spherical coordinates
if n=2,l=1,ml=x, add up all the probabiliity densities.
To find the probability densities for spherical coordinates with n = 2, l = 1, and ml = x, you will need to calculate the individual probability densities for each possible value of ml and then add them up.
In spherical coordinates, the probability density function (PDF) for an electron in an atom is given by the equation:
P(θ, φ, r) = R(r)^2 * Y(θ, φ)^2
Where:
- P(θ, φ, r) is the probability density at a given point in spherical coordinates
- R(r) is the radial part of the wave function that depends only on the distance from the nucleus
- Y(θ, φ) is the angular part of the wave function that depends on the spherical angles θ and φ
For n = 2, l = 1, and ml = x, the allowed values of ml are -1, 0, and 1. So you need to compute the probability densities for these three values separately.
1. For ml = -1:
- Calculate the radial part R(r) using the appropriate formula for n and l.
- Calculate the angular part Y(θ, φ) using the appropriate formula for ml = -1.
- Square both R(r) and Y(θ, φ) to get R(r)^2 and Y(θ, φ)^2.
- Multiply R(r)^2 by Y(θ, φ)^2 to get P(θ, φ, r) for ml = -1.
2. Repeat the above steps for ml = 0 and ml = 1.
3. Finally, add up the probability densities P(θ, φ, r) for all three values of ml to get the total probability density.
Please note that the specific formulas for R(r), Y(θ, φ), and their squares depend on the quantum numbers n, l, and ml. You may need to refer to appropriate tables or use specific equations for hydrogen-like atoms to determine these values accurately.