Consider two levels of the helium atom in both of which the...

Consider two levels of the helium atom in both of which the spins are antiparallel and one electron is in an $s$ state $(l=0)$. In the higher level the second electron occupies a $d$ state $(l=2)$, and in the lower level it occupies a $p$ state $(l=1)$. (a) Sketch the splitting of both levels resulting from a magnetic field along the $z$ axis. (b) Imagine a transition from one of the $d$ states, with $L_{z}=m_{i} \hbar$, to one of the $p$ states, with $L_{z}=m_{f} \hbar$. Since $m_{i}$ can be $2,1,0,-1$, or $-2$ and $m_{f}$ can be 1,0 , or $-1$, there are $5 \times 3$ or 15 distinct conceivable transitions. How many different photon energies would these 15 transitions produce? (c) Not all of these 15 transitions occur. In fact, it is found that the only transitions observed are those for which $$ \left(m_{f}-m_{i}\right)=1, \quad \text { or } 0, \quad \text { or }-1 \quad(9.36) $$ (A restriction like this on the transitions that take place is called a selection rule, as we discuss in Chapter 11.) Prove that because of the restriction (9.36), there are only three distinct photon energies produced in all possible transitions. (This means that the normal Zeeman effect always produces just three spectral lines, however large the angular momenta involved.)

Consider two levels of the helium atom in both of which the spins are antiparallel and one electron is in an $s$ state $(l=0)$. In the higher level the second electron occupies a $d$ state $(l=2)$, and in the lower level it occupies a $p$ state $(l=1)$. (a) Sketch the splitting of both levels resulting from a magnetic field along the $z$ axis. (b) Imagine a transition from one of the $d$ states, with $L_{z}=m_{i} \hbar$, to one of the $p$ states, with $L_{z}=m_{f} \hbar$. Since $m_{i}$ can be $2,1,0,-1$, or $-2$ and $m_{f}$ can be 1,0 , or $-1$, there are $5 \times 3$ or 15 distinct conceivable transitions. How many different photon energies would these 15 transitions produce? (c) Not all of these 15 transitions occur. In fact, it is found that the only transitions observed are those for which
$$
\left(m_{f}-m_{i}\right)=1, \quad \text { or } 0, \quad \text { or }-1 \quad(9.36)
$$
(A restriction like this on the transitions that take place is called a selection rule, as we discuss in Chapter 11.) Prove that because of the restriction (9.36), there are only three distinct photon energies produced in all possible transitions. (This means that the normal Zeeman effect always produces just three spectral lines, however large the angular momenta involved.)

Key Concepts

Zeeman Effect

The Zeeman effect is the phenomenon where spectral lines split into multiple components when an atom is placed in an external magnetic field. This splitting occurs because the magnetic field interacts with the magnetic dipole moments associated with the orbital angular momenta of the electrons. The effect can lead to a splitting of energy levels that depends on the magnetic quantum number, thereby influencing the transition energies between states.

Angular Momentum Quantization

In quantum mechanics, the orbital angular momentum of electrons in an atom is quantized, meaning electrons can only occupy states with specific angular momentum values characterized by the quantum number l. This quantization is fundamental in determining how the electrons’ energy levels split in a magnetic field, as well as the allowed transitions between these levels according to the differences in their angular momentum projections.

Magnetic Quantum Number

The magnetic quantum number, m, specifies the component of the orbital angular momentum along a chosen quantization axis, typically designated as the z-axis. In the presence of a magnetic field, the energy of an electron state is dependent on the value of m, causing different m states to shift by different amounts. This results in the fine splitting of energy levels observed in the Zeeman effect.

Selection Rules

Selection rules are criteria that dictate the allowed transitions between quantum states based on conservation laws and the symmetry of the system. In the context of atomic transitions influenced by a magnetic field, the selection rules, such as ?m = 0, ±1, restrict the changes in the magnetic quantum number during photon emission or absorption. This restriction is critical for understanding which transitions occur and how many distinct photon energies result from these transitions.

Photon Energy Differences

The energy of a photon emitted or absorbed during a transition between two atomic states is determined by the difference in energy levels, which, under a magnetic field, is influenced by the magnetic quantum number of the states involved. Due to the splitting of energy levels into sub-levels with different m values, the allowed transitions (as defined by the selection rules) produce specific and discrete energy differences. This concept explains why, even though multiple transitions may be conceivable, selection rules can constrain the system to yield only a few distinct photon energies.

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