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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:'''


== Outline of the Course ==


== Physical Basis of Quantum Mechanics ==
<b>Chapter 1: [[Physical Basis of Quantum Mechanics]]</b>


=== Basic concepts and theory of motion in QM ===
* [[Basic Concepts and Theory of Motion]]
In Quantum Mechanics, all of the information of the system of interest is contained in a wavefunction , <math>\Psi\,\!</math>. Physical properties of the system such as position, linear and angular momentum, energy, etc. can be represented via linear operators, called observables. These observables are a complete set of commuting Hermitian operators, which means that the common eigenstates (in the case of quantum mechanics, the wavefunctions) of these Hermitian operators form an orthonormal basis. Through these mathematical observables, a set of corresponding physical values can be calculated.
* [[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]]


In order to clarify the paragraph above, consider an analogous example: Suppose that the system is a book, and we characterize this book by taking measurements of the dimensions of this book and its mass (The volume and mass are enough to characterize this system). A ruler is used to measure the dimensions of the book, and this ruler is the observable operator. The length, width, and height (values) from the measurements are the physical values corresponding to that operator (ruler). For measuring the weight of the book, a balance is used as the operator. The measured mass of the book is the physical value for the corresponding observable. The two observable operators (the ruler and the mass scale) have to commute with each other, otherwise the system can not be characterized at the same time, and the two observables can not be measured with infinite precision.


In quantum mechanics, there are some measurements that cannot be done at the same time. For example, suppose we want to measure the position of an electron. What we would do is send a signal (a gamma ray, for example), which would strike the electron and return to our detectors. We have, then, the position of the electron. But as the photon struck the electron, the electron gained additional momentum, so then our simultaneous momentum measurement could not be precise. Therefore both momentum and position cannot be measured at the same time. These measurement are often called "incompatible observables." This is explained in the Heisenberg uncertainty principle and implies, mathematically, that the two operators do not commute.
<b>Chapter 2: [[Schrödinger Equation]]</b>
 
This concept contrasts with classical mechanics, where the two observables that do not commute with each other can still be measured with infinite precision. This is because of the difference in dimension of the object: macroscopic (classical mechanics) and microscopic scale (quantum mechanics). However, the prediction of quantum mechanics must be equivalent to that of the classical mechanics when the energy is very large (classical region). This is known as the Correspondence Principle, formally expressed by Bohr in 1923.
* [[Brief Derivation of Schrödinger Equation]]
* [[Relation Between the Wave Function and Probability Density]]
* [[Stationary States]]
* [[Heisenberg Uncertainty Principle]]
* [[Some Consequences of the Uncertainty Principle]]


We can explain this principle by the following:
In quantum mechanics, particles cannot have arbitrary values of energy, only certain discrete values of energy. There are quantum numbers corresponding to specific values of energy and states of the particle. As the energy gets larger, the spacing between these discrete values becomes relatively small and we can regard the energy levels as a continuum. The region where the energy can be treated as a continuum is what is called the classical region. 


=== UV Catastrophy (Blackbody Radiation) ===
<b>Chapter 3: [[Operators, Eigenfunctions, and Symmetry]]</b>
Imagine a perfect absorber cavity (i.e. it absorbs all radiation at all wavelengths, so that its spectral radiance is only going to depend on the temperature). This emission is called the blackbody radiation. This blackbody radiation experiment shows an important failure of classical mechanics. Lord Rayleigh (John William Strutt) and Sir James Jeans applied classical physics and assumed that the radiation in this perfect absorber could be represented by standing waves with nodes at the ends. The result predicts that the spectral intensity will increase quadratically with the increasing frequency, and will diverge to infinite energy or intensity squared at a UV frequency, or so called "Ultraviolet Catastophy." 


In 1900, Max Planck offered a successful explanation for blackbody radiation.  He also assumed the the radiation was due to oscillations of the electron, but the difference between his assumption and Rayleigh's was that he assumed that the possible energies of an oscillator were not continuous. He proposed that the energy of this oscillator would be proportional to the frequency of a constant, the Planck constant.
* [[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]]




=== Stability of Matter ===
<b>Chapter 4: [[Motion in One Dimension]]</b>
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 <math>r\,!</math> is the orbital radius, and we neglect the motion of the proton by assuming it is much much more massive than the electron.
* [[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]]


'''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 5: [[Discrete Eigenvalues and Bound States - The Harmonic Oscillator and the WKB Approximation]]</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:
* [[Harmonic Oscillator Spectrum and Eigenstates]]
* [[Analytical Method for Solving the Simple Harmonic Oscillator]]
* [[Coherent States]]
* [[Charged Particles in an Electromagnetic Field]]
* [[WKB Approximation]]


'''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 6: [[Time Evolution and the Pictures of Quantum Mechanics]]</b>


Knowing this much already imposes certain constraints on the possible dimensions. By using dimensional analysis, let's construct something with units of energy.
* [[The Heisenberg Picture: Equations of Motion for Operators]]
* [[The Interaction Picture]]
* [[The Virial Theorem]]


From potential energy for coulombic electrostatic attractions:


has to be with , multiply by , and divide  .
<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]]


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 8: [[Central Forces]]</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.
* [[General Formalism]]
 
* [[Free Particle in Spherical Coordinates]]
=== Stationary states and Heisenberg Uncertainty relations ===
* [[Spherical Well]]
 
* [[Isotropic Harmonic Oscillator]]
== Schrodinger equation and motion in one dimension ==
* [[Hydrogen Atom]]
=== The time dependent Schrodinger equation is===
* [[WKB in Spherical Coordinates]]
 
<math>i\hbar \frac{\partial}{\partial t}|\psi\rangle=\mathcal{H}|\psi\rangle</math>
 
where the state <math>|\psi\rangle</math> evolves in time according to the Hamiltonian operator <math>\mathcal{H}</math>.
 
Motion in 1D


== 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 ==
<b>Chapter 10: [[Continuous Eigenvalues and Collision Theory]]</b>
   
   
Commutation relations, angular momentum as generator of rotations in 3D, eigenvalue quantization
* [[Differential Cross Section and the Green's Function Formulation of Scattering]]
Orbital angular momentum eigenfunctions
* [[Central Potential Scattering and Phase Shifts]]
 
* [[Coulomb Potential Scattering]]
== Central forces  ==
Free particle in spherical coordinates
Hydrogen atom
 
== 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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