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''' Welcome to the Quantum Mechanics A PHY5645 Fall2008'''
[[Image:Quantum.png]]


This is the first semester of a two-semester graduate level sequence. Its goal is to explain the concepts and mathematical methods of Quantum Mechanics, and to prepare a student to solve quantum mechanics problems arising in different physical applications. The emphasis of the courses is equally on conceptual grasp of the subject as well as on problem solving. This sequence of courses builds the foundation for more advanced courses and graduate research in experimental or theoretical physics.  
''' Welcome to the Quantum Mechanics A PHY5645 Fall2008/2009'''
[[Image:SchrodEq.png|thumb|550px|<b>[[Schrödinger Equation]]</b><br/>The most fundamental equation of quantum mechanics; given a Hamiltonian <math>\mathcal{H}</math>, it describes how a state <math>|\Psi\rangle</math> evolves in time.]]
 
This is the first semester of a two-semester graduate level sequence, the second being [[phy5646|PHY5646 Quantum B]]. Its goal is to explain the concepts and mathematical methods of Quantum Mechanics, and to prepare a student to solve quantum mechanics problems arising in different physical applications. The emphasis of the courses is equally on conceptual grasp of the subject as well as on problem solving. This sequence of courses builds the foundation for more advanced courses and graduate research in experimental or theoretical physics.  
   
   
The key component of the course is the collaborative student contribution to the course Wiki-textbook. Each team of students is responsible for BOTH writing the assigned chapter AND editing chapters of others.
The key component of the course is the collaborative student contribution to the course Wiki-textbook. Each team of students is responsible for BOTH writing the assigned chapter AND editing chapters of others.


'''Team assignments:''' [[Phy5645_Fall09_teams|Fall 2009 student teams]]
'''Fall 2009 Midterm is October 15'''
== Outline of the Course ==


----
<b>Chapter 1: [[Physical Basis of Quantum Mechanics]]</b>
'''Outline of the course:'''


* [[Basic Concepts and Theory of Motion]]
* [[UV Catastrophe (Black-Body Radiation)]]
* [[Photoelectric Effect]]
* [[Stability of Matter]]
* [[Double Slit Experiment]]
* [[Stern-Gerlach Experiment]]
* [[The Principle of Complementarity]]
* [[The Correspondence Principle]]
* [[The Philosophy of Quantum Theory]]


== Physical Basis of Quantum Mechanics ==


=== Basic concepts and theory of motion in QM ===
<b>Chapter 2: [[Schrödinger Equation]]</b>
==== Stability of Matter ====
One of the most important problems to inspire the creation of Quantum Mechanics was the model of the Hydrogen Atom. After Thompson discovered the electron, and Rutherford, the nucleus (or Kern, as he called it), the model of the Hydrogen atom was refined to one of the lighter electron of unit negative elementary charge orbiting the larger proton, of unit positive elementary charge. However, it was well known that classical electrodynamics required that charges accelerated by an EM field must radiate, and therefore lose energy. For an electron that moves in circular orbit about the more massive nucleus under the influence of the Coulomb attractive force, here is a simple non-relativistic model of this classical system:
   
   
Where r is the orbital radius, and we neglect the motion of the proton by assuming it is much much more massive than the electron.
* [[Brief Derivation of Schrödinger Equation]]
* [[Relation Between the Wave Function and Probability Density]]
* [[Stationary States]]
* [[Heisenberg Uncertainty Principle]]
* [[Some Consequences of the Uncertainty Principle]]


'''So the question is:  What determines the rate <math>\rho</math> of this radiation?  and how fast is this rate?'''


The electron in the Bohr's model involves factors of:  radius  <math>r_0\,\!</math>, angular velocity <math>\omega\,\!</math>, charge of the particle <math>e\,\!</math>, and the speed of light, <math>c\,\!</math>: <math>\rho=\rho(r_0,\omega,e,c)\,\!</math>
<b>Chapter 3: [[Operators, Eigenfunctions, and Symmetry]]</b>
           
The radius and charge will not enter separately, this is because if the electron is far from the proton, then the result can only depend on the dipole moment, which is .


Therefore the above parameters is now:
* [[Linear Vector Spaces and Operators]]
* [[Commutation Relations and Simultaneous Eigenvalues]]
* [[The Schrödinger Equation in Dirac Notation]]
* [[Transformations of Operators and Symmetry]]
* [[Time Evolution of Expectation Values and Ehrenfest's Theorem]]


'''What are the dimensions of <math>\rho\,\!</math>?'''


Essentially, since light is energy, we are looking for how much energy is passed in a given time.
<b>Chapter 4: [[Motion in One Dimension]]</b>
* [[One-Dimensional Bound States]]
* [[Oscillation Theorem]]
* [[The Dirac Delta Function Potential]]
* [[Scattering States, Transmission and Reflection]]
* [[Motion in a Periodic Potential]]
* [[Summary of One-Dimensional Systems]]


Knowing this much already imposes certain constraints on the possible dimensions. By using dimensional analysis, let's construct something with units of energy.


