# 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 $$\pi$$.

Figure 1: Spherical Coordinates

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

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

$$z=rcos(\theta)$$

### 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.

### Problems

1. Which quantum numbers depend on the angular wavefunction?

2. Give the quantum numbers defined by the angular wavefunction for a $$d_{z^2}$$ orbital.

3. Given that the spherical representation of the $$d_{x^2-y^2}$$ orbital is $$r^2 sin^{2}(\theta) cos(2\phi)$$, show that this matches the label for the orbital, $$x^2-y^2$$. Hint: $$cos(2x)=cos^2(x)-sin^2(x)$$

Solutions:

1. $$l$$ and $$m_{l}$$

2. $$l=2$$ and $$m_{l}=2,1,0,-1,-2$$

3. $$r^2 sin^{2}(\theta)cos(2\phi)=r^2 sin^{2}(\theta)(cos^2(\phi)-sin^2(\phi))$$

$$r^2 sin^{2}(\theta)cos^2(\phi)-r^2 sin^{2}(\theta)sin^2(\phi)$$

Since $$x=rsin(\theta)cos(\phi)$$ and $$y=rsin(\theta)sin(\phi)$$,

$$r^2 sin^{2}(\theta) cos(2\phi)=x^2-y^2$$

• Bryn Ellison