Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze ar

Q: Why do we use the approximation $n \gg 1$?

The problem specifically mentions "highly excited states" or Rydberg states, where the electron is very far from the nucleus. In this regime, $n$ is a very large number, so terms like $1/n^2$ become negligible compared to $1/n$, allowing us to simplify the expressions significantly.

Q: Why does the relative change in radius not depend on $Z$?

The radius $r_n$ is proportional to $n^2/Z$. When we calculate the relative change $\frac{\Delta r}{r_n}$, the factor of $1/Z$ is present in both the numerator (the difference in radii) and the denominator (the initial radius). Thus, it cancels out completely.

Q: I got the relative change in energy as $1/n^3$. Where did I go wrong?

A common mistake is to just differentiate $E_n \propto 1/n^2$ to get $dE \propto -2/n^3 dn$. While the absolute change $\Delta E$ is indeed proportional to $1/n^3$, the *relative* change is $\Delta E / E_n$. Dividing $1/n^3$ by $1/n^2$ gives $1/n$.

Q: Is the relative change in angular momentum exact or an approximation?

It is exact! Since $L_n = \frac{nh}{2\pi}$, the difference $\Delta L = \frac{h}{2\pi}$. The relative change is exactly $\frac{\Delta L}{L_n} = \frac{1}{n}$. No approximation is needed here.

Q: Do these proportionalities hold for multi-electron atoms?

The problem specifies "hydrogen-like atoms," which means there is only one electron. For multi-electron atoms, shielding effects come into play, and the simple Bohr proportionalities do not hold perfectly, though Rydberg states in multi-electron atoms often behave very similarly to hydrogen-like atoms because the inner electrons effectively shield the nucleus, making the effective nuclear charge close to $+1e$.

Question

Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principle quantum number n, where n>>1. Which of the following statement(s) is (are) true?

Options:

(A) Relative change in the radii of two consecutive orbitals does not depend on Z

(B) Relative change in the radii of two consecutive orbitals varies as $1/ n$

(C) Relative change in the energy of two consecutive orbitals varies as $1/ n^3$

(D) Relative change in the angular momenta of two consecutive orbitals varies as $1/ n$

Sigma Percentile · Accepted Answer

### The Realm of Rydberg States
Imagine an atom where the electron is pushed so far away from the nucleus that it almost forgets it's bound to it. These are **Rydberg states**, characterized by a very large principal quantum number ($n \gg 1$). In this expansive quantum playground, the classical and quantum worlds begin to blur, and we can use powerful approximations to understand how physical quantities change as we hop from one orbit to the next.

### Analyzing the Orbital Radius
Let's start with the size of these orbits. According to Bohr's model, the radius of the $n^{	ext{th}}$ orbit is given by the proportionality:
$$r_n \propto \frac{n^2}{Z}$$
To find the **relative change** in radius between two consecutive orbitals ($n$ and $n+1$), we calculate:
$$\frac{\Delta r}{r_n} = \frac{r_{n+1} - r_n}{r_n}$$
Substituting our proportionality, the atomic number $Z$ beautifully cancels out:
$$\frac{\Delta r}{r_n} = \frac{(n+1)^2 - n^2}{n^2} = \frac{2n + 1}{n^2}$$
Since $n$ is incredibly large ($n \gg 1$), the $+1$ in the numerator becomes negligible compared to $2n$. Thus, the expression simplifies to:
$$\frac{\Delta r}{r_n} \approx \frac{2n}{n^2} = \frac{2}{n}$$
This elegant result tells us two things: the relative change is completely **independent of $Z$**, and it **varies as $1/n$**.

### The Energy Landscape
Next, we dive into the energy of these states. The energy of an electron in the $n^{	ext{th}}$ orbit is negative (indicating a bound state) and follows the relation:
$$E_n \propto -\frac{Z^2}{n^2}$$
The relative change in energy is the magnitude of the difference divided by the initial energy:
$$\left| \frac{\Delta E}{E_n} ight| = \left| \frac{E_{n+1} - E_n}{E_n} ight|$$
Again, the $Z^2$ terms vanish, leaving us with:
$$\left| \frac{\Delta E}{E_n} ight| = \frac{\frac{1}{n^2} - \frac{1}{(n+1)^2}}{\frac{1}{n^2}} = 1 - \frac{n^2}{(n+1)^2} = \frac{(n+1)^2 - n^2}{(n+1)^2} = \frac{2n+1}{(n+1)^2}$$
Applying our large $n$ approximation, the numerator is roughly $2n$ and the denominator is roughly $n^2$:
$$\left| \frac{\Delta E}{E_n} ight| \approx \frac{2n}{n^2} = \frac{2}{n}$$
Fascinatingly, the relative change in energy also **varies as $1/n$**, not $1/n^3$!

### Angular Momentum: The Simplest of All
Finally, let's look at the angular momentum, which Bohr famously quantized as:
$$L_n = \frac{nh}{2\pi} \propto n$$
The relative change here requires no approximations. It is exactly:
$$\frac{\Delta L}{L_n} = \frac{(n+1) - n}{n} = \frac{1}{n}$$
The relative change in angular momentum perfectly **varies as $1/n$**.

### Conclusion
By exploring the mathematics of Rydberg states, we've uncovered a beautiful symmetry: the relative changes in radius, energy, and angular momentum all scale as $1/n$. This harmony is a hallmark of highly excited quantum systems!

Animated Solution for Physics - Atoms and Nuclei: Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principle quantum number n, where n>>1. Which of the following statement(s) is (are) true?

Select Answer:

Visualized Solution

Radius of $n^{th}$ orbit

Relative change in radius

Energy of $n^{th}$ orbit

Relative change in energy

Angular momentum

Relative change in angular momentum

Conclusion

The Sigma Insight: Bohr's Atomic Model and Energy Levels

The Realm of Rydberg States Imagine an atom where the electron is pushed so far away from the nucleus that it almost forgets it's bound to it

Analyzing the Orbital Radius Let's start with the size of these orbits

The Energy Landscape Next, we dive into the energy of these states

Angular Momentum

Conclusion By exploring the mathematics of Rydberg states, we've uncovered a beautiful symmetry: the relative changes in radius, energy, and angular momentum all scale as $1/ n$

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