Mind Network - Samuel Solomon

Angular Momentum

Operators
          Experiments such as the Einstein-De Hass and Stern-Gerlach motivated a new quantum outlook on angular momentum. We know that mathematically it should be a conserved quantity (for no external torque), and that experimentally their seems to be an extrinsic rotational component related to visible gyrations and an intrinsic 'spin' component related to the atoms' dipole (with two quantized 'spin' values).
          Analyzing this intrinsic 'spin' component is not very intuitive; hence, we will need to build up our quantum mechanics toolbox just a little more (more information about spin in the course 'quantum mechanics II'). The extrinsic angular momentum, on the other hand, can be visibly and even classically observed. To formulate our quantum operators, let us analyze the classical definition of angular momentum:

Classical Angular Momentum
          As it turns out, the angular momentum is completely described in terms of the position and momentum (in the x, y, z directions) of the particle. We know the quantum operators of position and momentum and can just substitute them in to find the angular momentum quantum operator.
          We will now find the quantum operator of the angular momentum in the 'Z' direction ('L_z'). We will show the direct proof for the z component as it is the most common of the three seen in literature; however, all we are really doing is transforming between Cartesian and spherical coordinates (as you will see; it is more tedious work). We will start on the left with a small relation between spherical and Cartesian derivatives and use it to calculate the z component angular momentum on the right:

Lz Operator
Lz Operator
In order to make sure everyone is following, let us review some key steps below:
       2: Phi is defined to be the angle in the x-y plane (hence dependent on x and y). In equation 2, we apply the chain rule
       3 and 4: We take derivatives of the polar relationships between x, y and phi
       5: Plug in the polar relationships found in equations 3 and 4 into equation 2
       5 to 6: We can again use the relationships in equations 3 and 4 to get d/dphi in terms of 'x' and 'y'
       7: The classical definition of angular momentum (the extrinsic component for quantum mechanics)
       7 to 8: Plug in our quantum momentum operator
       8 to 9: plug in equation 6 to find the z component angular momentum operator in spherical coordinates

          For anyone interested in the other formulas. Through the exact same spherical substitutions, one can find all the spherical angular momentum operators below:

Quantum Mechanics: Angular Momentum Operators
          Remember, operators are mathematically defined to scale an eigenfunction by the real observed value. Now that we have the z component angular momentum operator, we can find the eigenfunction it acts on to produce the z component angular momentum eigenvalue:
Spherical harmonics: Y(phi) eigenfunction
Spherical harmonics: Y(phi) eigenfunction and Lz eigenvalue
In order to make sure everyone is following, let us review some key steps below:
       10: The L_z operator acting on some unknown eigenfunction
       10 to 11: We apply the separation of variables ansatz for a standing wave (for all three variables this time)
       11 to 12: The derivative only acts on the Y(phi) function
       12 to 13: The operator should scale the whole eigenfunction by some eigenvalue 'L_z' (to be determined)
       13 to 14: We set equations 12 and 13 equal to each other and solve for the eigenfunction Y(phi)
We can now apply the boundary condition that phi = 0 is the exact same as phi = 2pi in spherical coordinates (15)
       16: Therefore, Y(0) must be equal to Y(2pi). We plug this into our eigenfunction in equation 14
       16 to 17: In order for equation 16 to be true, we need the e^ix function to equal 1. e^ix is a sinusoidal function with a
                        period of 2pi. It starts at 1 where x = 0 and oscillates back every 2pi interval. Hence, when x = 2pi*m (for
                        some positive or negative integer m), e^ix is ALWAYS equal to 1.
       17 to 18: We can now solve for the eigenvalue 'L_z' and, by extension, the eigenfunction Y(phi)

          Note that we have solved for a generic z component angular momentum value (hence why we do not get a single angular momentum value, but a range of them for some integer 'm'). What is important to notice is that we have just proven that not all angular momentum values work (it is quantized, just like Bohr predicted).
          An additional note is that, while we have not discussed it yet, intrinsic angular momentum can not only have integer values of angular momentum but also half integer values (but that is another story. Just note that you may see h_bar/2 for an m_spin = 1/2). Besides this, intrinsic angular momentum actually follows a lot of the same rules as the extrinsic part (more on that is part II).

          Now that we have a great sense of the angular momentum operators, let us evaluate some important commutator relationships between them. Remember, commutators are expressions that allow us to switch the order of two operators as well as tell us information about the two observable's uncertainties (we can know two observables at the same time with certainty if they commute as then they can share a common eigenfunction; see commutator page if confused). Here are some important commutator relationships:

Angular Momentum Commutators
Levi-Civita
Let us discuss the importance of these commutators:
       Equation 19: As they do not commute, we cannot know for certainty L_x, L_y, or L_z at the same time.
       Equation 20: As they do commute, we can know the total angular momentum as well as one of its components at the same time. Their is an eigenfunction shared between the total angular momentum and each of its individual components.

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    • Introduction to Waves (The Wave Equation)
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    • 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)
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    • One Electron Atom (Radial Solution for S-orbital)
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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