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Advanced Quantum Mechanics

Syllabus

Chapter One:Quick Revision of Quantum Mechanics Concepts and

Some Applications1-1 Rules Of Quantum Mechanics1-2 Free Particle in a 1DPB1-3 Harmonic Oscillator

1-3-1 Classical Theory of L.H.O.1-3-2 Quantum Theory of L.H.O.

1-4 Central Potentials1-4-1 Spherical Harmonics1-4-2 Angular Momentum1-4-3 Particle in A potential Sphere1-4-4 Hydrogen Like Atoms

1-5 Rigid Rotator

1-6 Selection Rules

1-7 Matrix representation

1-7-1 Angular Momentum

1-7-2 Spin Angular Momentum

Chapter Two: Correction Methods

2-1 Time Independent None Degenerate Perturbation Theory

2-1-1 Stark Effect On Simple Harmonic Oscillator

2-1-2 Particle in Slanted Box

2-1-3 An Harmonic Oscillator

2-1-4 Third Order Correction

Physics Dep.,

Education Col.,

Al-Mustansyriyah Uni.

Ph.D.- Course

Semester- I, Oct. 2010

Prof. Dr. Hassan N. Al-Obaidi

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2-2 Time Independent Degenerate Perturbation Theory

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 Atom2-4 The WKB Method ( Approximation )

Chapter Three : Time Dependant Quantum Mechanics

3-1 Formal Theory

3-1-1 Schrdinger Picture

3-1-2 Heisenberg Picture

3-1-3 Interaction Picture3-2 Time Dependant Perturbation Theory

3-2-1 Step Perturbation

3-2-3 Sinusoidal Perturbation

3-3 Two Level Approximation

3-4 Rabi Solutions

3-5 Multi Level System

3-6 Adiabatic Perturbation

3-7 Fermi's Golden Rule

Chapter Four: Related Topics

4-1 Motion of Charge Particle In EM field

4-2 Propagators and Feynman Path Integrals

4-3 Potentials and Gauge Transformation

4-4 Interlude

4-5 Electric Dipole Approximation4-6 Radiation and Matter Interacting

4-7 Einstein A and B Coefficients

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References:

1- A Text Book of Quantum Mechanics by Mathews andVenkatesan.

2- Quantum Mechanics by Landau and Lifshits.3- Theory and Application of Quantum Mechanics by Ammon

Yariv.

4- Quantum Mechanics: Fundamental Principle and Applicationsby Dawson.

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 every one who need to explore quantum physics world, It

important to know the reasons that requiring to use such a physical-

mathematical tool. In other word, one have to answer questions like;

What is the QM?, Why?, etc.

1-1 Rules of Quantum Mechanics

Rule-1:Wavefunction

Given the DeBroglie 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 hencesatisfies three important conditions namely :

i- Finite

ii- Continuity

iii- Singularity

Accordingly, due to the "probabilistic or Boher interpretation of

wave function" one can define the probability density to be the

probability per unit length of finding the particle at a point x. In threedimension it may represent the probability of finding he particle per unit

volume:

2*d

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

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

If one extending the above integration over all of the system space , then

the finite condition requires the probability becomes certainty (unity). i.e .

This equation called thenormalization condition. However, any function

satisfy this condition called normalized. Elsewhere it must be

normalizable .i.e:

N being thenormalizationconstant.

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 solvethe Eigen value equation;

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

the system has state satisfying the Eigen value equation then themeasurement of A definitely yield to the number .

Notes1) Depending on position and momentum operators and.. respectively one often be able to set up a desirecorrespondence equation such as TDSE and TIDSE. (Show that)

dxdydzz)y,(x,P 2zyxv

=

dz)y,(x,vvP2

=

1a.s

d2z)y,(x,tP ==

+

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

nnn aA =

nan

na

xx =dxdh-ipx =

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2) Operator is said to be Hermitical when satisfying the relation:

i- Eigen value corresponds to any Hermitical operator must be realquantities. (Prove)ii- Eigen functions corresponding to different eigen values arealways orthogonal. i.e

(Prove and explain these facts ).

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

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

Completeness or Superposition Principle

4) It can be directly realized that the total probability is conserved. i.e.

Due to that a system is said to be in a stationary state and has a wavefunctions of the form .

5) The flow of probability density at a position x is given by theprobability current density:

Which satisfy the continuity equation:

d)A(dA-

*nm

-m

*n

+

+

=

mn0dA-

m*

n =+

mn1

mn0dA

-m

*n

=

==

+

nm

= af(x)n

nn

(Prove)o/dtdPt =

h-iEt/nn e(x)t)(x, =

(Prove))(2

=

rvhv

m

iS

(Prove)0S.dt

dPt =+vv

n

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Rule-3: Expectation Value

If the system is in state which is not an eigen state of a suchobservable, 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 definedmathematically as:

The probability that the measurement will yield the value is

defined by:

(Discuss)

The integration in last two equations calledoverlap integral. ( Explain)

Notes:

1) Expectation value of an observable A is the sum of the possible eigenvalues times the corresponding partial probability in that state .i.e.

(Prove)

2) One can easilyprove that:

Rule-4:Variance

When the function are a set of complete eigen functions of a twodifferent operators and . i.e.

and

=

>==