Mind Network - Samuel Solomon

Bohr Model of the Atom

The photoelectric effect, Compton scatter, and black body radiation taught physicists something: the man-written rules of physics must be edited. One of the first attempts was made by Niels Bohr. In order to start this new age in physics, he decided to make his own atomic model. Here are the key assumptions:

       1. Electrons move in circular orbits
       2. Bound electrons are standing waves
       3. Energy of photon's are quantized, with energy gaps = h*v.
       4. Electron's orbit around the nucleus with an angular momentum of L = r x P


Let us now analyze the physics of assumptions 2 and 4.

Bohr Model Assumptions
Now it is important to introduce a new topic in quantum mechanics. It is called the de Broglie relationship (which was presented as his thesis ... I can only imagine the thesis review that occurred...). It states that all matter has an associative wavelength and all waves have an associated mass. The simple proof (yes very simple) can be found below.
De Broglie Relation
In order to make sure everyone is following, let us review some key steps below:
       3: Start from our previous general energy relationship for particles; however, photon's are massless.
      4: Set out planck's energy and general energy equal to each other. Solve for the de Broglie relation.

Let us now plug this relation (4) back into our equation 1.

Quantization of Angular Momentum
In order to make sure everyone is following, let us review some key steps below:
       5: Take equation 1 and plug out de Broglie relationship into the equation. Additionally used a new h_bar symbol.
       6: Added equation 2 into the mix. Found a new equation for angular momentum.

          The significance of this calculation is that (while some of the assumptions are inaccurate) it is the first postulate that angular momentum of particles is quantized just like energy. Only certain angular momentum values are allowed and again it is some integer multiple (divided by 2*pi) of this strange constant h.

Let us continue and now use these principles on the hydrogen atom (ie one electron orbiting one nucleus). The first thing we are going to do is go back to Newton's second law for orbiting particles:

Bohr radius
In order to make sure everyone is following, let us review some key steps below:
       7: Newton's second law for orbiting particles = sum(forces) = electrostatic force of the electron and nucleus.
       7 to 8: plug in equation 6 (without angular momentum) for velocity.
       8 to 9: simplify the expression.

While this radial distance of the electron looks daunting, it actually is composed of essentially constants. We can therefore define a useful constant below (called the Bohr radius):

Bohr radius
Now that one has found the equilibrium radius of an electron calculated by Bohr (which is wrong for basically all cases but 1), it is time to determine the theoretical energy.

Total energy is the sum of kinetic and potential energy.

Energy of Hydrogen Atom
In order to make sure everyone is following, let us review some key steps below:
       12 to 13: Plugged in the equation 6 (without angular momentum) for velocity. Also made both denominators equal.
       13 to 14: Replaced the r^2 of each term with the one we solved for in equation 9.
       14 to 15: Simplified the expression.

Again, a lot of these terms are constants, so let's plug in those true values to get:
Energy of Hydrogen Atom
This is a great approximation of the energy gaps in the Hydrogen atom (where Z = 1) and a good takeaway from the Bohr model. It was very coincidental that this is a good model (given the false starting assumptions), but we will be able to go and develop a better equation for Hydrogen through our quantum mechanical expressions.

The last note about this equation is that the plus and minus sign are very case dependent as follows:

Energy of Hydrogen Atom
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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)
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    • 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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  • Special Relativity
    • Terminology and Notation
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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