Quantum Angular Momentum and H-Atom

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Introduction

Quantum angular momentum is a fundamental concept in quantum mechanics. It describes the rotational properties of quantum systems and plays an important role in atomic structure. The hydrogen atom provides an important application of angular momentum theory through the Schrödinger equation.

1. Orbital Angular Momentum

Orbital angular momentum is defined as:

L = r × p

Using the momentum operator p = −iℏ∇:

L = −iℏ(r × ∇)

The angular momentum components satisfy the commutation relations:

[Lₓ, Lᵧ] = iℏL_z

[Lᵧ, L_z] = iℏLₓ

[L_z, Lₓ] = iℏLᵧ

2. Angular Momentum Eigenvalues

The squared angular momentum and its z-component have eigenvalues:

L²|l,m⟩ = ℏ²l(l+1)|l,m⟩

L_z|l,m⟩ = mℏ|l,m⟩

Therefore, the magnitude of orbital angular momentum is:

|L| = ℏ√[l(l+1)]

For orbital angular momentum, l = 0, 1, 2, ... and m = −l, ..., 0, ..., +l.

3. Spherical Harmonics

Spherical harmonics, represented by Yₗᵐ(θ, φ), are the angular eigenfunctions of the angular momentum operators.

L²Yₗᵐ = ℏ²l(l+1)Yₗᵐ

L_zYₗᵐ = mℏYₗᵐ

They describe the angular dependence of atomic orbitals and are essential for solving the Schrödinger equation in spherical coordinates.

4. Hydrogen Atom Hamiltonian

The hydrogen atom consists of an electron and a proton interacting through the Coulomb potential. Its relative-motion Hamiltonian is:

H = −(ℏ²/2μ)∇² − e²/(4πε₀r)

Here, μ is the reduced mass:

μ = mₑmₚ/(mₑ + mₚ)

The stationary Schrödinger equation is:

Hψ = Eψ

Since the potential depends only on r, the equation can be separated into radial and angular parts.

5. Separation of Variables

The wavefunction is expressed as:

ψ(r,θ,φ) = R(r)Yₗᵐ(θ,φ)

The radial equation is:

−(ℏ²/2μ) d²u/dr² + V_eff(r)u = Eu

where u(r) = rR(r), and the effective potential is:

V_eff(r) = ℏ²l(l+1)/(2μr²) − e²/(4πε₀r)

The first term represents the centrifugal contribution, while the second is the attractive Coulomb potential.

6. Energy Eigenvalues

Solving the radial equation with the appropriate boundary conditions gives the hydrogen energy levels:

Eₙ = −13.6/n² eV

where n = 1, 2, 3, ... is the principal quantum number. This expression is an approximation that neglects fine structure and other small corrections.

7. Quantum Numbers and Degeneracy

The hydrogen atom is described by three spatial quantum numbers:

• Principal quantum number: n = 1, 2, 3, ...
• Orbital quantum number: l = 0, 1, ..., n − 1
• Magnetic quantum number: m = −l, ..., +l

For a given principal quantum number, the number of spatial states is:

gₙ = n²

This degeneracy arises because the nonrelativistic hydrogen energy depends only on n.

8. Ground-State Wavefunction

For the ground state, n = 1, l = 0, and m = 0. The normalized wavefunction is:

ψ₁₀₀(r) = (1/√πa₀³)e^(−r/a₀)

Here, a₀ is the Bohr radius, approximately 0.529 Å for hydrogen when the proton's finite mass is neglected.

9. Spectral Transitions

When an electron transitions between energy levels, a photon is emitted or absorbed. Its energy is:

hν = |Eᵢ − E𝒻|

For electric-dipole transitions, the orbital selection rules are:

Δl = ±1

Δm = 0, ±1

Conclusion

Angular momentum provides the mathematical foundation for understanding atomic orbitals and rotational symmetry. Its application to the hydrogen atom explains the quantization of energy levels and the structure of atomic wavefunctions. These concepts form the basis for further studies of spin, fine structure, and atomic spectroscopy.

References

• J. J. Sakurai and Jim Napolitano, Modern Quantum Mechanics.
• R. Shankar, Principles of Quantum Mechanics.
• David J. Griffiths and Darrell F. Schroeter, Introduction to Quantum Mechanics.

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