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Chapter 8: Problem 33

The wave function is evaluated at spherical coordinates \((r, \theta, \phi)=\left(\sqrt{3}, 45^{\circ}, 45^{\circ}\right), \quad\) where the value of the radial coordinate is given in arbitrary units. What are the rectangular coordinates of this position?

Short Answer

Expert verified
The rectangular coordinates of the position are \((x, y, z) = \left(\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}, \frac{3\sqrt{2}}{2}\right)\).

Step by step solution

01

Convert degrees to radians

First, we need to convert the angles from degrees to radians. To do this, we'll use the conversion factor \(\frac{\pi}{180}\). \(\theta = 45^{\circ} \times \frac{\pi}{180} = \frac{\pi}{4} \text{ radians}\) \(\phi = 45^{\circ} \times \frac{\pi}{180} = \frac{\pi}{4} \text{ radians}\) Now, we have the angles in radians as \(\theta=\frac{\pi}{4}\) and \(\phi=\frac{\pi}{4}\).
02

Apply conversion formulas

Next, we will use the conversion formulas mentioned in the analysis and plug in the values of \(r\), \(\theta\), and \(\phi\). \(x = r \sin\theta \cos\phi = \sqrt{3} \sin\frac{\pi}{4} \cos\frac{\pi}{4}\) \(y = r \sin\theta \sin\phi = \sqrt{3} \sin\frac{\pi}{4} \sin\frac{\pi}{4}\) \(z = r \cos\theta = \sqrt{3} \times \cos\frac{\pi}{4}\)
03

Calculate the rectangular coordinates

Finally, we will evaluate the sin and cos functions and simplify the expressions to get the rectangular coordinates. \(x = \sqrt{3} \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2} = \sqrt{3} \times \frac{1}{2}\) \(y = \sqrt{3} \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2} = \sqrt{3} \times \frac{1}{2}\) \(z = \sqrt{3} \times \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}\) So the rectangular coordinates are: \((x, y, z) = \left(\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}, \frac{3\sqrt{2}}{2}\right)\)

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