Questions: Zeeman Effect: Magnetic Field Splitting of Energy Levels

Question 1

An atomic state has total angular momentum quantum number j = 3/2. How many distinct energy sublevels does it split into when placed in an external magnetic field?

• A. 3 sublevels, one for each spatial dimension
• B. 4 sublevels, corresponding to m_j = −3/2, −1/2, +1/2, +3/2
• C. 6 sublevels, equal to twice the value of j
• D. 2 sublevels, corresponding to spin-up and spin-down only

Answer: (B)

A state with total angular momentum quantum number j splits into 2j+1 sublevels in a magnetic field, corresponding to the allowed values of m_j: −j, −j+1, …, +j. For j = 3/2, this gives 2(3/2)+1 = 4 sublevels. Option D (just spin up/down) confuses j with spin quantum number s = 1/2. The 2j+1 formula directly reflects space quantization — the angular momentum can only point in 2j+1 discrete directions relative to the field axis.

Question 2

A spectroscopist observes a spectral line splitting into more than three components in a magnetic field. The atom has nonzero electron spin. What is the most likely explanation?

• A. The field is strong enough that the energy shifts become proportional to B², adding extra sublevels
• B. The anomalous Zeeman effect — nonzero spin means the Landé g-factor differs from 1, causing different levels in the multiplet to shift by different amounts
• C. The photon carries spin-1, which contributes additional magnetic quantum numbers to the splitting
• D. More than three components indicate an experimental error, since selection rules always limit splitting to three lines

Answer: (B)

When electron spin is nonzero, orbital and spin angular momenta couple into total angular momentum J, with an effective magnetic moment governed by the Landé g-factor g_J = 1 + [J(J+1)+S(S+1)−L(L+1)]/[2J(J+1)]. Because g_J ≠ 1, different levels shift by different amounts, producing complex patterns — the 'anomalous' Zeeman effect. The normal effect (three lines) applies only when S=0 and g=1. A strong-field quadratic shift (option A) is a separate phenomenon (Paschen-Back limit) not expected at moderate fields.

Question 3

In the normal Zeeman effect, a single spectral line splits into exactly three lines due to the selection rule Δm_ℓ = 0, ±1.

• T. True
• F. False

Answer: True

In the normal Zeeman effect (S=0, so g=1), transitions between levels with Δm_ℓ = 0, +1, and −1 each produce a distinct photon frequency shifted by ±μ_B·B from the original line or unchanged. The three groups of transitions all merge into just three lines regardless of the ℓ values involved — a striking simplicity that historically made the normal effect easier to interpret than the anomalous case.

Question 4

The energy shift of an atomic sublevel in the Zeeman effect is proportional to the square of the external magnetic field strength.

• T. True
• F. False

Answer: False

The Zeeman energy shift is ΔE = g_J m_J μ_B B — linear in B, not B². This is the linear (or first-order) Zeeman effect. A quadratic dependence on B arises only in the quadratic Zeeman effect, which is a much smaller second-order correction relevant in extremely strong fields or for very weakly bound states. The linear dependence is what makes Zeeman splitting so useful for measuring magnetic fields: the splitting directly and simply encodes field strength.

Question 5

Why does the anomalous Zeeman effect produce more complex splitting patterns than the normal Zeeman effect? What role does the Landé g-factor play?

Model answer: In the anomalous Zeeman effect, orbital and spin angular momenta combine into total angular momentum J, and the electron's spin magnetic moment has g_s ≈ 2 rather than 1. This means the effective magnetic moment is not simply proportional to the angular momentum magnitude. The Landé g-factor g_J accounts for the mixture of orbital and spin contributions; it differs between different J levels of a multiplet. Because different levels shift by different amounts (g_J varies), the resulting spectral pattern is more complex than the simple three-line pattern of the normal effect.

In the normal Zeeman effect (S=0), g=1 for all levels, so the energy spacing between sublevels is the same in upper and lower states, and transitions neatly produce three lines. In the anomalous case, g_J depends on the specific values of L, S, and J for each level. Upper and lower states have different g_J values, so their sublevel spacings differ. Transitions between all pairs of m_J values that satisfy selection rules then produce a forest of lines at different frequencies, which was historically called 'anomalous' because it defied classical explanation — until electron spin and its g ≈ 2 were understood.