Perturbation Theory in the Linear Stark Effect for Hydrogen

ABSTRACT:
The Stark effect is the effect of a background electric field that shifts and distorts quantized energy states of atoms and molecules. It is observed through shifts and splits in spectral lines. This paper will cover the linear stark effect, thus using hydrogen's second energy level as the simplest level with degenerate subspaces. The perturbation can be observed by observing the splitting of spectral lines, as the perturbation would distort subspaces from their unperturbed, degenerate symmetry. Due to the quantum scale of this perturbation, it is used in high-precision instruments and tests.

1. Intro

The Stark effect is the shifting and splitting of the spectral lines of atoms and molecules in the presence of a background electric field. The field adds a new layer of complexity to the quantized energy states of electrons within these clouds. As the atomic structure still remains mostly unchanged, the field, intuitively, induces a dipole moment onto the atom, effectively polarizing it. The effects of this polarization are perturbations of quantized energy levels for electrons. Therefore, a new component will be incorporated into the Schrödinger equation for the electrons. This paper will primarily focus on the electron cloud for hydrogen. 

2. Theory

Once a constant electric field along the z-axis is applied to the electron in a hydrogen atom, the Hamiltonian becomes , while the unperturbed hydrogen electron is given by the Hamiltonian. The Stark perturbation then comes from the term eϵz, governed by the electron charge e, electric field strength ϵ, and the positional operator z.

2.1 Solving the Unperturbed Time-Independent Schrodinger Equation

To begin, the unperturbed Hamiltonian is used to solve the time-independent Schrodinger equation (TISE) before accounting for the added complexity of a background electric field.

Given that the potential term  is only dependent on r, which will be simply labeled as V (r), the wavefunction would be solved in polar coordinates. Therefore, the Laplacian operator would be                

To solve for the wavefunction ψ, a separation of variables is assumed to set ψ(r, θ, ϕ) = R(r)Y (θ, ϕ) with R(r) as the radial part and Y (θ, ϕ) as the angular part. After substituting ψ = RY and dividing through by RY , the equation rearranges to become

The orbital quantum number "l" first appears here, coming from the equation for angular momentum (consider l as determining s, p, d, and f orbitals in chemistry). Y turns out to be spherical harmonics, which gives a normalized             

Here m is defined as the magnetic quantum number, determining Lz (m can be considered as determining the orientation of orbitals in space).

To solve for the radial equation, it is helpful to substitute u(r) = rR(r), which results in

for u(0) = 0 and u(r) → 0 as r → ∞.

Solving for the differential equation results in                                      

where n is a positive integer corresponding to energy levels in atoms and molecules.

2.2 Solving for Perturbation

Assuming that the electric background remains constant, the TISE Hψ = Eψ is applicable as the Hamiltonian is plugged in.                                 

By removing the unperturbed Hamiltonian, the focus is centered on electric perturbation.

2.2.1 Linear Effect in n=2 Degenerate Subspace 

Moving on, using the quantum states |n, l, m⟩ would help differentiate the different quantum electron states and how they are affected at the first order. In hydrogen’s second energy level n = 2, there are four states which will be considered degenerate when unperturbed.

Angular selection rules also determine that ∆l = ±1 and ∆m = ±0. With limitations to ∆m and the odd property of z, the matrix becomes

To finish the matrix, ⟨2, 0, 0|z|2, 1, 0⟩ = 3a0, where a0 is the Bohr radius. The effective 2x2 Hamiltonian is in the {|2, 0, 0⟩|2, 1, 0⟩} subspace as

3 Solving for Linear Perturbation

Given the TISE, Hψ = Eψ, the first-order eigenvalues are 

The two states with a non-zero m value do not undergo first-order coupling and therefore do not experience linear shifting.

4 Physical implications

At the n=2 energy level, all of hydrogen’s subspaces were originally degenerate, having the same energy regardless of subspaces. However in the presence of an electric field, the pair of m=0 states couple, producing a linear Stark split ∆E ∝ E. Spectroscopy is a way to measure the energy difference between different energy states, and by observing the spectral lines characterized by the electric perturbation, it can be observed that the spherical symmetry of the electron cloud is broken. The broken symmetry agrees with the classical notion that a background electric field would tilt the potential landscape of a charged particle. Effectively, the hydrogen atom’s electron cloud is polarized, inducing a permanent dipole moment aligning with or against the field.

4.1 Practical applications

In real-world applications, a straightforward purpose for this shift is to measure unknown electric field strengths by observing spectral lines. Known as Stark spectroscopy, it is used in plasma diagnostics, astrophysics, and atomic beam experiments. The Stark effect has also seen use in precision tests for physics, observed for violations in parity or time as any deviation from ±3eϵa0 could indicate unexplored parts of physics.