Mind Network - Samuel Solomon

Patching Solutions

Finite, Infinite, and Delta Function Potentials
When patching solutions together, two different quantities about the wave function must be considered:
       1. The connecting point of the wave function on both sides of the boundary
       2. The slope (angle of attack) of the wave function on both sides of the boundary


          The wave function, to have real probabilistic interpretations, must be a continuous wave function, so that takes care of the first point. Whether the slope of the wave function has to be continuous is a separate issue that will be discussed in a few cases below.

          But first, let us quickly define mathematically what we mean when we ask whether the first derivative (slope) of the wave function is continuous. In order to for the first derivative (the slope) of the wave function to be continuous, we require the difference in the slope of either sides of a boundary (lets say at position 'a') to be zero. Mathematically we can therefore set up this question as:

Patching Solutions: Jump Condition
We find:
       1. If equation 3 equals zero: the slope is continuous
       2. If equation 3 is non-zero: the slope is discontinuous

CASE I: V(x) is finite and continuous
Patching Solutions: Jump Condition
In order to make sure everyone is following, let us review some key steps below:
       3 to 4: Plug in the time independent Schrodinger equation for the second derivative of the wave function
       4 to 5: The integral of a point (which is the case as delta goes to zero) is just the point itself (sum up 1 point)
       5 to 6: As the limit of delta goes to zero, we essentially have v*psi - v*psi = 0

This should not shock anyone as the wave function is continuous and finite and the potential is continuous and finite, so it should have a continuous derivative.

CASE II: V(x) is an infinite wall (discontinuous)
Patching Solutions: Jump Condition
In order to make sure everyone is following, let us review some key steps below:
       3 to 5: The same steps as the first case
       5 to 7: V is infinity on one side (which we previously found makes the wave function 0)
       7 to 8: Infinity * 0 is very ambiguous. Is it zero, is it still infinity. Either way, the whole equation is definitely not zero.

The slope is is not continuous, and it is ambiguous to what the difference in the slope is.      

CASE II: V(x) is a Delta Function (discontinuous, infinite point)
Patching Solutions: Jump Condition
In order to make sure everyone is following, let us review some key steps below:
       3 to 4: The same steps as the first two cases
       4 to 9: Plug in the delta function potential for V(x)
       9 to 10: Pull out the constants from the integral
       10 to 11: Solve the integral (Note: the delta function is zero in every place except when x = a)
       11 to 12: Evaluate the limit

When we have a delta function potential, we still do not have continuous first derivative of the wave function, but we do have a known finite difference that we can calculate.
SUMMARY OF RESULTS
When we patch together solutions for the wave function we will use the following rules:
       1. No matter what: the wave function psi(x) is always continuous and finite
       2. If the potential is finite, the slope of the wave function on either side of the boundary is the same.
       3. If the potential is infinite, the slope is discontinuous at the boundary (finite difference for delta functions)

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    • Introduction to Waves (The Wave Equation)
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    • Patching Solutions (Finite, Infinite, and Delta Function Potentials)
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    • 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)
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    • One Electron Atom (Radial Solution for S-orbital)
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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