Mind Network - Samuel Solomon

Quantum Tunneling

Alpha Decay Example
          Circling back to the introduction of tunneling, I previously stated that scientists can use quantum tunneling principles to push reactions forwards that would otherwise, classically, not have enough energy to precede. One example of this occurring in nature is the alpha decay process. The potential diagram for the alpha decay is as follow:
Alpha Decay Process, Quantum
Okay, the above picture might be a bit intimidating so lets break it down:
       Purple dotted line: The strong force holding the nucleus together (short range: only for close distances to nucleus)
       Blue dotted line: The coulombic force that the nucleus exerts at all distances (exponential decay of 1/r^2)
       Red straight line: The combination of the two energies (Force *distance = energy)
Notice that the strong nuclear force dominates early (but does NOT contribute at all very far away from the nucleus although it seems to diverge to infinity).

          Physically, the picture represents the energy barrier an alpha particle would feel leaving the nucleus. Think of it as a reaction barrier. The decay may be thermodynamically stable for heavy nuclei, but there is still an activation energy to jump over for the reaction to proceed. It turns out that the energy needed for the particle to decay out of the nucleus is generally GREATER than the energy of the particle. In fact, alpha decay is an inherent quantum tunneling phenomenon (would not exist without quantum tunneling).

          Let us now begin evaluating this problem. We notice that for certain energies, there is NO possibility of the alpha particle tunneling out of the nucleus. In this motivation, we will make a simplifying approximation to the potential shown below:

Alpha Decay Potential Approximation, Quantum
          To be clear: we are only analyzing the portion of the potential that the energy tunnels through. Specifically, this corresponds to the coulombic potential portion, which means for now we will ignore the strong force. The radius at which the coulombic potential matters is the radius of the nucleus itself (the strong force only exists inside the nucleus). Hence we will define 'R' as the nucleus's radius. To be clear: this is an approximation to the problem at hand, but a reasonable one.

          We can now bring back out total transmission coefficient found in the last page (for large energy-potential gaps like this one):
Alpha Decay: Transmission Coefficient
In order to make sure everyone is following, let us review some key steps below:
       2: Our previous formula for the total transmission coefficient
       2 to 3: Plug in our integral bound and V(x) equation
       3 to 4: Factor out the constant sqrt(2mE)

With the invention of computational online tools, one never need to get bogged down in long integrals. We can solve the integral online to yield the final equation below:

Alpha Decay: Transmission Coefficient
The radius 'R' of a nucleus is actually EXTREMELY small. We can therefore make the approximation:
Alpha Decay: Transmission Coefficient
'L' actually relates to the particles energy. We can substitute its full expression back in:
Alpha Decay: Transmission Coefficient
And that is alpha decay. Now we did make some approximation for the sake of this problem; however, in general, people find alpha decay as:
Alpha Decay: Transmission Coefficient
          With beta_1 and beta_2 both constants that relate to the Z (number of protons) of the nucleus. While our expression in equation 8 is NOT the exact expression for alpha decay, we did find the correct energy dependence of the system and only off by some constants in the exponent (taking a rigorous look at all the potentials the alpha particle faces will get you a better answer).
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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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    • Scatter States (A Touch on Dispersion)
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    • 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)
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    • One Electron Atom (Radial Solution for S-orbital)
    • Hydrogen Atom (Angular Solution; Spherically Symmetric)
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    • 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
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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