect were that the emission was instantaneous when

the radiation was applied, however low its intensity, but there was no emission,

whatever the intensity of the radiation, unless its frequency exceeded a threshold

value typical of each element. It was also known that the kinetic energy of the

ejected electrons varied linearly with the frequency of the incident radiation.Einstein pointed out that all the observations fell into place if the electromagnetic

field was quantized, and that it consisted of bundles of energy of magnitude h.

Einstein viewed the photoelectric effect as the outcome of a collision between an

incoming projectile, a photon of energy h, and an electron buried in the metal.This picture accounts for the instantaneous character of the effect, because even


40/69

one photon can participate in one collision. It also accounted for the frequency

threshold, because a minimum energy (which is normally denoted and called the'work function' for the metal) must be supplied in a collision before photoejection

can occur; hence, only radiation for which h> can be successful.

When a potential is applied so as to stop the electron flow, the kinetic energy is just

balanced by the potential energy of an electron in an electric field, thus:

e = electric charge

Plotting - should give a straight line of slop h/e and intercept asshown above.

Example: What is the minimum energy that photons must possrss in order to

produce photoelectric effect with platinum metal?? The threshold frequency for Pt

is 1.3x1015sec-1.

Solution:

E = = (6.625 x 10-27 egr sec)(1.3x1015 sec-1)= 8.6x10-12 erg


41/69

4-Atomic Spectra (Line Spectra)Most of what is known about atomic (and molecular) structure and mechanics has

been deduced from spectroscopy. Fig. 7 shows two different types of spectra. A

continuous spectrum can be produced by an incandescent solid or gas at high

pressure. Blackbody radiation, for example, is a continuum.An emission spectrum can be produced by a gas at low pressure excited by heat or

by collisions with electrons. An absorption spectrum results when light from acontinuous source passes through a cooler gas, consisting of a series of dark lines

characteristic of the composition of the gas. Frauenhofer between 1814 and 1823

discovered nearly 600 dark lines in the solar spectrum viewed at high resolution. It

is now understood that these lines are caused by absorption by the outer layers ofthe Sun.

Figure 7. Continuous spectrum and two types of line spectra

Gases heated to incandescence were found by Bunsen, Kirkhoff and others to emit

light with a series of sharp wavelengths. The emitted light analyzed by aspectrometer (or even a simple prism) appears as a multitude of narrow bands of

color. These so called line spectra are characteristic of the atomic composition ofthe gas. The line spectra of several elements are shown in Fig. 8.


42/69

Figure 8. Emission spectra of several elements

It is consistent with classical electromagnetic theory that motions of electrical

charges within atoms can be associated with the absorption and emission ofradiation. What is completely mysterious is how such radiation can occur for

discrete frequencies, rather than as a continuum. The breakthrough that explained

line spectra is credited to Neils Bohr in 1913. Building on the ideas of Planck andEinstein, Bohr postulated that the energy levels of atoms belong to a discrete set of

values En, rather than a continuum as in classical mechanics. When an atom makes

a downward energy transition from a higher energy level Em to a lower energy

level En, it caused the emission of a photon of energy

h = Em - En -------------------------(4)

This is what accounts for the discrete values of frequency in emission spectra ofatoms. Absorption spectra are correspondingly associated with the annihilation of a

photon of the same energy and concomitant excitation of the atom from En to Em.Fig. 9 is a schematic representation of the processes of absorption and emission of

photons by atoms. Absorption and emission processes occur at the same set

frequencies, as is shown by the two line spectra in Fig. 7.


43/69

Figure 9. Origin of line spectra. Absorption of the photon shown in bluecauses atomic transition from E0 to E2. Transition from E2 to E1 causes

emission of the photon shown in red.

Rydberg (1890) found that all the lines of the atomic hydrogen spectrum could be

fitted to a simple empirical formula

where R, known as the Rydberg constant, has the value 109,677 cm-1. This formulawas found to be valid for hydrogen spectral lines in the infrared and ultraviolet

regions, in addition to the four lines in the visible region.No analogously simple formula has been found for any atom other than hydrogen.

