Mind Network - Samuel Solomon

Dirac Notation

          Let us now move onto the most common notation for using quantum operators called Dirac Notation (though some may see the resemblance to an inner product). Let me quickly highlight the word notation. Quantum math can be very very (very) messy. So let us just start to introduce this nice notation here:
Dirac Notation
In order to make sure everyone is following, let us review:
       9: The left hand side is called Dirac notation and is always equivalent to the right hand side
       10: The operator is normally placed in the middle of the bracket, and acts on the function to the right.
       11: If the operator acts on the function to the left, it is called the "adjoint" operator (many just say "dagger")

          Expressions 10 and 11 are equivalent expressions; hence, many times the adjoint operator and the operator have different functional representation (in order to make sure that the ending expression is the same). Do note that I am assuming that both psi and phi are in the same basis set (like e^ikx).

          Like any mathematical approach to working in multiple dimensions, matrices are a very helpful tool. Some of you may recall the term adjoint in some linear algebra class. It is in fact the same adjoint. If you were in matrix format, you would take the adjoint of the matrix to solve for the "dagger" form of the operator.

          While <A|B> may appear as one object, it can actually be broken down (and appear in reverse order). In order to familiarize ourselves with these names, I broke the nomenclature down below:

Picture
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    • Introduction to Waves (The Wave Equation)
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    • Introduction to Commutation
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    • Free Particle
    • Infinite Cubic Well (3D Particle in a Box)
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    • Introduction to Fission (Energy Extraction)
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  • Home
  • Quantum Mechanics I
    • Introduction to Waves (The Wave Equation)
    • Introduction to Waves (The Wave Function)
    • Motivation for Quantum Mechanics (Photoelectric effect)
    • Motivation for Quantum Mechanics (Compton Scattering)
    • Motivation for Quantum Mechanics (Black Body Radiation)
    • Bohr Model of the Atom
    • Wave-Particle Duality (The Wave Function Motivation)
    • Problems with the Wave Function
    • Introduction to Quantum Operators (The Formalism)
    • Introduction to Dirac Notation
    • Introduction to Quantum Operators (The Hermitian and the Adjoint)
    • Introduction to Commutation
    • Expectation Values of Operators
    • Quantum Uncertainty (Defining Uncertainty)
    • Quantum Uncertainty (Heisenberg's Uncertainty Principle)
    • The Schrödinger Equation (The "Derivation")
    • The Schrödinger Equation (How to use it)
    • No Degeneracy in 1-Dimension
    • Parity Operator
    • Quantum Mechanics' Core Postulates
    • Free Particle (In a Vacuum)
    • Particle in a Box (Infinite Square Well)
    • Bound States (The Mathematical Setup)
    • Bound States (The Shooting Method)
    • Bound States (Patching Solutions Together)
    • Patching Solutions (Finite, Infinite, and Delta Function Potentials)
    • Delta Function Potential Well
    • Scatter States (A Touch on Dispersion)
    • Scatter States (Reflection, Transmission, Probability Current)
    • Scatter States (Worked Example)
    • Scatter States (Elastic Collision)
    • Quantum Tunneling (Constant Potential)
    • Quantum Tunneling (Changing Potential)
    • Quantum Tunneling (Alpha Decay Example)
    • Quantum Harmonic Oscillator (Classical Mechanics Analogue)
    • Quantum Harmonic Oscillator (Brute Force Solution)
    • Quantum Harmonic Oscillator (Ladder Operators)
    • Quantum Harmonic Oscillator (Expectation Values)
    • Bringing Quantum to 3D (Cartesian Coordinates)
    • Free Particle
    • Infinite Cubic Well (3D Particle in a Box)
    • Quantum Harmonic Oscillator
    • Schrödinger Equation (Spherical Coordinates)
    • Angular Momentum (Experiments)
    • Angular Momentum (Operators)
    • Angular Momentum (Ladder Operators)
    • Schrödinger Equation (Spherical Symmetric Potential)
    • Infinite Spherical Well (Radial Solution)
    • One Electron Atom (Radial Solution for S-orbital)
    • Hydrogen Atom (Angular Solution; Spherically Symmetric)
    • Hydrogen Atom (Radial Solution; Any Orbital)
    • Hydrogen atom (Recap)
  • Quantum Mechanics II
  • Nuclear Fusion
    • Introduction to Fission (Energy Extraction)
    • Introduction to Fusion (Applications and Challenges)
    • Choosing Fusion Reactants
  • Special Relativity
    • Terminology and Notation
    • Galilean Transformation
  • Statistical Thermodynamics
  • Chemical Thermodynamics
  • Ionization Radiation
  • Multivariable Calculus
    • Vectors
    • Dot Product
    • Cross Product
    • Rotating Vectors
    • Level Curves
    • Gradients
    • Directional Derivatives
  • Differential Equations
  • Contact