Quantum Mechanics - Commutation Relationship and Angular Momentum

Kinetic Energy Operator

• The kinetic energy operator in Cartesian coordinates (x, y, z) is given by -\frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right).
• This operator is then converted into spherical coordinates (r, θ, φ).
• The conversion results in terms involving r and terms involving θ and φ.

Spherical Coordinates

• The kinetic energy operator in spherical coordinates can be separated into radial and tangential components.
• The tangential kinetic energy is identified as \frac{L^2}{2mr^2}, where L^2 is the angular momentum operator.
• The radial kinetic energy is related to the radial momentum squared over 2m.

Angular Momentum Operator

• The angular momentum operator L^2 can be derived by comparing the kinetic energy operator in Cartesian and spherical coordinates.

Commutation Relationship

• The LX, LY, and LZ operators represent the components of angular momentum along the x, y, and z axes, respectively.
• The LX operator does not commute with the LY operator, meaning you cannot simultaneously specify two components of angular momentum.
• Measuring one component (e.g., LX) scrambles the others (LY and LZ), resulting in a distribution of possible values.
• Only one component of angular momentum can be precisely specified at a time in quantum mechanics.

Magnitude and Z-Component

• It is possible to know the total magnitude of the angular momentum vector (L) and one component, typically the z-component (LZ), simultaneously.
• The x and y components (LX and LY) will have a distribution of values.

Visual Representation

• Angular momentum vector can be visualized as a cone where the magnitude (L) and the z-component (LZ) are known.
• The x and y components are distributed around the base of the cone.
• The relationship between the components is given by Lx^2 + Ly^2 = L^2 - L_z^2, derived from the Pythagorean theorem.

Importance of Commutation Relations

• Commutation relations impose restrictions on what can be measured simultaneously in nature.
• This is analogous to the Heisenberg Uncertainty Principle, which restricts simultaneous measurements of position and momentum.

Proof of Commutation Relation

• LX is defined as Lx = ry pz - rz py, and LY is defined as Ly = rz px - rx pz.
• The commutator [Lx, Ly] is calculated as Lx Ly - Ly Lx.
• Expanding the commutator results in eight terms, which are then simplified.

Simplification of Terms

• Terms are simplified using the commutation relations between position and momentum operators.
• For example, [x, py] = 0 and [x, px] = i\hbar.
• After simplification, the commutator [Lx, Ly] is shown to be equal to i\hbar L_z.

Uncertainty Principle for Angular Momentum

• The uncertainty in measuring LX and LY is related to the value of LZ by \Delta Lx \Delta Ly \geq \frac{1}{2} |\langle i\hbar L_z \rangle| .
• This means that the product of the standard deviations of LX and LY must be greater than or equal to half the magnitude of the average value of LZ.
• As LZ increases, the uncertainties in LX and LY also increase, making the circle bigger.

Observations

• None of the components of angular momentum can be greater than the magnitude of L.

Eigenfunctions and Eigenvalues

• The L^2 operator has certain eigenfunctions, each with its own eigenvalue.
• The case where l = 0 is allowed, representing a system with no angular momentum.

Particle in a Box Analogy

• In the particle in a box problem, the n = 0 solution was rejected because it implied the particle had zero energy, which didn't make sense.
• However, the l = 0 solution is allowed because it corresponds to a particle moving directly towards or away from the origin, which is physically plausible.

Magnitude of L Squared

• The magnitude of L^2 is given by L^2 = \hbar^2 l(l+1), where l is an integer (0, 1, 2, …).

Eigenfunctions of LZ

• The eigenvalue problem for the LZ operator is given by Lz \Psi{lz} = constant \times \Psi_{lz}.
• Solving this problem yields the possible values that can be measured.
• The LZ operator is related to the partial derivative with respect to the azimuthal angle φ.

Boundary Conditions

• The solution to the eigenvalue problem is \Psi = e^{i \frac{L_z}{\hbar} \phi}.
• The values that are allowed need to satisfy that \Psi(\phi) = \Psi(\phi + 2\pi). This implies that e^{i \frac{Lz}{\hbar} \phi} = e^{i \frac{Lz}{\hbar} (\phi + 2\pi)}.
• Which then can be reduced to, 1 = e^{i \frac{L_z}{\hbar} (2\pi)}.

Quantization

• Because of the boundary condition requiring the wave function to return to the same value after a full rotation (\phi + 2\pi), the values of LZ are quantized.
• The allowed values of LZ are given by L_z = m \hbar, where m is an integer.

Restrictions

• The conditions lead to L_z = m \hbar, where m can be any integer but must be less than l

Special Cases

• If l = 0, then the magnitude of angular momentum is zero, and LZ must also be zero.
• If l = 1, then the magnitude of angular momentum is \sqrt{2} \hbar, and LZ can be 0, +1, or -1.

Angle Quantization

• The angle between the angular momentum vector L and the z-axis is quantized.
• For example, the cosine of the angle is equal to 1/1.4 \approx 45 degrees.

Eigenfunctions of L Squared and LZ

• The discussion focuses on finding the eigenfunctions of the L^2 operator and then finding the eigenfunctions of L_z.

Solving Problems

• The next step is to solve a problem involving a certain potential is solving the eigenvalue problem for the full Hamiltonian.

Hamiltonian

• To deal with the Hamiltonian, all three terms (r, theta, phi) must be accounted for to create one differential equation.

Commuting Operators

• If two operators commute, they share eigenfunctions.
• Since the Hamiltonian includes the L^2 operator, it means the eigenfunctions of the L^2 operator should be considered when solving the full equation.

Practice Material

• Practice problems for quiz four have been provided, along with solutions.