A wave function is evaluated at the rectangular coordinates 𝑥, 𝑦, 𝑧 equals two, one, one in arbitrary units. What are the spherical coordinates of this position?
The spherical coordinates 𝜚, 𝜙, and 𝜃 are an equivalent way of describing position in three-dimensional space. In this exercise, we’ll convert from rectangular coordinates to spherical coordinates. And to do that, we can recall or look up the conversion steps from Cartesian 𝑥, 𝑦, 𝑧 rectangular coordinates to spherical coordinates 𝜚, 𝜙, and 𝜃.
𝜚 gives the overall vector magnitude and is defined as the square root of 𝑥 squared plus 𝑦 squared plus 𝑧 squared. 𝜙, the vector’s angle in the 𝑧-plane, is defined as the inverse cos of 𝑧 divided by 𝜚, the vector magnitude. And 𝜃, the vector’s angle in the 𝑥𝑦-plane, is defined as the arctangent of 𝑦 over 𝑥.
Let’s apply these coordinate transformations to our particular coordinate values. In our case, 𝜚, the square root of 𝑥 squared plus 𝑦 squared plus 𝑧 squared, equals the square root of two squared plus one squared plus one squared, or the square root of six.
Now we’ll move on to solving for 𝜙, which in our case equals the arc cosine of one divided by the square root of six. When we evaluate this term on our calculator, we find that, to two significant figures, 𝜙 is 66 degrees.
Lastly, we’ll solve for the angle 𝜃. 𝜃, the arctangent of 𝑦 divided by 𝑥, in our case is the arctangent of one divided by two, which equals 27 degrees. So the spherical coordinates of the Cartesian coordinates two, one, one are equal to the square root of six, 66 degrees, and 27 degrees.