Bohr proposed a model for the energy levels of a hydrogen atom which agreedwith Rydberg's formula for radiative transition frequencies.

Inspired by Rutherford's nuclear atom, Bohr suggested a planetary model for the

hydrogen atom in which the electron goes around the proton in one of a set ofallowed circular orbits, as shown in Fig 8. A more fundamental understanding ofthe discrete nature of orbits and energy levels had to await the discoveries of 1925-

26, but Bohr's model provided an invaluable stepping-stone to the development ofquantum mechanics.


44/69

Figure 8. Bohr model of the hydrogen atom showing three lowest-energy

orbits.

Figure 9. A stylized representation of the Bohr model for a multielectronatom.


45/69

5- Structure of atomsRutherford had proposed the nuclear model of the atom. In this model, the

positive charge, and most of the mass, is concentrated in the center of the atom,

called nucleus, electrons were postulated to revolve around it like planets around

the sun.A simple consequence of Newtons law of motion that for an electron subject to the

attractive force from the nucleus to move in a stable orbit, it must be accelerated.And according to the classical thermodynamics, an accelerated charge must

continuously lose energy by radiating.

The paradox in this view of the structure of atoms is that an atom should not be

stable. That is, as the electrons radiate, it will lose energy and spiral inward till itcollide with nucleus. But we see that atoms are stable, and they radiate only when

they are execited.

Bohr try to explain the differences between experimental and theoretical ideas.Bohrs hypothesis was that the lines in an atom spectrum come from a transition ofan electron between two discrete states in an atom, and he made some

assumptions:

1- The Plank-Einstein relation: transition between two energy levels is relatedto the frequency of spectral line.

2- In the discrete states, the magnitude of the angular momentum of theelectron can only have the values

L (angular momentum) = 3- The behavior of electron transition can not be explained classically.

To calculate the allowed orbits by Bohr, we start with the 2nd law of Newton:-

The force is the columbic force between the ( +ve) nucleus and (-ve) electron:-

Z = nuclear charge


46/69

From the assumption of Bohr:

So the value of ( r ) is restricted to certain orbits, that is

So the smallest allowed orbit in the hydrogen atom, Z=1 , n=1

= 0.529 AoThe total energy E= T + V

Since the system is conservative .


47/69

it mean only discrete energy levels appear because (n) is

Integer value.

Bohrs theory was successful for hydrogen atom, but failed to account for thespectrum of any atom having more than one electron.More general equations were needed.

Louis de Broglie when, in 1924, he suggested that any particle, not only photons,

travelling with a linear momentum p should have (in some sense) a wavelengthgiven by the de Broglie relation:

This idea was confirmed by the electron diffraction experiments made by Davisson

&Germer.

Diffraction is a property that is only associated with wave motion.

.[Enistein]


48/69

and

r= distance between e- and nucleusN = principle quantum number

Example: Estimate the wavelength of electrons that have been accelerated from

rest through a potential difference of 1 kV.

p=mv or mc

The Debroglie wavelength is

= 3.88x10-11 m


49/69

The formulation of Quantum Mechanics

It was studied in two different ways:-

1- Schrodinger wavefunction ( wave mechanics).2- Heisenbeg matrix property ( matrix mechanism).

In the more general treatments of quantum mechanics by Dirac & Neuman,showed that Schrodinger and Heisenberg approches are specific cases of a general

theory.

The Postulates of Quantum Mechanics

Postulate I :-

(a) is a wavefunction that contain all informations about the state of adynamical system. It ( ) could be time dependent (if the observableproperties do not change with time) and the system is said to be stationery

state.(b)The quantity * d is proportional to the probability of finding ( q1)

between ( q1 and q1+dq1) , q2 between q2 and q2+dq2.

* is a product of and its complex conjugate and it is called probability

density.

Properties of :1- The fuction should be continous, and its 1st and 2nd derivative should be

continous too.

