# Electronic Angular Wavefunction

The electronic angular wavefunction is one spatial component of the electronic Schrödinger wave equation, which describes the motion of an electron. It depends on angular variables, $$\theta$$ and $$\phi$$, and describes the direction of the orbital that the electron may occupy. Some of its solutions are equal in energy and are therefore called degenerate.

## Introduction

Electrons can be described as a particle or a wave. Because they exhibit wave behavior, there is a wavefunction that is a solution to the Schrödinger wave equation:

$\hat{H}\Psi(r,\phi,\theta,t)=E\Psi(r,\phi,\theta,t)$

This equation has eigenvalues, $$E$$, which are energy values that correspond to the different wavefunctions.

## Spherical Coordinates

To solve the Shrödinger equation, spherical coordinates are used. Spherical coordinates are in terms of a radius $$r$$, as well as angles $$\phi$$, which is measured from the positive x axis in the xy plane and may be between 0 and $$2\pi$$, and $$\theta$$, which is measured from the positive z axis towards the xy plane and may be between 0 and $$\pi$$.

$$x=rsin(\theta)cos(\phi)$$

$$y=rsin(\theta)sin(\phi)$$

$$z=rcos(\theta)$$

## Electronic Wavefunction

The electronic wavefunction, $$\Psi(r,\phi ,\theta ,t)$$, describes the wave behavior of an electron. Its value is purely mathematical and has no corresponding measurable physical quantity. However, the square modulus of the wavefunction, $$\mid \Psi(r,\phi ,\theta ,t)\mid ^2$$ gives the probability of locating the electron at a given set of values. To use separation of variables, the wavefunction can be expressed as

$$\Psi(r,\phi, \theta ,t)=R(r)Y_{l}^{m}(\phi, \theta)$$

$$R(r)$$ is the radial wavefunction and $$Y_{l}^{m}(\phi, \theta)$$ is the angular wavefunction. Separating the angular variables in $$Y_{l}^{m}(\phi, \theta)$$ gives

$$Y_{l}^{m}(\phi, \theta)=\left[\dfrac{2l+1}{4\pi}\left(\dfrac{(l-\mid m \mid)!}{(l+\mid m \mid)!}\right)\right]^{\frac{1}{2}}P_l^{\mid m \mid}(cos(\theta))e^{im\phi}$$

where $$P_l^{\mid m \mid}(cos(\theta))$$ is a Legendre polynomial and is only in terms of the variable $$\theta$$. The exponential function, which is only in terms of $$\phi$$, determines the phase of the orbital.

For the angular wavefunction, the square modulus gives the probability of finding the electron at a point in space on a ray described by $$(\phi, \theta)$$. The angular wavefunction describes the spherical harmonics of the electron's motion. Because orbitals are a cloud of the probability density of the electron, the square modulus of the angular wavefunction influences the direction and shape of the orbital.

## Quantum Numbers and Orbitals

There are 3 quantum numbers defined by the Schrodinger wave equation. They are $$n$$, $$l$$, and $$m_{l}$$. Each combination of these quantum numbers describe an orbital. Values for $$n$$ come from from the radial wavefunction. $$n$$ may be 1, 2, 3... Because they evolved from the separation of variables performed to solve the wavefunction, solutions to the angular wavefunction are quantized by the values for $$l$$ and $$m_{l}$$. Acceptable values for $$l$$ are given by $$l=n-1$$. The corresponding values for $$m_{l}$$ are integers between $$-l$$ and $$+l$$.

## Degeneracy and p, d and f Orbitals

Orbitals descrbed by the same $$n$$ and $$l$$ values but different $$m_{l}$$ values are degenerate, meaning that they are equal in energy but vary in their direction and, sometimes, shape. For $$p$$ orbitals, $$l=1$$, giving three $$m_{l}$$ values and thus, 3 degenerate states. They are $$p_{x}$$, $$p_{y}$$and $$p_{z}$$. $$d$$ orbitals have $$l=2$$, giving 5 degenerate states. These are $$d_{xy}$$, $$d_{xz}$$, $$d_{yz}$$, $$d_{z^2}$$, $$d_{x^2-y^2}$$. $$f$$ orbitals have $$l=3$$, giving a total of 7 degenerate states.

## References

1. McMahon, David. (2006) Quantum Mechanics Demystified New York: McGraw-Hill.
2. McGervey, J. D. (1995) Quantum Mechanics:concepts and applications San Diego:Academic Press.