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Chapter 8: Problem 25

The wave function is evaluated at rectangular coordinates \((x, y, z)=(2,1,1)\) in arbitrary units. What are the spherical coordinates of this position?

Short Answer

Expert verified
The spherical coordinates of the given rectangular coordinates (2, 1, 1) are: \(r = \sqrt{6}\), θ = \(\arccos{\frac{1}{\sqrt{6}}}\), and φ = \(\arctan{\frac{1}{2}}\).

Step by step solution

01

Calculate r (the radial distance)

To find the radial distance, we will apply the formula r = \(\sqrt{x^2 + y^2 + z^2}\) using the given rectangular coordinates (2, 1, 1): r = \(\sqrt{2^2 + 1^2 + 1^2}\) = \(\sqrt{6}\)
02

Calculate θ (the polar angle)

To calculate the polar angle, we will use the formula \(θ = \arccos{\frac{z}{r}}\): θ = \(\arccos{\frac{1}{\sqrt{6}}}\)
03

Calculate φ (the azimuthal angle)

To find the azimuthal angle, we need to use the formula \(φ = \arctan{\frac{y}{x}}\): φ = \(\arctan{\frac{1}{2}}\) Now we have all the required spherical coordinates:
04

Spherical Coordinate Values

r = \(\sqrt{6}\), θ = \(\arccos{\frac{1}{\sqrt{6}}}\), and φ = \(\arctan{\frac{1}{2}}\).

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