3.1 Orbital angular momentum operators and eigenfunctions

Orbital angular momentum is a key concept in quantum mechanics, describing how particles rotate in space. It's quantized, meaning it can only take specific values, which affects particle behavior and energy levels.

Understanding orbital angular momentum is crucial for grasping atomic structure and spectroscopy. The operators, eigenfunctions, and quantum numbers we'll explore help explain electron orbitals and their interactions with magnetic fields.

Orbital Angular Momentum Operators

Definition and Notation

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• The orbital angular momentum operators are denoted as $L_x$, $L_y$, and $[L_z](https://www.fiveableKeyTerm:l_z)$, corresponding to the x, y, and z components of the orbital angular momentum
• The orbital angular momentum operators are defined in terms of the position and momentum operators: $L_x = yP_z - zP_y$, $L_y = zP_x - xP_z$, and $L_z = xP_y - yP_x$

Commutation Relations

• The commutation relations between the orbital angular momentum operators are: $[L_x, L_y] = i\hbar L_z$, $[L_y, L_z] = i\hbar L_x$, and $[L_z, L_x] = i\hbar L_y$, where $\hbar$ is the reduced Planck's constant and $i$ is the imaginary unit
• The commutation relation between any orbital angular momentum operator and the total orbital angular momentum operator (###[l](https://www.fiveableKeyTerm:l)^2_0### = L_x^2 + L_y^2 + L_z^2) is zero: $[L_x, L^2] = [L_y, L^2] = [L_z, L^2] = 0$
• These commutation relations are fundamental to the quantum mechanical description of angular momentum and lead to the quantization of angular momentum

Eigenvalue Problem for Angular Momentum

Eigenfunctions and Eigenvalues

• The eigenvalue problem for the orbital angular momentum operators is solved by finding the eigenfunctions and eigenvalues that satisfy the equation: $L^2 \psi(\theta, \phi) = \lambda \psi(\theta, \phi)$, where $\psi(\theta, \phi)$ is the eigenfunction and $\lambda$ is the eigenvalue
• The eigenfunctions of the orbital angular momentum operators are the spherical harmonics, denoted as ###y_{l,[m](https://www.fiveableKeyTerm:m)}_0###(\theta, \phi), where $l$ is the orbital angular momentum quantum number and $m$ is the magnetic quantum number
• The spherical harmonics are complex-valued functions that depend on the polar angle $\theta$ and the azimuthal angle $\phi$ and form a complete orthonormal basis for functions on the unit sphere

Quantization of Angular Momentum

• The eigenvalues of the total orbital angular momentum operator ($L^2$) are given by $\lambda = \hbar^2 l(l+1)$, where $l$ is a non-negative integer ($l = 0, 1, 2, ...$)
• The eigenvalues of the z-component of the orbital angular momentum operator ($L_z$) are given by $\lambda_z = \hbar m$, where $m$ is an integer ranging from $-l$ to $+l$ in steps of 1 ($m = -l, -l+1, ..., l-1, l$)
• The quantization of angular momentum is a consequence of the boundary conditions imposed on the wavefunctions, which require the eigenfunctions to be single-valued and continuous

Physical Meaning of Quantum Numbers

Orbital Angular Momentum Quantum Number ($l$)

• The orbital angular momentum quantum number ($l$) determines the magnitude of the total orbital angular momentum of a particle, given by $|L| = \hbar\sqrt{l(l+1)}$
• The orbital angular momentum quantum number ($l$) also determines the shape of the electron orbitals in atoms, with increasing $l$ corresponding to more complex shapes (s, p, d, f, etc.)
  • $l = 0$ corresponds to s orbitals, which are spherically symmetric
  • $l = 1$ corresponds to p orbitals, which have a dumbbell shape
  • $l = 2$ corresponds to d orbitals, which have more complex shapes, such as cloverleaf or double dumbbell

Magnetic Quantum Number ($m$)

• The magnetic quantum number ($m$) determines the z-component of the orbital angular momentum, given by $L_z = \hbar m$
• The magnetic quantum number ($m$) describes the orientation of the orbital angular momentum vector with respect to the z-axis
  • $m = 0$ corresponds to an orbital angular momentum vector perpendicular to the z-axis
  • $m = \pm l$ corresponds to an orbital angular momentum vector aligned with the z-axis
• The quantum numbers $l$ and $m$ arise from the quantization of the orbital angular momentum in quantum mechanics, which is a consequence of the boundary conditions imposed on the wavefunctions

Ladder Operators and Eigenfunctions

Definition and Action

• The ladder operators, also known as raising and lowering operators, are denoted as $L_+$ and $L_-$ and are defined as $L_\pm = L_x \pm iL_y$
• The ladder operators act on the eigenfunctions of the orbital angular momentum operators to raise or lower the magnetic quantum number ($m$) by one unit:
  • $L_+ Y_{l,m}(\theta, \phi) = \hbar\sqrt{(l-m)(l+m+1)} Y_{l,m+1}(\theta, \phi)$
  • $L_- Y_{l,m}(\theta, \phi) = \hbar\sqrt{(l+m)(l-m+1)} Y_{l,m-1}(\theta, \phi)$

Generating Eigenfunctions

• The ladder operators can be used to generate the eigenfunctions (spherical harmonics) for a given orbital angular momentum quantum number ($l$) by starting with the eigenfunction with the lowest magnetic quantum number ($m = -l$) and successively applying the raising operator ($L_+$)
• The normalization of the eigenfunctions generated by the ladder operators is ensured by the coefficients involving the square roots of the magnetic quantum number-dependent factors
• This method provides a systematic way to construct the complete set of eigenfunctions for a given $l$, starting from a single eigenfunction and using the ladder operators to generate the rest