Question

The wave function for the \(2 p_{z}\) orbital in the hydrogen atom is $$ \psi_{2 p_{i}}=\frac{1}{4 \sqrt{2 \pi}}\left(\frac{Z}{a_{0}}\right)^{3 / 2} \sigma \mathrm{e}^{-\alpha / 2} \cos \theta $$ where \(a_{0}\) is the value for the radius of the first Bohr orbit in meters \(\left(5.29 \times 10^{-11}\right), \sigma\) is \(Z\left(r / a_{0}\right), r\) is the value for the distance from the nucleus in meters, and \(\theta\) is an angle. Calculate the value of \(\psi_{2 p^{2}}\) at \(r=a_{0}\) for \(\theta=0^{\circ}\left(z \text { axis) and for } \theta=90^{\circ}\right.\) (xy plane).

Short answer

The value of the wave function \(\psi_{2p_{z}}\) at \(r=a_{0}\) and \(\theta=0^{\circ}\) is \(\frac{1}{4 \sqrt{2 \pi}}\left(\frac{1}{5.29 \times 10^{-11}}\right)^{3 / 2} \frac{r}{a_0} \mathrm{e}^{-\alpha / 2}\), and the value at \(r=a_{0}\) and \(\theta=90^{\circ}\) is 0.

Step-by-step solution

Write down the given wave function equation

The wave function for the 2pz orbital in the hydrogen atom is given as: \[ \psi_{2 p_{z}}=\frac{1}{4 \sqrt{2 \pi}}\left(\frac{Z}{a_{0}}\right)^{3 / 2} \sigma \mathrm{e}^{-\alpha / 2} \cos \theta \]

Plug in the values of r, a0, and Z

It is given that: \(r = a_{0}\), \(a_{0} = 5.29 \times 10^{-11} \:m\), \(\sigma = Z\left(r / a_{0}\right)\), Since it is a hydrogen atom, Z=1. Then, \(\sigma = \frac{r}{a_0}\) Plug these values into the wave function equation: \[ \psi_{2 p_{z}}=\frac{1}{4 \sqrt{2 \pi}}\left(\frac{1}{5.29 \times 10^{-11}}\right)^{3 / 2} \frac{r}{a_0} \mathrm{e}^{-\alpha / 2} \cos \theta \]

Calculate the value of the wave function at \(\theta=0^{\circ}\)

Plug the value of \(\theta=0^{\circ}\) into the equation and solve for the value of the wave function: \[ \psi_{2p_{z}} = \frac{1}{4\sqrt{2\pi}}\left(\frac{1}{5.29\times10^{-11}}\right)^{3/2} \frac{r}{a_0} \mathrm{e}^{-\alpha/2} \cos(0^{\circ}) \] Since \(\cos(0^{\circ}) = 1\), \[ \psi_{2 p_{z}}=\frac{1}{4 \sqrt{2 \pi}}\left(\frac{1}{5.29 \times 10^{-11}}\right)^{3 / 2} \frac{r}{a_0} \mathrm{e}^{-\alpha / 2} \] Now calculate the value of the wave function \(\psi_{2p_{z}}\) at \(r=a_{0}\) and \(\theta=0^{\circ}\).

Calculate the value of the wave function at \(\theta=90^{\circ}\)

Plug the value of \(\theta=90^{\circ}\) into the equation and solve for the value of the wave function: \[ \psi_{2 p_{z}}=\frac{1}{4 \sqrt{2 \pi}}\left(\frac{1}{5.29 \times 10^{-11}}\right)^{3 / 2} \frac{r}{a_0} \mathrm{e}^{-\alpha / 2} \cos(90^{\circ}) \] Since \(\cos(90^{\circ}) = 0\), \[ \psi_{2 p_{z}} = 0 \] So, the value of the wave function \(\psi_{2 p_{z}}\) at \(r=a_{0}\) and \(\theta=90^{\circ}\) is 0.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation for understanding the behavior and interaction of matter and energy at the quantum level, and it significantly departs from classical mechanics by introducing new phenomena such as quantization, wave-particle duality, and uncertainty.

At the heart of quantum mechanics is the concept of the quantum state, represented by a wave function. This wave function provides the probabilities of the outcomes of measurements on a system, rather than determining the outcomes with certainty as in classical physics. Understanding wave functions and how they describe the quantum state of a particle, like an electron in the hydrogen atom, is key to solving problems in quantum mechanics.

Atomic Orbitals

Atomic orbitals are the regions in space around the nucleus of an atom where there is a high probability of finding an electron. According to quantum mechanics, electrons do not travel in fixed paths as they would in a planetary model, but instead exist in orbitals that are defined by the wave functions of the electrons.

These orbitals come in different shapes such as spherical (s orbitals), dumbbell-shaped (p orbitals), and more complex patterns for d and f orbitals. Each orbital corresponds to a particular energy level and quantum numbers that describe the distribution of an electron's probability density. The 2pz orbital mentioned in the exercise is one such example, characterized by a specific set of quantum numbers and a unique spatial orientation along the z-axis.

Wave Functions

A wave function is a mathematical function that describes the quantum state of a particle or a system of particles in quantum mechanics. The value of the wave function at a given point in space and time correlates to the probability amplitude for a particle's position or momentum. When squared, it gives the probability density, showing where an electron is likely to be found.

To fully grasp the concept in exercises, it's important to remember that wave functions are integral in determining the allowed energy levels of electrons in atoms and the likelihood of their presence in particular regions of space. The wave function for the 2pz orbital, provided in the exercise, is complex and comprises terms involving the Bohr radius and angular dependencies, reflecting the intricate nature of quantum state description.

Bohr Model

The Bohr model is an early model of the atomic structure proposed by Niels Bohr in 1913. While it is a simplification compared to modern quantum mechanics, it introduced the idea of electrons orbiting the nucleus at fixed energy levels or shells, without radiation emission, a contrast from classical notions.

The model's highlight is quantization, which stipulates that electrons have quantized energy levels and can jump from one level to another by absorbing or emitting photons corresponding to the energy difference between levels. Notably, the concept of the Bohr radius, a key variable in the wave function for the 2pz orbital, originated from this model, which provides a measure for the smallest orbit radius for an electron orbiting a hydrogen nucleus. Despite its limitations, the Bohr model serves as a foundation in the historical progression toward more sophisticated quantum theories.