Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004)

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LECTURE 1

• Source D. Griffiths, Introduction to Quantum
  Mechanics (Prentice Hall, 2004)
• R. Scherrer, Quantum Mechanics An Accessible
  Introduction (Pearson Intl Ed., 2006)
• R. Eisberg R. Resnick, Quantum Physics of
  Atoms, Molecules, Solids, Nuclei and Particles
  (Wiley, 1974)

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Topics Today

• Quantum Mechanics and Classical Physics.
• Waves Plane Waves, Wave Packets.
• Wavefunction Properties, Normalization,
  Expectation Values, Scroedinger Equation.
• Operators
• Eigenvalues and Eigenfunctions.
• Uncertainty Principle.

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Classical Physics

• Classical Mechanics
• Electricity and Magnetism
• Thermodynamics

Modern Physics Areas of Physics emerging from
Quantum Mechanics

• Atomic Physics
• Nuclear Physics
• Particle Physics
• Condensed Matter Physics

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THE WAVE FUNCTION
Wavefunction of a physical system contains the
measurable information about the system.
Quantum Mechanics Approach Particle Wave
Function Y(x,t) Y (x,t) is obtained by solving
Schroedinger Equation
V(x) Potential Energy
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Probability in Quantum Mechanics Wavefunction
Y The probability amplitude for finding a
particle at a given point in space at a given
time. Actual probability YY                  
                     The sum of the
probabilities for all of space must be equal to
Normalization of Wave Functions To obtain
the physically applicable probability amplitudes.
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Operators

• To obtain specific values for physical
  parameters, the quantum mechanical operator
  associated with that parameter is operated on the
  wavefunction.

When an operator operates on the wave function,
a number is obtained. This number corresponds to
a possible result of a physical measurement of
that quantity. Operators in Quantum Mechanics are
linear operators. If P is a linear operator,
where a and b are constants.
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Operators in Quantum Mechanics                   
                                                  
                                         
Observables
Operators
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Expectation Value
For a physical system described by a
wavefunction Y , the expectation value of any
physical observable q can be expressed in terms
of the corresponding operator Q as follows
                                                  
     The wavefunction must be normalized and
that the integration is over all of space. The
function can be represented as a linear
combination of eigenfunctions of Q, and the
results of the operation gives the physical
values times a probability coefficient. Expectati
on value gives a weighted average of the possible
observable values.
(Standard Deviation)2
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Expectation Values
Expectation Value To relate a quantum mechanical
calculation to something you can observe in the
laboratory. For the position x, the expectation
value is defined as                             
                               Expectation
Value of x The average value of position for a
large number of particles which are described by
the same wavefunction. Example The expectation
value of the radius of the electron in the ground
state of the hydrogen atom is the average value
you expect to obtain from making the measurement
for a large number of hydrogen atoms.
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Expectation Value of Momentum
The expectation value of momentum involves the
representation of momentum as a quantum
mechanical operator.                          
                                          Where
                                 is the
operator for the x component of momentum.
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Hamiltonian
The Hamiltonian contains the operations
associated with the kinetic and potential
energies and for a particle in one dimension can
be written                                       
                                                
                    The Hamiltonian operator
also generates the time evolution of the
wavefunction in the form                        
     The full role of the Hamiltonian is shown
in the time dependent Shrodinger equation where
both its spatial and time operations manifest
themselves.
Y

• The operator associated with energy is the
  Hamiltonian, and the operation on the
  wavefunction is the Schrodinger equation.

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• Eigenvalues and Eigenfunctions
• Solutions exist for the time independent
  Schrodinger equation only for certain values of
  energy, and these values are called
  "eigenvalues" of energy.
• Corresponding to each eigenvalue is an
  "eigenfunction".
• The solution to the Schrodinger equation for a
  given energy    involves also finding the
  specific function    which describes that
  energy state.
• The solution of the time independent Schrodinger
  equation takes the form
•                                    

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The eigenvalue concept is not limited to energy.
When applied to a general operator Q, it can take
the form                                  
                                   if the
function    is an eigenfunction for that
operator. The eigenvalues qi may be discrete, and
in such cases we can say that the physical
variable is "quantized" and that the index i
plays the role of a "quantum number" which
characterizes that state.
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The Wavefunction

• Each particle is represented by a wavefunction,
• Y(position, time).
• YY The probability of finding a particle at
  that time.
• Wavefunction is the solution of the Schroedinger
  Equation.
• Schroedinger Equation plays the role of Newtons
  Law in classical mechanics.
• It predicts the future behaviour of a dynamic
  system.
• It predicts analytically and precisely the
  probability of events or outcome.
• The detailed outcome depends on chance.
• Large number of events The Schroedinger
  Equation predicts the distribution of results.

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The Uncertainty Principle

• The position and momentum of a particle cannot be
  simultaneously measured with arbitrarily high
  precision.
• There is a minimum for the product of the
  uncertainties of these two measurements.

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UNCERTAINTY PRINCIPLE
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WAVEFUNCTION PROPERTIES
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Problem 1
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Problem 2
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Problem 3