Mind Network - Samuel Solomon

Expectation Values of Operators

          When performing a quantum experiment (or any experiment for that matter), it is important to have some general sense of what the result will be. While results may vary (sometimes drastically), we can always find some average value to expect. We call this the expectation value.

          To calculate the expectation / average value for quantum operators, let us revisit the general definition of average values. The average of any quantity is defined (with some example value C_1 and C_2) as:

Expectation Values of Operators
          Equation 2 is the general definition of the average (total sum / number summed) for an example value of C_1 and C_2. In this case, there is 1 C_1 for every C_2, and hence the probability of getting a C_1 or a C_2 is 1/2. If we had 3 C_1 for 1 C_2, then we would have 3/4 C_1 + 1/4 C_2. This weighted average approach is represented mathematically in equation 3.

          Now let us translate this into quantum mechanics. Let us start with a given wave function psi, which is composed of basis functions phi. When applying an operator A to psi, the operator acts on every single basis phi, producing a series of possible eigenvalues depending on which basis we are in. This is shown visually below:
Picture
          a_n are the values we want to add up. The probability of getting an a_n value depends on the probability we are operating on its corresponding basis function phi_n. We have previously computed this probability (of being in any basis state) as |C_n|^2. Hence, the average value of an operator is given by:
Expectation Values of Operators
          We were able to previously find |C_n|^2 through normalizing the probability density function |psi|^2. Therefore, we can insert the integral of psi^2 in place of the C_n^2. This yields the average value equation below:
Expectation Values of Operators
          Just like every average value, it is important to note that the average value is not necessarily a real value. For example, the average value of 3 and 5 is 4, but 4 is not one of the initial value options. The same holds for quantum mechanics. An average value of 0 does not necessarily mean that the system can ever have a value of 0.
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  • 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
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    • Free Particle (In a Vacuum)
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    • Quantum Tunneling (Constant Potential)
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    • 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)
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    • 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)
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    • Choosing Fusion Reactants
  • Special Relativity
    • Terminology and Notation
    • Galilean Transformation
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    • Dot Product
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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