2- It should be single valued.3- It should have an integrable square.

d = volume element = dx dy dzWhen the above equation is true, then the function is said to be normalized.( 1 ) means 100% probability, in such case the is siad to be normalized.


50/69

Postulate II

For any observable property of a system, there exists a coresponding linearHermitian operator, and the physical properties of the observable can be inffered

from the mathematical properties of its associated operator.

A Hermitian opeator is defined by the :-

Let us constract the quantum mechanical operator for the kinetic energy T.

The classical form for a particle in cartisian coordinate is :-


51/69

The classical form for the total energy is Hamiltonian function, and the

corresponding operator is called the Hamiltonian.For a single particle :

H = T + V ------- ( 2 )

vvv

Postulate III:-

Suppose that

v

operatorcorresponding to an observable and there is a set of

identical systems in state is an eigenfunction ofv

that is

v

.as is a number.

Then if we make a series of experiments of the quantity corresponding to on

different members of the set, we will always get the result ( as), ( like px, py and pz

orbitals) ( different wave functions that give the same eigenvalue).

Substitute eq. 3 in 4

or

This is Schrodinger equation for a single particle in a stationary state.


52/69

Application of Postulates to Simple Systems

The free particle motion

The three basic modes of motion-translation (motion through space), vibration,and rotation-all play an important role in chemistry because they are ways in which

molecules store energy. Gas-phase molecules, for instance, undergo translationalmotion and their kinetic energy is a contribution to the total internal energy of a

sample. Molecules can also store energy as rotational kinetic energy and transitions

between their rotational energy states can be observed spectroscopically. Energy is

also stored as molecular vibration and transitions between vibrational states areresponsible for the appearance of infrared and Raman spectra.

Consider a particle in a box, in which a particle of mass (m) is confined between

two walls at x = 0 and x = L: the potential energy (V) is zero inside the box butrises abruptly to infinity at the walls. This model is an idealization of the potential

energy of a gas-phase molecule that is free to move in a one-dimensionalcontainer. However, it is also the basis of the treatment of the electronic structure

of metals and of a primitive treatment of conjugated molecules. The particle in abox is also used in statistical thermodynamics in assessing the contribution of the

translational motion of molecules to their thermodynamic properties.

Etot = Ekin

Shrodinger eq. for one dimensional system.

V= zero ( zero pot. Energy inside the box)


53/69

The solution for he above equation is :-

side of the box.

x: distance along the x-axisA: amplitude of wave

n=1,2,3 .. eigenvalueThe particle in a box assumes its lowest possible energy when n = 1, namely


54/69

To complete the calculation of the wavefunctions, they should be normalized, thisrequires that :-

When the equation is evaluated, ( A ) must have the value A=

So, the allowed wavefunctions and energies for the particle in a box are:-


55/69

On x-axis

On y-axis

On z-axis

If nx = ny = nz = 1

And ax = ay = az = a

= (

If nx =2 ny = nz = 1So

( 4 + 1 + 1 ) =

= 6 x 27.8 x

If nx = nz = 1 and ny = 2

= 6 x 27.8 x


56/69

Example:- Calculate the E between ground state and E2,2,1.

While

E =

When more than one state has the same energy, the state is said to be degenerate.The number of states with the same energy is the degree of degeneracy.

Example 2:- Calculate the energy in cm-1 of the first two energy levels of a particle

in a box, and the energy difference E2-1 = E2 - E1 for :-a/ an electron in a box 2Ao length.

b/ a ball bearin of mass 1g in a box 10cm length.

E2-1 = E2-1 = (

a/ E=

b/ E =


57/69

Example 3:-An electron is confined to a molecule of length (1nm)(about 5 atoms long). What is

a. Its minimum energy?b. Its minimum excitation energy from that state?

Minimum energy is zero point energy when n=1

For a= 1 nm

The minimum excitation state

E2-1 =

= 1.8 x

A significant feature of the particle-in-a-box quantum states is the occurrence

of nodes. These are points, other than the two end points (which are fixed by the

boundary conditions), at which the wavefunction vanishes.

