Does a particular p-orbital orientation (px, py, pz) correspond to a particular value of m (magnetic quantum number 0, +1, -1)?
Does a particular p-orbital orientation (px, py, pz) correspond to a particular value of m (magnetic quantum number 0, +1, -1)?
Solving Schrodinger's equation for hydrogen atom we obtain the solutions , the wave functions which are also called orbitals. These orbitals are, in general, complex functions of the position coordinates r, theta and phi in the spherical polar coordinate system. The spherical polar coordinates r, theta, and phi are related to the Cartesian coordinates x,y and z as follows. x = r Sin(theta) Cos(phi), y = r Sin(theta) Sin(phi), and z = r Cos(theta).
These wave functions depend parametrically on the quantum numbers n, l and m. Wave functions with l=0 are real functions. Wave functions with m=0 for any value of l are also real functions.
The imaginary 'i' is present in the wave functions as exp( i*m*phi) . The wave functions are complex if m is not zero.
The p wave functions are of the form R(r) Sin(theta) exp(i*m*phi) if m= +1 or -1. For m=0, the p function is R(r) Cos theta, which is designated as Pz. Px and Py are real functions generated by the linear combination of P(m=+1) and P(m= -1). Px = R(r) Sin theta (e^i*phi + e^-i*phi) = R(r) Sin theta Cos phi. Py = R(r)Sin theta (e^i*phi - e^-i*phi) = R(r) Sin theta Sin phi. ( Recall from your mathematics classes, e^(i*x) = Cos x + i* Sin x) These linear combinations are also solutions of the Schrodinger equation. We cannot assign any value to the magnetic quantum number m for Px and Py orbitals.
Solving Schrodinger's equation for hydrogen atom we obtain the solutions , the wave functions which are also called orbitals. These orbitals are, in general, complex functions of the position coordinates r, theta and phi in the spherical polar coordinate system. The spherical polar coordinates r, theta, and phi are related to the Cartesian coordinates x,y and z as follows. x = r Sin(theta) Cos(phi), y = r Sin(theta) Sin(phi), and z = r Cos(theta).
These wave functions depend parametrically on the quantum numbers n, l and m. Wave functions with l=0 are real functions. Wave functions with m=0 for any value of l are also real functions.
The imaginary 'i' is present in the wave functions as exp( i*m*phi) . The wave functions are complex if m is not zero.
The p wave functions are of the form R(r) Sin(theta) exp(i*m*phi) if m= +1 or -1. For m=0, the p function is R(r) Cos theta, which is designated as Pz. Px and Py are real functions generated by the linear combination of P(m=+1) and P(m= -1). Px = R(r) Sin theta (e^i*phi + e^-i*phi) = R(r) Sin theta Cos phi. Py = R(r)Sin theta (e^i*phi - e^-i*phi) = R(r) Sin theta Sin phi. ( Recall from your mathematics classes, e^(i*x) = Cos x + i* Sin x) These linear combinations are also solutions of the Schrodinger equation. We cannot assign any value to the magnetic quantum number m for Px and Py orbitals.
