3.2 Angular Momentum and Spin in Hydrogen Atom

Angular momentum in the hydrogen atom is a crucial concept in atomic physics. It's quantized, meaning it can only take on specific values, which leads to distinct energy levels and sublevels in the atom.

Electrons also have an intrinsic spin, adding another layer of complexity. The interaction between orbital and spin angular momentum results in fine structure splitting, revealing the atom's intricate energy landscape.

Angular Momentum in the Hydrogen Atom

Quantization of Angular Momentum

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• Angular momentum, a vector quantity describing the rotational motion of a particle or system, is represented by the symbol $L$
• In quantum mechanics, angular momentum is quantized and can only take on discrete values
• The magnitude of the orbital angular momentum is given by $|L| = \sqrt{l(l+1)}\hbar$, where $l$ is the orbital quantum number (a non-negative integer) and $\hbar$ is the reduced Planck's constant
• The z-component of the orbital angular momentum is given by $L_z = m_l\hbar$, where $m_l$ is the magnetic quantum number, which can take on integer values from $-l$ to $+l$
• The quantization of angular momentum in the hydrogen atom leads to the formation of distinct energy levels ($n=1,2,3...$) and sublevels ($s,p,d...$)

Impact on Energy Levels

• The quantization of angular momentum results in the hydrogen atom having discrete energy levels and sublevels
• Each energy level is associated with a principal quantum number $n$, and within each level, there are sublevels corresponding to different values of the orbital quantum number $l$
• The allowed values of $l$ for a given $n$ are $0, 1, 2, ..., n-1$, and each value of $l$ corresponds to a specific sublevel ($s,p,d,f...$)
• The energy of an electron in the hydrogen atom depends on both $n$ and $l$, with the energy decreasing as $n$ increases and the sublevels within a given $n$ having slightly different energies due to the electron's orbital angular momentum

Electron Spin Angular Momentum

Intrinsic Spin of Electrons

• Electrons possess an intrinsic angular momentum called spin, which is not related to their orbital motion
• Spin is a fundamental property of electrons and other subatomic particles, characterized by the spin quantum number $s$, which has a value of $1/2$ for electrons
• The magnitude of the spin angular momentum is given by $|S| = \sqrt{s(s+1)}\hbar$, where $s$ is the spin quantum number
• The z-component of the spin angular momentum is given by $S_z = m_s\hbar$, where $m_s$ is the spin magnetic quantum number, which can take on values of $+1/2$ or $-1/2$
• Electron spin is an intrinsic form of angular momentum, similar to a particle spinning on its own axis (although this is a simplified classical analogy)

Magnetic Moments

• The intrinsic spin of electrons is associated with a magnetic moment $\mu_s$, which is proportional to the spin angular momentum
• The magnetic moment is given by $\mu_s = -g_s\mu_B S/\hbar$, where $g_s$ is the electron spin g-factor (approximately 2) and $\mu_B$ is the Bohr magneton
• The spin magnetic moment interacts with external magnetic fields, leading to the splitting of energy levels (Zeeman effect)
• The orientation of an electron's spin magnetic moment can be either parallel (spin-up, $m_s = +1/2$) or antiparallel (spin-down, $m_s = -1/2$) to an external magnetic field
• The interaction between the electron's spin magnetic moment and the magnetic field generated by its orbital motion gives rise to the fine structure of atomic energy levels

Adding Angular Momentum

Rules for Adding Angular Momentum

• The total angular momentum of the hydrogen atom, $J$, is the vector sum of the orbital angular momentum, $L$, and the spin angular momentum, $S$: $J = L + S$
• The magnitude of the total angular momentum is given by $|J| = \sqrt{j(j+1)}\hbar$, where $j$ is the total angular momentum quantum number, which can take on values of $|l-s|, |l-s|+1, ..., l+s-1, l+s$
• The z-component of the total angular momentum is given by $J_z = m_j\hbar$, where $m_j$ is the total magnetic quantum number, which can take on integer values from $-j$ to $+j$
• The addition of angular momentum follows the triangular inequality: $|l-s| \leq j \leq l+s$
• The allowed values of $j$ for a given $l$ and $s$ can be determined using the Clebsch-Gordan coefficients or by constructing a vector model (vector addition of $L$ and $S$)

Total Angular Momentum in Hydrogen

• In the hydrogen atom, the electron's orbital angular momentum ($l = 0, 1, 2, ...$) and spin angular momentum ($s = 1/2$) couple to form the total angular momentum $J$
• For the ground state ($n=1, l=0$), the total angular momentum quantum number can only be $j=1/2$, as $l=0$ and $s=1/2$
• For excited states ($n>1$), the possible values of $j$ depend on the orbital quantum number $l$:
  • For $l=0$ (s orbitals), $j=1/2$
  • For $l=1$ (p orbitals), $j=1/2$ or $3/2$
  • For $l=2$ (d orbitals), $j=3/2$ or $5/2$
• The different possible values of $j$ for a given $l$ lead to the fine structure splitting of energy levels

Coupling of Orbital and Spin Angular Momentum

LS Coupling Scheme

• The coupling of orbital and spin angular momentum in the hydrogen atom leads to the fine structure of energy levels
• The fine structure arises from the interaction between the electron's orbital angular momentum and its spin angular momentum, as well as from relativistic corrections to the electron's energy
• The coupling of $L$ and $S$ is described by the LS coupling scheme (also known as Russell-Saunders coupling), which is appropriate for light atoms like hydrogen
• In the LS coupling scheme, the orbital angular momenta of individual electrons are first coupled to form a total orbital angular momentum, $L$, and the spin angular momenta are coupled to form a total spin angular momentum, $S$. Then, $L$ and $S$ are coupled to form the total angular momentum, $J$
• The LS coupling scheme is applicable when the spin-orbit interaction is weaker than the electrostatic interactions between electrons

Fine Structure Splitting

• The fine structure splitting of energy levels depends on the values of $l$, $s$, and $j$, and is characterized by the fine structure constant, $\alpha \approx 1/137$
• The energy shift due to fine structure is given by the formula: $\Delta E = (\alpha^2/2n^3) * [j(j+1) - l(l+1) - s(s+1)] * ([Rydberg constant](https://www.fiveableKeyTerm:Rydberg_Constant))$, where $n$ is the principal quantum number
• The fine structure splitting increases with increasing atomic number, as the spin-orbit interaction becomes stronger
• In the hydrogen atom, the fine structure splitting is much smaller than the energy differences between principal quantum levels ($\Delta E_{fine} \ll \Delta E_{n}$)
• The fine structure splitting can be observed using high-resolution spectroscopy, as it leads to the splitting of spectral lines (e.g., the famous sodium doublet lines)