At a node there is exactly zero probability of finding the particle. The nth quantum

state has, in fact, n 1 nodes. It is generally true that the number of nodes

increases with the energy of a quantum state, which can be rationalized by the

following qualitative argument. As the number of nodes increases, so does the

number and steepness of the `wiggles' in the wavefunction. It's like skiing down a

slalom course. Accordingly, the average curvature, given by the second derivative,

must increase. But the second derivative is proportional to the kinetic energy

operator. Therefore, the more nodes, the higher the energy. This will prove to be an

invaluable guide in more complex quantum systems.


58/69

Free-Electron ModelThe simple quantum-mechanical problem we have just solved can provide aninstructive application to chemistry: the free-electron model (FEM) for delocalized

-electrons. The simplest case is the 1,3-butadiene molecule

The four-electrons are assumed to move freely over the four-carbon frameworkof single bonds. We neglect the zig-zagging of the C-C bonds and assume a one-

dimensional box. We also overlook the reality that

-electrons actually have a

node in the plane of the molecule. Since the electron wavefunction extends beyondthe terminal carbons, we add approximately onehalf bond length at each end. This

conveniently gives a box of length equal to the number of carbon atoms times the

C-C bond length, for butadiene, approximately 4x1.40Ao. Recall that 1 Ao=10-10 m,

Now, in the lowest energy state of butadiene, the 4 delocalized electrons will fill

the two lowest FEM \molecular orbitals." The total -electron density will begiven (as shown in Fig. 4) by

A chemical interpretation of this picture might be that, since the -electron densityis concentrated between carbon atoms 1 and 2, and between 3 and 4, the

predominant structure of butadiene has double bonds between these two pairs of

atoms. Each double bond consists of a - bond, in addition to the underlying -


59/69

bond. However, this is not the complete story, because we must also take account

of the residual -electron density between carbons 2 and 3. In the terminology ofvalence-bond theory, butadiene would be described as a resonance hybrid with thecontributing structures CH2=CH-CH=CH2 (the predominant structure) and

oCH2-

CH=CH-CH2

o (a secondary contribution). The reality of the latter structure issuggested by the ability of butadiene to undergo 1,4-addition reactions.

The free-electron model can also be applied to the electronic spectrum of butadieneand other linear polyenes. The lowest unoccupied molecular orbital (LUMO) in

butadiene corresponds to the n = 3 particle-in-abox state. Neglecting electron-electron interaction, the longest-wavelength (lowest-energy) electronic transition

should occur from n = 2, the highest occupied molecular orbital (HOMO).

The energy difference is given by

Here m represents the mass of an electron (not a butadiene molecule!),

9.1x10-31 Kg, and L is the effective length of the box, 4x1.40x10-10 m.

By the Bohr frequency condition

The wavelength is predicted to be 207 nm. This compares well with the

experimental maximum of the first electronic absorption band, max 210 nm, inthe ultraviolet region.We might therefore be emboldened to apply the model to predict absorption

spectra in higher polyenes CH2=(CH-CH=)n-1CH2. For the molecule with 2n carbon

atoms (n double bonds), the HOMO LUMO transition corresponds to n n + 1,thus


60/69

A useful constant in this computation is the Compton wavelength 2.426x10-12m. For n = 3, hexatriene, the predicted wavelength is 332nm,

while experiment gives max = 250 nm. For n = 4, octatetraene, FEM predicts460nm, while

max= 300 nm. Clearly the model has been pushed beyond it range

of quantitative validity, although the trend of increasing absorption bandwavelength with increasing n is correctly predicted. Incidentally, a compound

should be colored if its absorption includes any part of the visible range 400-700

nm. Retinol (vitamin A), which contains a polyene chain with n = 5, has a pale

yellow color. This is its structure:

Example :What is the probability of locating an electron between x=0 and x=0.2 nm in itslowest energy state in a molecule of 1nm length??

n=1 , z= 0.2nm

p=0.05, or a chance of 1 in 20 of finding the electron in that region.


