Question

The kinetic energy of hydrogen atom wave functions for which $E$ is its minimum value of 0 is all radial. This is the case for the $\mathrm{Is}$ and 2 r states. The $2\mathrm{p}$ state has some rotational kinetic energy and some radial. Show that for very large $n$, the states of largest allowed $\ell$ have essentially no radial kinetic energy. Exercise 55 notes that the expectation value of the kinetic energy (including both rotational and radial) equals the magnitude of the total energy. Compare this magnitude with the rotational energy alone, $L^2/2m{r}^2$. assuming that $n$ is large. that $\ell$ is as large as it can be, and that $r\equiv n^2{a}_0$.

Short answer

When $n$ is extremely large and $\ell =n-1$, we found that the rotational kinetic energy $T_{\text{rot}}$ is not equal to the total kinetic energy $T_{\text{kinetic}}$. This shows that the kinetic energy isn't entirely due to rotation, suggesting that there is some radial kinetic energy present. Therefore, the claim that the states of largest allowed $\ell$ have no radial kinetic energy is not valid in this context.

Step-by-step solution

Referring to relevant equations

First, recall the equation for the rotational kinetic energy: $$T_{\text{rot}}=\frac{L^2}{2m{r}^2}$$ where $L$ is the angular momentum, $m$ is the mass of the electron, and $r$ is the radius. Knowing that $L=\hslash \sqrt{\ell (\ell +1)}$ and that $r=n^2{a}_0$, where $a_0$ is the Bohr radius, we can insert these values into the equation: $$T_{\text{rot}}=\frac{{\hslash }^2\ell (\ell +1)}{2m(n^2{a}_0{)}^2}$$ Because $\ell$ can be as large as $n-1$ for a hydrogen atom (if n is large), we replace $\ell$ by $n-1$ to get: $$T_{\text{rot}}=\frac{{\hslash }^2(n-1)n}{2m(n^2{a}_0{)}^2}$$ This equation gives us the rotational kinetic energy when $\ell$ is at its maximum.

Doing further simplifications

First, simplify $T_{\text{rot}}$: $$T_{\text{rot}}=\frac{{\hslash }^2{n}^2-{\hslash }^{2}n}{2m(n^2{a}_0{)}^2}$$ $$T_{\text{rot}}=\frac{{\hslash }^{2}n}{2m{a}_0^2}(1-\frac{1}{n})$$ For large $n$, $\frac{1}{n}\approx 0$, so: $$T_{\text{rot}}\approx \frac{{\hslash }^{2}n}{2m{a}_0^2}$$ This simplification shows that the rotational kinetic energy $T_{\text{rot}}$ becomes directly proportional to $n$ when $n$ is extremely large and $\ell =n-1$.

Comparing rotational kinetic energy to total energy

Next, remember that the expectation value of the kinetic energy equals the magnitude of the total energy: $$|E_{\text{total}}|=T_{\text{kinetic}}$$ The total energy of a hydrogen atom in any state is given by: $$E_{\text{total}}=-\frac{13.6\text{eV}}{n^2}$$ Therefore, the expectation value of the kinetic energy is equal to the absolute value of $E_{\text{total}}$: $$T_{\text{kinetic}}=|E_{\text{total}}|=\frac{13.6\text{eV}}{n^2}$$ With $T_{\text{rot}}$ obtained in Step 2, we see that $T_{\text{kinetic}}\ne T_{\text{rot}}$ which means the total kinetic energy isn't entirely due to rotation when $n$ is large and $\ell =n-1$. This is contrary to the initial claim that the states of largest allowed $\ell$ have essentially no radial kinetic energy.

Key concepts

Rotational Kinetic Energy

In physics, the concept of 'rotational kinetic energy' refers to the energy an object possesses due to its rotation. It's a component of the total kinetic energy and is particularly essential in systems where objects spin about an axis.

For a hydrogen atom's electron, when we think about rotational kinetic energy, we're considering the electron's motion along a circular path around the nucleus. The faster the electron spins, or the larger the path it takes, the greater this energy becomes. In quantum mechanics, this is quantified using the angular momentum quantum number 'l'.

The kinetic energy associated with the rotation of an electron in a hydrogen atom is given by the formula: $$T_{\text{rot}}=\frac{L^2}{2m{r}^2}$$ where 'L' stands for angular momentum, 'm' is the mass of the electron, and 'r' is the radius of the electron's orbit. As per the quantum mechanical treatment, L is quantized and depends on the integer 'l', presenting a discrete spectrum for 'T_rot'. When 'n' is large, 'l' can be as large as 'n-1', leading to the electrons in these states having negligible radial kinetic energy and primarily possessing rotational kinetic energy.

Quantum Mechanics

Quantum mechanics is a foundational pillar of modern physics that explains the behavior of particles at the atomic and subatomic levels. It departs from classical mechanics by introducing the quantization of energy, probabilistic nature of particle properties, and wave-particle duality.

In the context of a hydrogen atom, quantum mechanics describes the electron's behavior and interactions using wave functions. These functions determine the probability of finding an electron in a particular region of space. The total energy of an electron in a hydrogen atom is expressed with quantum numbers, which include the principal quantum number (n), angular momentum quantum number (l), and magnetic and spin quantum numbers.

Understanding how quantum mechanics governs the hydrogen atom's electron provides insights into the electron's allowed energy levels and the shapes of the atomic orbitals. It explains why certain states have specific energy values, such as why the 1s and 2r states have minimal energy and why, at very high 'n' levels, the rotational kinetic energy dominates the electron's behavior.

Angular Momentum

Angular momentum is a crucial concept in both classical and quantum mechanics, related to the rotation or circular motion of an object. In quantum mechanics, the angular momentum of particles like electrons is quantized, meaning it can only take on certain discrete values.

For an electron in an atom, the angular momentum depends on the angular momentum quantum number 'l', which determines the shape of the orbital and the electron's rotation around the nucleus. The allowed values of 'l' range from 0 to 'n-1', with 'n' being the principal quantum number associated with energy levels of the electron.

As an exercise in hydrogen atom wave functions exemplifies, for large 'n' values, the states with the largest allowed 'l' value exhibit significant rotational kinetic energy compared to radial. The formula $L=\hslash \sqrt{l(l+1)}$ is used to calculate the magnitude of the angular momentum, illustrating the relationship between 'l' and the rotational characteristics of an electron within an atom.