Question

Carry out the following coordinate transformations: a. Express the point \(x=3, y=1,\) and \(z=1\) in spherical coordinates. b. Express the point \(r=5, \theta=\frac{\pi}{4},\) and \(\phi=\frac{3 \pi}{4}\) in Cartesian coordinates.

Short answer

a. The spherical coordinates are \(r = \sqrt{11}\), \(\theta = \arccos{\frac{1}{\sqrt{11}}}\), and \(\phi = \arctan{\frac{1}{3}}\). b. The Cartesian coordinates are \(x = -5\), \(y = 5\), and \(z = \frac{5\sqrt{2}}{2}\).

Step-by-step solution

Part (a): Convert Cartesian to Spherical Coordinates

We must use the conversion formulas to change the Cartesian coordinates (x, y, z) = (3, 1, 1) to spherical coordinates (r, θ, φ). 1. Calculate r: \(r = \sqrt{x^2 + y^2 + z^2} = \sqrt{3^2 + 1^2 + 1^2} = \sqrt{11}\) 2. Calculate θ: \(\theta = \arccos{\frac{z}{r}} = \arccos{\frac{1}{\sqrt{11}}}\) 3. Calculate φ: \(\phi = \arctan{\frac{y}{x}} = \arctan{\frac{1}{3}}\) So the spherical coordinates are \(r = \sqrt{11}\), \(\theta = \arccos{\frac{1}{\sqrt{11}}}\), and \(\phi = \arctan{\frac{1}{3}}\).

Part (b): Convert Spherical to Cartesian Coordinates

We must use the conversion formulas to change the spherical coordinates (r, θ, φ) = (5, π/4, 3π/4) to Cartesian coordinates (x, y, z). 1. Calculate x: \(x = r\sin{\theta}\cos{\phi} = 5\sin{\frac{\pi}{4}}\cos{\frac{3\pi}{4}} = 5\frac{\sqrt{2}}{2} \times (-\frac{\sqrt{2}}{2}) = -\frac{10}{2} = -5\) 2. Calculate y: \(y = r\sin{\theta}\sin{\phi} = 5\sin{\frac{\pi}{4}}\sin{\frac{3\pi}{4}} = 5\frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2} = 5\) 3. Calculate z: \(z = r\cos{\theta} = 5\cos{\frac{\pi}{4}} = 5\frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2}\) So the Cartesian coordinates are \(x = -5\), \(y = 5\), and \(z = \frac{5\sqrt{2}}{2}\).

Key concepts

Cartesian to Spherical Coordinates

When we talk about Cartesian to Spherical Coordinates, we're looking at a way to redefine a point in space from the familiar x, y, and z axes to a set involving a radius and two angles, known as spherical coordinates. This transform involves measuring the radial distance from the origin to the point, the angle relative to the positive z-axis, and the azimuthal angle in the x-y plane from the positive x-axis.

For our exercise, we used the transformation formulas as follows:

• The radial distance r is calculated as the square root of the sum of the squares of the x, y, and z coordinates.
• The polar angle θ (theta) is determined using the inverse cosine (arccos) of z/r.
• The azimuthal angle φ (phi) is found with the inverse tangent (arctan) of y/x.

Once these values are computed, we have effectively re-described the location of our point in three-dimensional space using spherical coordinates.

Spherical to Cartesian Coordinates

Conversely, converting Spherical to Cartesian Coordinates involves re-expressing a point defined by radial distance and two angles back to the rectangular coordinates of x, y, and z. This conversion utilizes spherical coordinates (r, θ, φ) and translates them through trigonometry to find the Cartesian equivalents.

In our example, the conversion process is as follows:

• The x-coordinate is found by multiplying the radius with the sine of the polar angle and the cosine of the azimuthal angle.
• To obtain the y-coordinate, we multiply the radius with the sine of both angles.
• The z-coordinate is computed by multiplying the radius with the cosine of the polar angle.

This step is crucial for problems needing a switch back to Cartesian coordinates for further processing or visualization.

Trigonometric functions in coordinate conversion

Understanding Trigonometric functions in coordinate conversion is key in moving between Cartesian and spherical coordinates. These functions translate the angular relationships of the spherical coordinate system into the linear measures used in the Cartesian system.

The sine and cosine functions relate the angles θ and φ to the x, y, and z coordinates. The arc functions (arccos and arctan) are used to determine the angles when converting from Cartesian to Spherical. Such functions are fundamental because they maintain the geometric integrity of the point's position during the conversion process.

Coordinate systems in physical chemistry

In the context of Coordinate systems in physical chemistry, the choice of coordinate system can significantly simplify the mathematical treatment of molecules and reactions. While Cartesian coordinates are straightforward and are widely used for defining locations on a grid, spherical coordinates are much better suited for problems where symmetry about a point is evident, such as in orbital shapes and angular momentum in atoms.

The understanding and manipulation of these coordinate systems are essential, particularly in quantum chemistry where the Schrödinger equation is often solved in terms of spherical coordinates due to the spherical nature of atom. The conversions between the coordinate systems enable chemists and physicists to select the most convenient mathematical approach for the problem at hand.