61/69

Example:-The ground state energy for an electron in a 3Ao one dimentional box is 4 eV, theradius of hydrogen atom is 1Ao , suppose that the hydrogen electron is thought of

as being in a three dimensional cubic box (1Ao one side ). Estimate the energy of

H-electron on this basis.

n=1

= 27.8 eV for one dimensionE = 3x27.8 = 83.4 eV for three dimension

Example :- calculate the wavelength of an electron having a kinetic energy of

4.55x10-25j.


62/69

TunnellingIf the potential energy of a particle does not rise to infinity when it is in the walls

of the container, and E < V, the wave function does not decay abruptly to zero. If

the walls are thin (so that the potential energy falls to zero again after a finite

distance), then the wavefunction oscillates inside the box, varies smoothly insidethe region representing the wall, and oscillates again on the other side of the wall

outside the box (Fig. 9.9).Hence the particle might be found on the outside of a container even though

according to classical mechanics it has insufficient energy to escape. Such leakage

by penetration through a classically forbidden region is called tunnelling.

The Schrodinger equation Can be used to calculate the probability of tunnelling ofa particle of mass m incident on a finite barrier from the left. On the left of the

barrier (for x < 0) the wavefunctions are those of a particle with V = 0, so from eqn9.2 we can write

The Schrodinger equation for the region representing the barrier (for 0 x L),

where the potential energy is the constant V, is

We shall consider particles that have E < V (so, according to classical physics, the

particle has insufficient energy to pass over the barrier), and therefore V - E ispositive. The general solutions of this equation are


63/69

Uncertainty Principle of Heisenberg

This is an important feature of wave mechanics and discuses the relation

between a pair of conjugate properties (those properties that are interdependent) of

substance.According to the uncertainty principle, it is impossible to know simultaneously

both the conjugate properties accurately.For example, the position and momentum of a moving particle are interdependent

and thus conjugate properties also, they can not be determined with absolute

exactness or certainty at the same time.

We have to point that there is a clear difference between the behavior of largeobjects like a stone and small particles like electron.

The uncertainty product is negligible for large objects.For a moving ball of iron (wt=500g), the uncertainty expression.

Which is very small and negligible.

But for electron of m = 9.1 x10-28g

This value is large in comparison with the size of electron and cannot benegligible.


64/69

Example:- Calculate the uncertainty in position of an electron if the uncertainty in

velocity is 5.7x105 m.sec-1.

Solution:

= 1x10-10 m

Example:- The uncertainty in the position and velocity of a particle are 10-10m and

5.27x10-24 m.sec-1 respectively. Calculate the mass of the particle.

Solution:

=0.1 Kg = 100 g


65/69

One dimensional box model is used to correlate the wavelength of maximum

absorption of certain long chain dye molecules.A two dimensional box model gives quite good electron density distribution maps

in aromatic systems.

Three dimensional box problems are used in the derivation of the expression for

the translational partition function in statistical mechanics.

Hermitian operatorsHermitian operators are very important in quantum mechanics because their

eigenvalues are real. As a result, hermitian operators are used to represent

observables since an observation must result in a real number.Examples of hermitian operators include position, momentum, and kinetic and

potential energy.

OPERATORS AND EXPECTATION VALUES

Any hermitian operator, signifies a mathematical operation to be done on a

wavefunction, which will yield a constant, o, if the wavefunction is an

eigenfunction of the operator.

------- (1)

Next the complex conjugate of the wavefunction, is multiplied to both sides of

Equation -1 and integrated over all space.

If the wavefunction is normalized, then the integral is equal to one. This

leads directly to the value of the constant o.

