Simultaneous Eigenstates

Overview

• Aim to find functions that are eigenstates of the operators L and L squared.
• This leads to simplification of the Schrödinger equation to a radial equation.
• The Schrödinger equation includes variables r, theta, and phi; theta and phi deal with angular dependence.

Simultaneous Eigenstates of L and Lz

• Define eigenstates as ( \psi_{lm}(\theta, \phi) ).
• Eigenstates are characterized by quantum numbers l and m, which are related to the eigenvalues.
• Condition: ( L_z \psi_{lm} = m \hbar \psi_{lm} ).
• ( m ) must be in the real numbers due to the properties of Hermitian operators.
• Condition from L squared: ( L^2 \psi_{lm} = \lambda \psi_{lm} ), where ( \lambda ) is positive.

Eigenvalue Expressions

• Write ( \lambda = l(l + 1) \hbar^2 ).
• ( l ) is a real number. The eigenvalue form ensures positivity of ( \lambda ).
• Eigenvalues of Lz must be quantized, resulting in ( m ) being an integer.

First Equation: Finding Eigenstates

• Begin with: [ \frac{\hbar}{i} \frac{d}{d\phi} \psi_{lm} = \hbar m \psi_{lm} ]
• Result leads to ( \psi_{lm}(\theta, \phi) = e^{i m \phi} f(\theta) ).
• Requirement for periodicity implies ( m \in \mathbb{Z} ) (integers).

Second Equation: Complicated Differential Operator

• The equation is: [ -\hbar^2 \left( \frac{1}{\sin \theta} \frac{d}{d\theta} \sin \theta \frac{d}{d\theta} + \frac{1}{\sin^2 \theta} \frac{d^2}{d\phi^2} \right) \psi_{lm} = \hbar^2 l(l+1) \psi_{lm} ]
• Substitute the known ( \psi_{lm} ) form to reduce the equation to: [ \sin \theta \frac{d^2}{d\theta^2} \psi_{lm} + (l(l + 1) \sin^2 \theta - m^2) \psi_{lm} = 0 ]

Changing Variables

• Substitute ( x = \cos \theta ) to simplify the equation.
• The resulting differential equation involves only ( x ).

Recursive Relations

• Analyze the case when m = 0: [ (1 - x^2) \frac{dP_{L}}{dx} + l(l + 1) P_{L} = 0 ]
• Use series solutions for ( P_L(x) = \sum_{k} a_k x^k ).
• Obtain a recursion relation leading to termination conditions for series, indicating that ( l ) must be a non-negative integer (0, 1, 2, ...).

Conclusion

• Eigenvalues of ( L^2 ) are quantized, corresponding to the magnitude of angular momentum.
• Solutions are related to Legendre polynomials, indicating that the physical constraints of the problem enforce quantization.
• Further exploration will ensure that ( m ) cannot exceed ( l ).

Key Takeaways

• The study of simultaneous eigenstates leads to quantization in quantum mechanics.
• Eigenstates are essential for simplifying the Schrödinger equation and understanding angular momentum in quantum systems.