An equivalent statement is that these two orbitals do not lie on the $x$- and $y$-axes, but rather bisect them.Thus it is typical to take linear combinations of them to make the equation look prettier.(v) The coupling of spin and orbital angular momentum.
The applet below indicates how this can be done (click here for information concerning applets; the filenames here are tutorials/atomic/at Spec Fit/at Spec and tutorials/atomic/at Spec Fit/At Spec Fit Applet J.html).
The applet uses these principles and you can use it to answer several of the problems presented below. These lines are already drawn in the applet for a series of lines that have been input as data.
The degeneracy of the angular wavefunctions and space quantization of the angular momentum. A program that simulates the spectra of the alkali metals and other elements is used below to calculate the spectra of lithium and hydrogen.
The selection rules for l in determining the spectrum. You can observe the atomic spectrum of sodium in the work for a second year tutorial on electronic spectroscopy.
The different series of lines and their relation to the energy level diagram (Grotrian diagram).
Interpretation of the levels in terms of the Bohr model of the atom. Some quantum mechanics follow, but the TL; DR version is that while $m_l=0$ corresponds to $p_z$, the orbitals for $m_l= 1$ and $m_l=-1$ lie in the $xy$-plane, but not on the axes.Thus, it is not possible to directly correlate the values of $m_l=\pm1$ with specific orbitals.For hydrogen and hydrogen-like atoms it is often easy to guess the starting number of the m series.If the guess is correct a graph of line wavenumber against 1/m = 0 (m = ∞) gives the ionization energy of the fixed level.In switching from spherical to Cartesian coordinates, we make the substitution $z=r\cos$, so: $$\Psi_=zf(r)$$ This is $\Psi_$ since the value of $\Psi$ is dependent on $z$: when $z=0;\ \Psi=0$, which is expected since $z=0$ describes the $xy$-plane.The other two wavefunctions are unhelpfully degenerate in the $xy$-plane.The determination of the ionization potential from the line spectrum. (ii) The nature of the wavefunctions of the hydrogen atom, especially the form of the radial wavefunction. The effect on the energy levels of penetration of inner electron shells by the outer electrons.Quantization of angular momentum and the shape of wavefunctions with different values of l. The physical origin of the doublet structure in the alkali spectra.Linear combinations are allowed by the maths of quantum mechanics.If any set of wavefunctions is a solution to the Schrödinger equation, then any set of linear combinations of these wavefunctions must also be a solution.