Animated Solution for Physics - Atoms and Nuclei: Radiation coming from transitions $n=2$ to $n=1$ of hydrogen atoms fall on ${\text{He}}^{+}$ ions in $n=1$ and $n=2$ states. The possible transition of helium ions as they absorb energy from the radiation is

A Tale of Two Atoms

Imagine a microscopic laboratory where we have isolated two distinct quantum systems: a neutral hydrogen atom and a singly ionized helium atom (${\text{He}}^{+}$). The hydrogen atom is in an excited state and is about to undergo an electron transition. As its electron falls to a lower energy orbit, it will release a burst of energy in the form of a photon.

This photon will then travel through space and strike the helium ion. Our mission is to determine exactly how the helium ion reacts to this incoming radiation. Will it ignore the photon, or will it absorb it and jump to a higher energy state? To solve this, we must act as quantum detectives and calculate the exact energy exchange.

Calculating the Photon's Energy

First, we need to find out exactly how much energy the hydrogen atom releases. When an electron jumps from a higher orbit ($n_2$) to a lower orbit ($n_1$), it loses energy. This lost energy is emitted as a single photon. The energy of this photon is governed by the famous Rydberg energy formula:

The problem explicitly states that the transition in the hydrogen atom is from the second orbit ($n=2$) down to the ground state ($n=1$). Let's carefully substitute these values into our master equation:

This calculation reveals that the hydrogen atom fires a photon with a precise energy of $10.2\text{ eV}$. This photon is now hurtling towards the helium ion.

Mapping the Energy Levels

To know what happens when the photon hits the helium ion, we must deeply understand the energy architecture of ${\text{He}}^{+}$. Unlike hydrogen, which has only one proton, helium has two protons in its nucleus. This means its atomic number $Z$ is $2$.

Because of this stronger nuclear pull, the electron is bound much more tightly. The energy formula for a hydrogen-like ion is modified by a factor of $Z^2$:

Substituting $Z=2$, the numerator becomes $13.6\times 4=54.4$. So, the specific energy formula for the helium ion is:

Let's quickly calculate the first few energy levels to build our quantum ladder. These are the only allowed "landing spots" for an electron in the helium ion:

Ground State ($n=1$): $E_1=-\frac{54.4}{1^2}=-54.4\text{ eV}$ First Excited State ($n=2$): $E_2=-\frac{54.4}{2^2}=-13.6\text{ eV}$ Second Excited State ($n=3$): $E_3=-\frac{54.4}{3^2}=-6.04\text{ eV}$ Third Excited State ($n=4$): $E_4=-\frac{54.4}{4^2}=-3.4\text{ eV}$

Testing for Resonance

In quantum mechanics, absorption is an "all-or-nothing" process. An atom will only absorb a photon if the photon's energy perfectly matches the exact gap between the electron's current state and a higher allowed state. This is known as the resonance condition.

The question states that the helium ions in our sample are present in both $n=1$ and $n=2$ states. We must test both scenarios.

If an electron sitting in the ground state ($E_1=-54.4\text{ eV}$) absorbs the $10.2\text{ eV}$ photon, its new energy would be:

We must now check our calculated ladder. Is there an energy level at $-44.2\text{ eV}$? No. Therefore, an electron in the $n=1$ state cannot absorb this photon. It's a quantum mismatch.

Now let's test the $n=2$ state. The initial energy of an electron here is $E_2=-13.6\text{ eV}$. If it successfully absorbs the photon, its energy increases:

Does this energy level exist? Yes, it absolutely does! If you look at our ladder, $-3.4\text{ eV}$ corresponds exactly to the $n=4$ state.

We have found a perfect resonance. The electron in the $n=2$ state of the helium ion perfectly absorbs the $10.2\text{ eV}$ photon and makes a quantum jump directly to the $n=4$ state.

Therefore, the correct possible transition for the helium ion is $n=2\to n=4$. This problem beautifully illustrates the strict, quantized nature of atomic energy levels and how light and matter interact in perfect mathematical harmony.