Chapter 4: Problem 13

Show that the expectation value of the potential energy of an electron in the $n$ th quantum state of a hydrogen atom is $-Z^{2} e^{2} / a_{0} n^{2} .$ From this result, find the expectation value of the kinetic energy.

Short Answer

Expert verified

The potential energy expectation value is $ -\frac{Z^2 e^2}{a_0 n^2} $, and the kinetic energy expectation value is $ \frac{Z^2 e^2}{2 a_0 n^2} $.

Step by step solution

Understand the Problem

We need to find the expectation values of the potential energy and kinetic energy for the electron in the nth quantum state of a hydrogen atom.

Recall the Potential Energy in a Hydrogen Atom

The potential energy for an electron in a hydrogen atom is given by $V(r) = -\frac{Z e^2}{r}$, where $Z$ is the atomic number, $e$ is the electron charge, and $r$ is the radial distance from the nucleus.

Use the Radial Wavefunction

In the nth quantum state, the radial wavefunction $R_{n}(r)$ for a hydrogen atom is used to find the expectation value of the potential energy. This will involve integrating over the probability density.

Compute the Expectation Value

The expectation value of the potential energy $ \langle V \rangle $ is calculated using: \[ \langle V \rangle = \int_{0}^{\infty} R_{n}^{*}(r) V(r) R_{n}(r) r^2 dr \] Substituting $V(r) = -\frac{Z e^2}{r}$, we can show that: \[ \langle V \rangle = -\frac{Z^2 e^2}{a_{0} n^2} \] where $a_{0}$ is the Bohr radius.

Use the Virial Theorem

The Virial theorem states that for a hydrogen-like atom, the expectation value of the kinetic energy $T$ is given by \[ \langle T \rangle = -\frac{1}{2} \langle V \rangle \]. Using the result from Step 4: \[ \langle T \rangle = -\frac{1}{2} \left( -\frac{Z^2 e^2}{a_{0} n^2} \right) = \frac{Z^2 e^2}{2 a_{0} n^2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

hydrogen atom

The hydrogen atom consists of a single electron orbiting a single proton within the nucleus. In quantum mechanics, the behavior of the electron is described by the Schrödinger equation, which accounts for its wave-like nature. The electron's allowed energy states are quantized, meaning it can only occupy certain discrete energy levels, denoted by quantum numbers such as $n$ (the principal quantum number). These energy levels determine the electron's behavior and interactions, including its potential and kinetic energy.

potential energy

In the context of a hydrogen atom, the potential energy of the electron is associated with its position relative to the nucleus. It's given by the Coulomb attraction between the negatively charged electron and positively charged proton. Mathematically, this is expressed as $V(r) = -\frac{Z e^2}{r}$, where $Z$ is the atomic number (1 for hydrogen), $e$ is the electron charge, and $r$ is the radial distance from the nucleus. This negative potential energy indicates an attractive force pulling the electron toward the nucleus.

kinetic energy

The kinetic energy of the electron in a hydrogen atom is related to its motion. According to the Virial theorem in quantum mechanics, the kinetic energy and potential energy are related in bound systems like the hydrogen atom. The theorem states that for a system bound by inverse square forces (like the Coulomb force), the expectation value of the kinetic energy $T$ is half the magnitude of the potential energy $V$: \ \ \raggedright $\langle T \rangle = -\frac{1}{2} \langle V \rangle$. Using this, once the potential energy is known, we can easily find the kinetic energy.

radial wavefunction

The radial wavefunction $R_{n}(r)$ describes the probability distribution of the electron's position in the hydrogen atom. It depends on the principal quantum number $n$. To find the expectation value of the potential energy, we integrate the product of the radial wavefunction, the potential energy function, and the radial distance squared over all space: \ \ $$\langle V \rangle = \int_{0}^{\infty} R_{n}^{*}(r) V(r) R_{n}(r) r^2 dr$$. This integral accounts for all possible positions of the electron, weighted by the likelihood of finding it at each point.

Virial theorem

The Virial theorem provides an important relation between potential and kinetic energy for bound systems. For a hydrogen-like atom, it tells us that the expectation value of the potential energy $V$ and kinetic energy $T$ are related by: \ \ \raggedright $\langle T \rangle = -\frac{1}{2} \langle V \rangle$. This means that once the potential energy is determined, the kinetic energy can be calculated directly. Applying this theorem simplifies the process of finding the kinetic energy of an electron in a hydrogen atom, making it a powerful tool in quantum mechanics.

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Show that the expectation value of the potential energy of an electron in the \(n\) th quantum state of a hydrogen atom is \(-Z^{2} e^{2} / a_{0} n^{2} .\) From this result, find the expectation value of the kinetic energy.

Short Answer

Step by step solution

Understand the Problem

Recall the Potential Energy in a Hydrogen Atom

Use the Radial Wavefunction

Compute the Expectation Value

Use the Virial Theorem

Key Concepts

hydrogen atom

potential energy

kinetic energy

radial wavefunction

Virial theorem

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