From potential energy for coulombic electrostatic attractions:
<b>Chapter 5: [[Discrete Eigenvalues and Bound States - The Harmonic Oscillator and the WKB Approximation]]</b>


has to be with , multiply by  , and divide  .
* [[Harmonic Oscillator Spectrum and Eigenstates]]
* [[Analytical Method for Solving the Simple Harmonic Oscillator]]
* [[Coherent States]]
* [[Charged Particles in an Electromagnetic Field]]
* [[WKB Approximation]]


The angular velocity is in frequency, so to get the above equations in energy/time, just multiply it with the angular velocity.


(Here, it is seen that the acceleration of the electron will increase with decreasing orbital radius. The radiation due to the acceleration a is given by the Larmor Formula:
<b>Chapter 6: [[Time Evolution and the Pictures of Quantum Mechanics]]</b>
It was known that the hydrogen atom had a certain radius on the order of .5 angstroms. Given this fact it can easily be seen that the electron will rapidly spiral into the nucleus, in the nanosecond scale. Clearly, the model depicts an unstable atom which would result in an unstable universe.  A better representation of of an electron in an atom is needed.


=== Stationary states and Heisenberg Uncertainty relations ===
* [[The Heisenberg Picture: Equations of Motion for Operators]]
* [[The Interaction Picture]]
* [[The Virial Theorem]]


== Schrodinger equation and motion in one dimension ==
=== The time dependent Schrodinger equation is===


<math>i\hbar \frac{\partial}{\partial t}|\psi\rangle=\mathcal{H}|\psi\rangle</math>
<b>Chapter 7: [[Angular Momentum]]</b>
* [[Commutation Relations]]
* [[Angular Momentum as a Generator of Rotations in 3D]]
* [[Spherical Coordinates]]
* [[Eigenvalue Quantization]]
* [[Orbital Angular Momentum Eigenfunctions]]


where the state <math>|\psi\rangle</math> evolves in time according to the Hamiltonian operator <math>\mathcal{H}</math>.


Motion in 1D
<b>Chapter 8: [[Central Forces]]</b>
* [[General Formalism]]
* [[Free Particle in Spherical Coordinates]]
* [[Spherical Well]]
* [[Isotropic Harmonic Oscillator]]
* [[Hydrogen Atom]]
* [[WKB in Spherical Coordinates]]


== Operators, eigenfunctions, symmetry, and time evolution ==


<b>Chapter 9: [[The Path Integral Formulation of Quantum Mechanics]]</b>


Commutation relations and simulatneous eigenvalues
* [[Feynman Path Integrals]]
Heisenberg and interaction picture: Equations of motion for operators
* [[The Free-Particle Propagator]]
Feynman path integrals
* [[Propagator for the Harmonic Oscillator]]
 
== Discrete eigenvalues and bound states. Harmonic oscillator and WKB approximation ==
Harmonic oscillator spectrum and eigenstates
Coherent states
Feynman path integral evaluation of the propagator
Motion in magnetic field
WKB


== Angular momentum ==
Commutation relations, angular momentum as generator of rotations in 3D, eigenvalue quantization
Orbital angular momentum eigenfunctions


== Central forces  ==
<b>Chapter 10: [[Continuous Eigenvalues and Collision Theory]]</b>
   
   
Free particle in spherical coordinates
* [[Differential Cross Section and the Green's Function Formulation of Scattering]]
Hydrogen atom
* [[Central Potential Scattering and Phase Shifts]]
 
* [[Coulomb Potential Scattering]]
== Continuous eigenvalues and collision theory ==
 
Differential cross-section and the Green's function formulation of scattering
Central potential scattering and phase shifts
Born approximation and examples of cross-section calculations
Coulomb potential scattering
Two particle scattering

Latest revision as of 14:59, 8 April 2014

Quantum.png

Welcome to the Quantum Mechanics A PHY5645 Fall2008/2009

Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian , it describes how a state evolves in time.

This is the first semester of a two-semester graduate level sequence, the second being PHY5646 Quantum B. Its goal is to explain the concepts and mathematical methods of Quantum Mechanics, and to prepare a student to solve quantum mechanics problems arising in different physical applications. The emphasis of the courses is equally on conceptual grasp of the subject as well as on problem solving. This sequence of courses builds the foundation for more advanced courses and graduate research in experimental or theoretical physics.

The key component of the course is the collaborative student contribution to the course Wiki-textbook. Each team of students is responsible for BOTH writing the assigned chapter AND editing chapters of others.

Team assignments: Fall 2009 student teams

Fall 2009 Midterm is October 15

Outline of the Course

Chapter 1: Physical Basis of Quantum Mechanics


Chapter 2: Schrödinger Equation


Chapter 3: Operators, Eigenfunctions, and Symmetry


Chapter 4: Motion in One Dimension


Chapter 5: Discrete Eigenvalues and Bound States - The Harmonic Oscillator and the WKB Approximation


Chapter 6: Time Evolution and the Pictures of Quantum Mechanics


Chapter 7: Angular Momentum


Chapter 8: Central Forces


Chapter 9: The Path Integral Formulation of Quantum Mechanics


Chapter 10: Continuous Eigenvalues and Collision Theory

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