Question

If one increases the angle away from the \(Z\) axis, how does the wave function for a \(2 \mathrm{P}_{Z}\) electron change?

Short answer

As the angle away from the Z-axis increases, the wave function for a 2Pz electron goes from a maximum positive value along the Z-axis to zero at the X-Y plane and then to a minimum negative value opposite to the Z-axis, which can be represented by the spherical harmonics \(Y_{1}^{0}(\theta, \phi) = \sqrt{\cfrac{3}{4\pi}}\cos(\theta)\).

Step-by-step solution

Identify the wave function for a 2Pz electron

We need to find the wave function for a 2Pz electron in spherical coordinates. The wave function for a 2Pz electron, in general, can be written as: \( \psi_{2Pz}(\vec{r}) = R_{nl}(r) Y_{l}^{m}(\theta, \phi) \) where \(R_{nl}(r)\) is the radial part, and \(Y_{l}^{m}(\theta, \phi)\) is the angular part or spherical harmonics of the wave function. For a 2Pz electron, we have the quantum numbers \(n=2, l=1, m=0\).

Determine the angular part of the wave function for a 2Pz electron

Since we are interested in the change of the wave function with the angle away from the Z-axis, we only need to focus on the spherical harmonics (angular part) of the wave function. For a 2Pz electron (\(l=1, m=0\)), the spherical harmonics can be expressed as: \( Y_{1}^{0}(\theta, \phi) = \sqrt{\cfrac{3}{4\pi}}\cos(\theta) \)

Analyze how the angular part changes with the angle away from the Z-axis

The angle away from the Z-axis is represented by the polar angle \(\theta\) in spherical coordinates. We can now analyze how the spherical harmonics change as \(\theta\) increases: \( Y_{1}^{0}(\theta, \phi) = \sqrt{\cfrac{3}{4\pi}}\cos(\theta) \) As \(\theta\) increases from 0 to \(\pi\), \(\cos(\theta)\) decreases from 1 to -1. Therefore, the angular part of the wave function for a 2Pz electron decreases from its maximum value at the Z-axis (\(\theta = 0\)) to zero at the X-Y plane (\(\theta = \cfrac{\pi}{2}\)), and then to its minimum negative value in the opposite direction of the Z-axis (\(\theta = \pi\)). In conclusion, as the angle away from the Z-axis increases, the wave function for a 2Pz electron goes from a maximum positive value along the Z-axis to zero at the X-Y plane and then to a minimum negative value opposite to the Z-axis.

Key concepts

Wave Function

In quantum mechanics, the wave function is a fundamental concept that describes the quantum state of a particle. It is a mathematical function that provides information about the probability amplitude of a particle's position and momentum. The wave function is usually denoted by the Greek letter \( \psi \), and it can depend on various variables such as time and spatial coordinates. For a hydrogen-like atom, the wave function is expressed in spherical coordinates to match the symmetry of the atomic orbits.
The general form of the wave function for an electron in an atom is:

• \( \psi(\vec{r}) = R_{nl}(r) Y_{l}^{m}(\theta, \phi) \)

Here, \( R_{nl}(r) \) is the radial part, which depends on the distance \( r \) from the nucleus, and \( Y_{l}^{m}(\theta, \phi) \) is the angular part, known as spherical harmonics. Understanding how a wave function describes an electron's behavior is crucial in quantum mechanics, as it provides insights into the electron's location and movement.

Spherical Harmonics

Spherical harmonics are a set of orthogonal functions defined on the surface of a sphere. They play a crucial role in the solution of the wave equation for electrons in atoms. When dealing with problems in quantum mechanics, spherical harmonics are used to describe the angular part of the wave function. These functions depend on the angles \( \theta \) (polar angle) and \( \phi \) (azimuthal angle), making them ideal for systems with spherical symmetry.
The spherical harmonics \( Y_{l}^{m}(\theta, \phi) \) are characterized by two quantum numbers:

• \( l \): the azimuthal quantum number
• \( m \): the magnetic quantum number

For a 2Pz electron, the relevant spherical harmonic is \( Y_{1}^{0}(\theta, \phi) = \sqrt{\frac{3}{4\pi}}\cos(\theta) \). This function describes how the probability amplitude of finding the electron changes with direction. As you move away from the Z-axis, the value of \( \cos(\theta) \) changes, altering the angular part of the wave function accordingly.

2Pz Electron

A 2Pz electron is an electron in a hydrogen-like atom that occupies the 2p orbital, specifically in the z-direction. The designation 2Pz indicates:

• The principal quantum number \( n = 2 \)
• The azimuthal quantum number \( l = 1 \)
• The magnetic quantum number \( m = 0 \)

These quantum numbers determine the shape and orientation of the electron's orbital. In spherical coordinates, the 2Pz electron's wave function can be broken into radial and angular parts. While the radial part \( R_{nl}(r) \) defines how the probability density varies with distance from the nucleus, the angular part \( Y_{1}^{0}(\theta, \phi) \) focuses on directional dependence.
For the 2Pz electron, the wave function is most positive along the Z-axis and changes as you move towards other directions. As the angle \( \theta \) increases from the Z-axis, the value of the wave function decreases from a maximum to zero in the X-Y plane, demonstrating the directional properties unique to the 2Pz state. This behavior reflects the spatial distribution and nodal properties of p orbitals in quantum mechanics.