The wave function for the 2 pz orbital in the hydrogen atom is ψ2 pz=(1)/(4 √(2 π))((Z)/(a0))^3

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The wave function for the 2 pz orbital in the hydrogen atom...

The wave function for the $2 p_{z}$ orbital in the hydrogen atom is $$ \psi_{2 p_{z}}=\frac{1}{4 \sqrt{2 \pi}}\left(\frac{Z}{a_{0}}\right)^{3 / 2} \sigma \mathrm{e}^{-\sigma / 2} \cos \theta $$ 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 $\psi_{2 p_{p}}^{2}$ at $r=a_{0}$ for $\theta=0^{\circ}\left(z\right.$ axis) and for $\theta=90^{\circ}(x y$ plane).

The wave function for the $2 p_{z}$ orbital in the hydrogen atom is
$$
\psi_{2 p_{z}}=\frac{1}{4 \sqrt{2 \pi}}\left(\frac{Z}{a_{0}}\right)^{3 / 2} \sigma \mathrm{e}^{-\sigma / 2} \cos \theta
$$
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 $\psi_{2 p_{p}}^{2}$ at $r=a_{0}$ for $\theta=0^{\circ}\left(z\right.$ axis) and for $\theta=90^{\circ}(x y$ plane).

Key Concepts

Angular Dependence in Atomic Orbitals

The angular dependence in atomic orbitals stems from the spherical harmonics, which are functions defined on the sphere that describe the angular part of the wave function. They determine the directional characteristics of the electron density, as seen in the distinct shapes and nodal patterns of the different orbitals, such as p orbitals.

Spherical Coordinates

Spherical coordinates are a natural choice for problems with central potentials, such as the hydrogen atom. In these coordinates, the position of a particle is described by a radial distance and two angles. This coordinate system simplifies the separation of variables in the Schrödinger equation, leading to distinct radial and angular components in the wave function.

Wave Function

The wave function is a core concept in quantum mechanics that describes the quantum state of a particle, where its square modulus gives the probability density of finding the particle in a particular region of space. It encapsulates all the available information about the system and evolves according to the Schrödinger equation.

Probability Density

Probability density is obtained by squaring the absolute value of the wave function, providing the likelihood of locating an electron at a specific point in space. It is a fundamental quantity used to predict measurable outcomes in quantum mechanics and plays a crucial role in interpreting electron distributions in atomic orbitals.

Hydrogen-like Atomic Orbitals

Hydrogen-like atomic orbitals are solutions to the Schrödinger equation for a one-electron atom or ion. These orbitals are characterized by quantum numbers that determine their size, shape, and orientation. They occur as a product of a radial part and an angular part, each reflecting different physical aspects of the electron’s behavior.

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