Mind Network - Samuel Solomon

Introduction to Quantum Operators

The Hermitian and the Adjoint
          After discussing quantum operators, one might start to wonder about all the different operators possible in this world. I claimed that the requirement is only that the wave function is an eigenfuntion of the operator. While this leads to many types of operators, not all of them are experimentally observed. These specific type of operators are called hermitian operators.

          Hermitian operators are operators that correspond to eigenvalues that we can physically observe. If we can physically observe the eigenvalue, then the eigenvalue must be real. Hence, hermitian operators are defined as operators that correspond to real eigenvalues. Mathematically this is represented below:

Hermitian Operator
          Note that the complex conjugate of a function is represented with a star (*) above it. In equation 1 we can appreciate that because the eigenvalue is real, the complex conjugate of the real eigenvalue is just the real eigenvalue (no imaginary term to take the complex conjugate of). We can now expand the expression above:
Hermitian Operator Adjoint
In order to make sure everyone is following, let us review some key steps below:
       2: We took the expression from equation 1 and wrote it in integral format
       2 to 3: took the complex conjugate of every variable
       3 to 4: Rearranged the order of the wave functions into a familiar format
       4 to 5: switched back to Dirac notation.


While this expression may not seem so significant at first, we can recall that
the inner product (<psi|O|psi>) is equal to its complex conjugate. Hence:
Hermitian Operator Adjoint
          This is a non-trivial solution found based on one simple assumption: the eigenvalue is real (and observable). What we have shown is that for any hermitian eigenfunction (real eigenvalue, observable quantity), the operator is equal to its own adjoint. We call these operators self-adjoint.

A summary about hermitian operators is shown below:
       1. Hermitian operators are defined to have real observables and real eigenvalues.
       2. Hermitian operator's are self-adjoint.
       3. Hermitian operators, in matrix format, are diagonalizable.
       4. The transpose of the transpose of an operator is just the operator. Hence the adjoint of the adjoint is the operator.
       4. If the adjoint of an operator is the negative of the operator, we call these anti-hermitian
.
             Example: i = sqrt(-1) -> not real. Note that two antihermitian operators can combine to make a hermitian operator


          Operators that are hermitian (observable) include the position, momentum, and energy. Here are a list of common adjoint operators:
Common Hermitian Operator (i, derivative, position, and momentum)
          There are two ways of finding an adjoint of an operator. You could either start with the Dirac notation and get from the adjoint of the operator to a function without any adjoints, or you could break the operator up into smaller pieces and take the adjoint of all the pieces individually.

I tried to show the method for a few common examples above, but let us quickly talk about the math used for each case:
       i: I start from the adjoint operator of i (sqrt(-1)). When I act on the function to the left, I take the adjoint and complex
           conjugate of i. Th adjoint of the adjoint returns my intiial operator and the complex conjugate of i is -i. Hence the
           adjoint of i is just -i (anti-hermitian).

       d/dx: I start the exact same way as I do for "i", but now d/dx is a real function. The difference is that "i" is a scalar and
                   can commute with any function like psi, while d/dx cannot. The first step is to use integration by part, which                    takes the form integral (d/dx*(f*g) )=integral( (d/dx(f) * g + f * d/dx (g). Next, we realize that the integral of the
                   derivative of |psi|^2 is just zero because |psi|^2 decays at +/- infinity to zero.

       x: The x operator on psi = x, which is a real value; hence the complex conjugate does nothing to it.

       p: We can replace p with its functional form and notice that we already solved for their adjoint in the above.

For completeness, here are the adjoint of these operators listed neatly below:

Common Hermitian Operator (i, derivative, position, and momentum)
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  • Quantum Mechanics I
    • Introduction to Waves (The Wave Equation)
    • Introduction to Waves (The Wave Function)
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    • 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")
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    • Bringing Quantum to 3D (Cartesian Coordinates)
    • Free Particle
    • Infinite Cubic Well (3D Particle in a Box)
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  • Quantum Mechanics II
  • Nuclear Fusion
    • 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