Assigning Quantum Numbers

Assigning Quantum Numbers-74
We can do this because orbitals and the wavefunctions that describe them are not real physical objects. In the equations below, we're going to make use of some trigonometry, notably Euler's formula: $$\mathrm^=\cos \mathrm\sin$$ $$\sin = \frac$$ $$\cos = \frac$$ We're also going to use $x=\sin\cos$ and $y=\sin\sin$.$$\begin \Psi_=\frac\left(\Psi_-\Psi_\right)=\frac\left(\mathrm^ \mathrm^ \right)r\sinf(r)=r\sin\cosf(r)=xf(r) \ \Psi_=\frac\left(\Psi_ \Psi_\right)=\frac\left(\mathrm^-\mathrm^ \right)r\sinf(r)=r\sin\sinf(r)=yf(r)\ \end$$ So, while $m_l=0$ corresponds to $\Psi_$, $m_l= 1$ and $m_l=-1$ cannot be directly assigned to $\Psi_$ and $\Psi_$. Put another way, I suppose we could say that $m_l= 1$ might correspond to $\Psi_$ and $m_l=-1$ might correspond to $\Psi_$.

We can do this because orbitals and the wavefunctions that describe them are not real physical objects. In the equations below, we're going to make use of some trigonometry, notably Euler's formula: $$\mathrm^=\cos \mathrm\sin$$ $$\sin = \frac$$ $$\cos = \frac$$ We're also going to use $x=\sin\cos$ and $y=\sin\sin$.$$\begin \Psi_=\frac\left(\Psi_-\Psi_\right)=\frac\left(\mathrm^ \mathrm^ \right)r\sinf(r)=r\sin\cosf(r)=xf(r) \ \Psi_=\frac\left(\Psi_ \Psi_\right)=\frac\left(\mathrm^-\mathrm^ \right)r\sinf(r)=r\sin\sinf(r)=yf(r)\ \end$$ So, while $m_l=0$ corresponds to $\Psi_$, $m_l= 1$ and $m_l=-1$ cannot be directly assigned to $\Psi_$ and $\Psi_$. Put another way, I suppose we could say that $m_l= 1$ might correspond to $\Psi_$ and $m_l=-1$ might correspond to $\Psi_$.

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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.

(v) The coupling of spin and orbital angular momentum.

Russell-Saunders coupling of L and S to give the total angular momentum J . (vi) Application of the results to the understanding of the periodic table.

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 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 assignment of the correct values of the quantum numbers to the lines in an atomic spectrum requires an approach of a type that you have not come across before.

A given series of lines involves one fixed energy level and a set of levels with a succession of principal quantum numbers, the wavenumber (cm) being given by the equation where n is fixed and m varies.

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