Mind Network - Samuel Solomon

Quantum Harmonic Oscillator

Expectation Values
          While I could never cover every example of QHOs, I think it is important to understand the mathematical technique in how they are used. To do this, we will solve for the expectation values of x, p, x^2, and p^2 for a wave function in a SINGLE basis state 'n.' Let us start with the x and p values below:
Quantum Harmonic oscillator: Position Expectation Value
Quantum Harmonic oscillator: Momentum Expectation Value
In order to make sure everyone is following, let us review some key steps below:
       1: Plug in the ladder operator version of the position operator
       1 to 2: Pull out the constant and split the Dirac notation in two
       2 to 3: We know how the ladder operators act on QHO states
       3 to 4: Each QHO basis state must be orthonormal to each other; zero inner product (no net overlap)
Equations 6 - 10 are the same but for the momentum operator


          Hopefully this isn't two shocking of a result. If you have ONE basis state in a symmetric potential well, then the basis state is either even or odd. Mathematically, this means that the total magnitude (amount either negative or positive) of the wave function is the same on both sides of the well. Hence, we should not expect the particle to favor either side (averages to x = 0). Physically, we also note that the particle never gets out of the well, so even though it has some velocity, it must average out to a net velocity of zero (or else it would get out of the well and keep going never to return). Momentum is m*v, so average momentum is zero.
          While our classical intuition leads us to the correct answer for the one basis state expectation values, it is important to note that the x and p expectation values are not always zero for the QHO. This is because with two basis states, the waves interfere with each other constructively and destructively. We can no longer think about basis state one and two separately as their is a portion of the probability density that would NEVER have been their if we evaluated the basis waves on their own. You can see this extra term below:

Wave Interference
          The yellow term would not normally be their if we only considered the probabilities of A and B separately. The fact that they are in 1 combined wave function changes the math just enough for our classical intuition to be mixed up.

          Let us now move onto the expectation values for x^2 and p^2 for ONE basis state:

Quantum Harmonic oscillator: x^2 and p^2 Expectation Value
In order to make sure everyone is following, let us review some key steps below:
       12: Plug in the ladder operator version of the position operator (in the QHO state)
       12 to 13: Pull out the constant and distribute the ladder operators
       13 to 14 We know how the ladder operators act on QHO states (plug in the eigenvalues)
       14 to 15: Each QHO basis state must be orthonormal to each other; zero inner product (no net overlap)

       15 to 16: Simplify the expression
Equations 17 - 21 follows the exact same logic, but now with momentum

A last check to make sure nothing is sketchy is to calculate the uncertainty in x and p:

Quantum Harmonic oscillator: Position Uncertainty
Quantum Harmonic oscillator: Position Uncertainty
Quantum Harmonic oscillator: Heisenberg's Uncertainty Principle
It checks that Heisenberg's uncertainty principle holds for every nth excited state of the QHO, so we are not in violation of any known principles.
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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
    • 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
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  • Multivariable Calculus
    • Vectors
    • 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