Mind Network - Samuel Solomon

Quantum Harmonic Oscillator

Classical Mechanics Analogue
          In classical mechanics we define a harmonic oscillator as a system that experiences a restoring force when perturbed away from equilibrium. Classical examples include:
       1. Spring: when displaced from the natural length, the spring either pushes or pulls the system back to equilibrium
       2. Pendulum: When displaced from the natural hanging position, gravity pulls the system back down to equilibrium
Nature finds this equilibrium position to be the most stable form of the system. It is found to be the local minimum on potential energy diagram:

Hamonic Oscillator: Potential
          The purple dots in the picture above represent points that exhibit this restorative force. If the particle moves to the left or to the right of the purple dot, the particle faces a greater energy barrier and is pulled back down ("falls back down") to reach its local minimum energy state again.
          Depending on one's familiarity with harmonic oscillators, one might start to be correctly questioning how one system can approximate different energy minimums (the two local minimums in the diagram above are obviously NOT the same). To address this question, we must quickly note that harmonic oscillators APPROXIMATE the local minimum for SMALL oscillations. In fact, the mathematical representation for harmonic oscillators clearly show this assumption, which is actually why the harmonic oscillators approximation can only be used for local minimum that show this characteristic concave up parabola (though applications to local maximum, concave down parabolas, are possible).
          The mathematical way of approximating functions is to Taylor expand around a point. Let us Taylor expand around one of the local minimum points 'x' that has a corresponding potential energy value of 'V_0'

Harmonic Oscillator Potential: Taylor Expansion
          While the whole Taylor expansion to the infinity term will give us exactly the whole function, we only care about small oscillations around the local minimum. Hence, we will make a few approximations:
       1. For a small X-X_0 perturbation, ( X-X_0 )^3 or higher (order three terms) will essentially be zero
                Think about (0.01)^3 vs. (0.01)^2. Both may be small, but (0.01)^3 is much smaller (less experimentally relevant)
       2. Because we are always around a local minimum, the first derivative of the potential is zero
                For small perturbations, the first derivative does not change much (still zero). Force = - dV/dx = 0 at equilibrium
       3. We can set the reference potential to be V_0 = 0
                A potential without a reference is meaningless (V = mgh is really
ΔV= mg*Δh). The reference is arbitrary

We can now approximate the potential for small perturbations (for small oscillations) as:

Harmonic Oscillator Potential
          The next obvious question is what is the second derivative of the potential around the local minimum. To answer this question, let us apply the harmonic oscillator to a problem. In quantum mechanics we will generally represent electrostatic forces as springs osculating, so let us use a spring as an example. The restorative force on a spring is F = -kx. We also know that F = - dV/dx in one dimension (it is the gradient for 3 dimensions). Let us combine these two facts below:
Harmonic Oscillator 'k' Value
Hence the potential energy of a spring is:
Harmonic Oscillator Potential
          Except knowing the 'k' constant (the force constant) between atoms is not always a trivial matter. However, we can experimentally find the frequency of oscillation. We can find 'k' in terms of the frequency 'w' below:
Harmonic Oscillator Function
In order to make sure everyone is following, let us review some key steps below:
       7: Newton's second law. The sum of all forces = mass*acceleration
       7 to 8: divide both sides by 'm'
       8 to 9: set w^2 = k/m
       9 to 10: solve the differential equation and notice that 'w' is the oscillation frequency

Replacing 'k' for 'w' in the harmonic oscillator potential will give us our final general harmonic oscillator form:

Harmonic Oscillator of Waves
This V(x) potential holds for any small oscillation around a local minimum potential with variables defined as:
       m = mass of the particle
       w = oscillation frequency of the particle in radians / second (pronounced 'omega')
       x = the displacement from the equilibrium x coordinate (used to be x - x_0 where 'x' was the equilibrium position)

A last final word about the harmonic oscillator is a warning. Harmonic oscillators only describe small oscillations. A pictorial representation of the harmonic oscillator's fit to a local minimum can be seen below:

Harmonic Oscillator Approximations
       To fit the curve better, we can only change the width of the parabola. For the same particle (same mass) the only free parameter is the frequency of oscillation. Higher frequency would mean a smaller width parabola; lower frequency would mean a higher width of the parabola.
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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
  • 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