--------- (2)

As mentioned previously, the constant o corresponds to some physically

observable quantity such as position, momentum, kinetic energy, or total energy ofthe system, and it is called the expectation value. Since the expression in Equation-

2 is being integrated over all space, the value obtained for the physically


66/69

observable quantity corresponds to the average value of that quantity. This leads to

the fourth postulate of quantum mechanics.

Example:-

Determine the average position

for the particlein- a Box model problem for

any state (n).

Solution/

This states that the average position of the particle is at the center of the box as it ispredicted by classical mechanics.

Example:- A particle limited to x-axis with between x=0 and x=1, elswhere.1- Find the probability that the particle can be found between x= 0.45 and

x=0.55.

2- Find the expectation value

of the particles position.Solution 1/ probability = dx

Probability = =

= 0.0251 a2

Solution 2/


67/69

Techniques of ApproximationThere are very few problems for which the Schroedinger equation can be solved

for exactly, so methods of approximation are needed in order to tackle these

problems. The two basic methods of approximation are variation andperturbation

theories.In variation theory, an initial educated guess is made as to the shape of the

wavefunction, which is then optimized to approximate the true wavefunction forthe problem. In perturbation theory, the Schroedinger equation is separated into

parts in which the solution is known (from previously solved problems or model

problems) and parts that represent changes or perturbations from the knownproblem.The wavefunctions from the part of the Schroedinger equation in which the

solution is known are used as a starting point and then modified to approximate thetrue wavefunction for the Schroedinger equation of interest.

Both theories are important and powerful problem solving techniques that will beused throughout the rest of the text.

VARIATION THEORYThe first step is to write the Hamiltonian for the problem. Then an educated guessis made at a reasonable wavefunction called formally the trial wavefunction, The

trial wavefunction will have one or more adjustable parameters, that will be usedfor optimization. An energy expectation value ( eq. 2) in terms of the adjustable

parameters, is obtained by using expectation equation (1) below:-

-------- (1)

----------(2)

The term in the denominator of Equation -2 is needed since the trial wavefunction

is most likely not normalized.Variation theory states that the energy expectation value is greater than or equal to

the true ground-state energy, Eo of the system. The equality occurs only when thetrial wavefunction is the true ground-state wavefunction of the system.


68/69

Since is a function of the yet undetermined adjustable parameters the value ofcan be optimized by taking the derivative of with respect to each adjustableparameter and setting it equal to zero. A value for each parameter is then obtained

for the optimized energy of the ground-state. -------------(3)

Variation theory can be proven as follows. Take the trial wavefunction, as alinear combination of the true eigenfunctions of the Hamiltonian .

Since is an eigenfunction of the Hamiltonian of the system, applying to will result in an energy eigenvalue En.

Completing calculation will give .

Variation theory states that the energy calculated from any trialwavefunction will never be less than the true ground-state energy of the system.

This means that the smaller the value of , the closer it is to the true ground-state

energy of the system and the more represents the true ground-statewavefunction.


69/69

TIME INDEPENDENT NON-DEGENERATE

PERTURBATION THEORY

The idea behind perturbation theory is that the system of interest is perturbed orchanged slightly from a system whereby the solution is known. This can occur intwo different ways: a) a new problem that has similarities to another system of

which the solution is known (this happens often in chemistry) or b) the molecule oratom experiences some type of external perturbation such as a magnetic field or

electromagnetic radiation (this is important in the case of spectroscopy). At thispoint the discussion will be limited to time-independent systems with non-

degenerate quantum states. A time-independent perturbation is one in which theperturbation is not a function of time.

The Hamiltonian for the system of interest is divided into parts: the part

representing system with a known solution, and then into a number of additionalparts that correspond to perturbations from the known system to the system of

interest.

The term in the equation above with a superscript zero corresponds to the

Hamiltonian for the system with a known solution (unperturbed system),

(0). The

rest of the terms correspond to additional terms that perturb the known system. The

term (1) is a first-order perturbation, the term (2) is a second-order perturbation,and so on. The idea is that each order of perturbation is a slight change from the

previous order.

Documents

Quantum Mechanics Lectures Note