Department of Physics/College of Education

Advanced Quantum MechanicsSyllabus

Chapter One: Quick Revision of Quantum Mechanics Concepts and Some Applications

1-1 Rules Of Quantum Mechanics 1-2 Free Particle in a 1DPB 1-3 Harmonic Oscillator 1-3-1 Classical Theory of L.H.O. 1-3-2 Quantum Theory of L.H.O. 1-4 Central Potentials

1-4-1 Spherical Harmonics 1-4-2 Angular Momentum 1-4-3 Particle in A potential Sphere 1-4-4 Hydrogen Like Atoms 1-5 Rigid Rotator

Chapter Two: Correction Methods 2-1 Time Independent None Degenerate Perturbation Theory

2-1-1 Stark Effect On Simple Harmonic Oscillator2-1-2 Particle in Slanted Box2-1-3 An Harmonic Oscillator2-1-4 Third Order Correction

2-2 Time Independent Degenerate Perturbation Theory

Physics Department, Education College, Al-Mustansiriyah University

of Education 2014-2015 Quantum Mechanics/Ph.D. Course

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Advanced Quantum Mechanics

Quick Revision of Quantum Mechanics Concepts and Some Applications

Rules Of Quantum Mechanics Free Particle in a 1DPB

Harmonic Oscillator Classical Theory of L.H.O. Quantum Theory of L.H.O.

Central Potentials

Spherical Harmonics Angular Momentum

Particle in A potential Sphere ike Atoms

: Correction Methods Time Independent None Degenerate Perturbation Theory

Stark Effect On Simple Harmonic Oscillator Particle in Slanted Box An Harmonic Oscillator Third Order Correction

Time Independent Degenerate Perturbation Theory

versity

Ph.D.- Course Semester- II, Mar. 2015Prof. Dr. Hassan N. Al

Ph.D. Course

Quick Revision of Quantum Mechanics Concepts and

Time Independent None Degenerate Perturbation Theory

, Mar. 2015 Hassan N. Al-Obaidi

Department of Physics/College of Education 2014-2015 Quantum Mechanics/Ph.D. Course

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2-2-1 Stark's Effect on Hydrogen Like Atoms 2-2-2 Zeeman Effect and Double Degenerate States

2-3 Variation Method 2-3-1 Hydrogen Atom

2-4 The WKB Method ( Approximation )

Chapter Three: Matrix Formulation of Quantum Mechanics 3-1 Abstract View of Quantum Mechanics 3-2 The Projection Operator 3-3 Matrix Representation of Operator 3-4 Space Transformation of Operators 3-5 Matrix Representation of Angular Momentum 3-5-1 Review for Some Basics 3-5-2 Creative and Destructive Operator 3-5-3 Matrices of Angular Momentum Operators 3-6 Matrices Representation of wave Functions 3-7 Selection Rules

Chapter Four: Time Dependant Quantum Mechanics 4-1 Formal Theory

4-1-1 Schrdinger Picture 4-1-2 Heisenberg Picture 4-1-3 Interaction Picture

4-2 Time Dependant Perturbation Theory 4-2-1 Step Perturbation 4-2-3 Sinusoidal Perturbation

4-3 Two Level Approximation 4-4 Rabi Solutions 4-5 Multi Level System

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4-6 Adiabatic Perturbation 4-7 Fermi's Golden Rule

Chapter Five: Related Topics 5-1 Motion of Charge Particle In EM field 5-2 Propagators and Feynman Path Integrals 5-3 Potentials and Gauge Transformation 5-4 Interlude 5-5 Electric Dipole Approximation

5-6 Radiation and Matter Interacting 5-7 Einstein A and B Coefficients

References: 1- A Text Book of Quantum Mechanics by Mathews and Venkatesan. 2- Quantum Mechanics by Landau and Lifshits. 3- Theory and Application of Quantum Mechanics by Ammon Yariv. 4- Quantum Mechanics by A.S. Davydov. 5- Quantum Mechanics by Schiff. 6- Quantum Theory by Bohm. 7- Quantum Mechanics- An Introduction by Greiner. 8- Modern Quantum Mechanics by Sakurai.

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Chapter One Quick Revision of Quantum Mechanics Concepts and Some

Applications

For everyone who needs to explore the quantum physics world, its important to know the reasons that requiring to use such a physical-

mathematical tool. In other word, one has to answer questions like; What is the QM?, Why?, etc.

1-1 Rules of Quantum Mechanics Rule-1:Wavefunction

Given the De Broglie wave-particle duality it turns out that one may mathematically express a particle like a wave using a "wave function"

usually denoted by ( (r,t) ). Consequently, in Q.M. the dynamical state of a particle (system) is described by this wave function which replace the classical concept of a trajectory and contain all what can be known about the particle (system). This wave function must be well behaved and hence satisfies three important conditions namely : i- Finite ii- Continuity iii- Singularity

Accordingly, due to the "probabilistic or Boher interpretation of the wave function" one can define the probability density to be the probability per unit length of finding the particle at a point x. In three dimensions it may represent the probability of finding he particle per unit volume:

2*d

z)y,(x,P z)y,(x, z).y,(x, ==

Hence, the probability of finding the particle within the volume V is :

Department of Physics/College of Education 2014-2015 Quantum Mechanics/Ph.D. Course

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dxdydzz)y,(x,P 2zyxv=

dz)y,(x,vv

P 2=

If one extends the above integration over all of the system space , then the finite condition requires the probability becomes certainty (unity). i.e .

1a.s

d2z)y,(x,tP ==

This equation called the normalization condition. However, any function satisfys this condition called normalized. Elsewhere it must be normalizable .i.e:

+

= 1dz)y,(x,2 2 N

N being the normalization constant.

Rule-2: Observables In Q.M. every observable quantity A like position, velocity, energy,etc, is represent by a correspondence mathematical operator . Accordingly, in order to measure the observable A it is necessary to solve the Eigen value equation;

nnn aA =

Where, na are the possible results of the measurement that doing and n are possible states of the system which called Eigen functions. If the

system has state satisfying the Eigen value equation then the

measurement of A definitely yield to the number na .

Notes

1) Depending on position and momentum operators xx = and dxdh-ipx = respectively one often be able to set up a desire correspondence quantum

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mechanical equation for any classical one such as TDSE and TIDSE. (Reveal that) 2) Operator is said to be Hermitical when satisfying the relation:

d ) A( d A -

*

nm

-

m

*

n +

+

=

i- Eigen value corresponds to any Hermitical operator must be real quantities. (Prove) ii- Eigen functions corresponding to different eigen values are always orthogonal. i.e

mn 0d A -

m

*

n =+

(Prove and Explain)

iii- Hence, one can directly define the orthonormality condition as:

mn 1 mn 0

d A -

m

*

n=

==

+

nm

3) The functions form a complete set of functions which in their terms any arbitrary function f(x) can be expand: = af(x)

nnn Completeness or Superposition Principle

4) It can be directly realized that the total probability is conserved. i.e. (Prove) o/dt dPt =

Due to that a system is said to be in a stationary state and has a wave

functions of the form h-iEt/nn e (x)t)(x, = . 5) The flow of probability density at a position x is given by the probability current density:

(Prove) )(2

= rvhv

m

iS

Which satisfy the continuity equation: (Prove) 0S.dtdPt

=+vv

n

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Rule-3: Expectation Value If the system is in state which is not an eigen state of a such

observable, then it is not possible to say with certainty what measured value will be found for A. Therefore, one has to use the average value which called in Q.M. expectation value of A. It is defined mathematically as:

=

>==