suppose you know the values for r, theta, and phi which specify a point. what are the corresponding values of x,y,z? Write an expression for x as a function or r, theta,and phi; then do the same for x,y,z. Using the answer to the previous question write the angular parts of the three l=1 wavefuncions in Cartesian coordinates and what can you say about the symmetries of each of these wavefunctions?
To find the corresponding values of x, y, and z based on the given values of r, theta, and phi, we can use the following expressions:
x = r * sin(theta) * cos(phi)
y = r * sin(theta) * sin(phi)
z = r * cos(theta)
These expressions represent the conversion from spherical coordinates (r, theta, phi) to Cartesian coordinates (x, y, z).
Now, let's focus on the angular parts of the three l=1 wavefunctions in Cartesian coordinates. The three spherical harmonics corresponding to l=1 are:
Y₁₋₁ = √(3/8π) * sin(theta) * sin(phi)
Y₁₀ = √(3/4π) * cos(theta)
Y₁₁ = -√(3/8π) * sin(theta) * cos(phi)
To convert these spherical harmonics to Cartesian coordinates, we use the following expressions:
x = rsinθcosϕ
y = rsinθsinϕ
z = rcosθ
For Y₁₋₁:
Y₁₋₁_cartesian = √(3/8π) * (rsinθcosϕ) * (rsinθsinϕ)
For Y₁₀:
Y₁₀_cartesian = √(3/4π) * (rcosθ)
For Y₁₁:
Y₁₁_cartesian = -√(3/8π) * (rsinθcosϕ) * (rsinθcosϕ)
From these expressions, we can observe the symmetries of the wavefunctions. The wavefunction Y₁₋₁ has symmetry about the x and y axes but is antisymmetric about the z-axis. The wavefunction Y₁₀ is symmetric about the z-axis but antisymmetric about the x and y axes. The wavefunction Y₁₁ has symmetry about the x and y axes but is antisymmetric about the z-axis.