Quantum Numbers

What is the role of quantum numbers?

Quantum numbers allow us to ``predict'' where an electron will be found moving around the nucleus, i.e. a zip code for any given electron.

Can two electrons have all of the same quantum numbers?

No. One zip code, one electron: The unique set of quantum numbers which are assigned to each electron is also known as the Pauli exclusion principle. This means that, of the four quantum numbers that an electron can have, two electrons can share up to three similar quantum numbers but not the fourth.

What is the Heisenberg Uncertainty Principle?

The concept which states that one cannot simultaneously predict both the momentum and the location of an electron.

Why?

Well not only is that definitely beyond the scope of the MCAT but it is hard to visualize in a macroscopic, physical frame of mind. Read on only to see the wonders of science:

Briefly, on such a small scale as with electrons, things do not act as they would in our macroscopic Newtonian world, nor are they ``seen'' similarily. Quantum mechanics sees the world in two simultaneous ways: waves and particles, while we see one or the other. For example, imagine taking a whole bunch oranges and stringing them onto a rope. Bring along two friends and ask them to hold the rope in between them and oscillate the oranges-on-a-rope up and down between them. Now, to see the world through are ``limited'' way, you can stand in such a way to see the oranges oscillating between them, or, stand behind one of your friends and, seeing through them, only see a vertical line of oranges going up and down. In quantum mechanics, you would be able to see both at the same time. Cool, right?

What are the four categories of quantum numbers (symbols)?

1. Principle quantum numbers ($n$)
   • This is the energy level indicator (shells).
   • The difference between lower energy levels is greater than that between the higher energy levels. For example, the difference between n = 1 and n = 2 is greater than n = 4 and n = 5. Note: This is a logarithmic relationship (See How to Take MCAT Math Without a Calculator on page ).
2. Angular momentum quantum numbers ($l$)
   • This is the subshell indicator.

3. Magnetic quantum numbers ($m_{l}$)
   • This indicates the slots for electrons available to be filled in each of the subshells.
   • Think of $m_{l}$ as being a ``P.O. Box<sup>29.1</sup>'' with each $m_{l}$ holding two ``letters,'' or in this case two electrons.
4. Spin quantum numbers ($m_{s}$)
   • This indicates the spin orientation of the electron

What are the possible values that each quantum number can possess?

Table 28.1: Possible values of quantum numbers.

Principle quantum numbers ($n$)	1, 2, 3, 4, 5 $\dots$n
Angular momentum quantum numbers ($l$)	0, 1, 2, 3, 4$\dots$n-1
Magnetic quantum numbers ($m_{l}$)	$l$ to $-l$
Spin quantum numbers ($m_{s}$)	$+\frac{1}{2}$ and $-\frac{1}{2}$

What is the real world corralary of the range of values that each of the four quantum numbers can take?

Table 28.2: Principle ($n$) and angular momentum quantum numbers ($l$).

	Theoretical
	Value
Principle quantum numbers ($n$)	1, 2, 3, 4, 5 $\dots$n	Denotes the distance from the nucleus.
		A log relationship exists between energy values of increasing n's.
Angular momentum quantum numbers ($l$)	0, 1, 2, 3, 4$\dots$n-1
		Denotes type of subshell ($l$):
		0 = s subshell
		1 = p subshell
		2 = d subshell
		3 = f subshell

Table 28.3: Magnetic ($m_{l}$) and spin quantum numbers ($m_{s}$).

	Theoretical
	Value
Magnetic quantum numbers ($m_{l}$)	$l$ to$-l$
		$0 = s = 0$
		$1 = p = -1, 0, +1$
		$2 = d = -2, -1, 0, +1, +2$
		$3 = f =$
		$-3, -2, -1, 0, +1, +2, +3$
Spin quantum numbers ($m_{s}$)	$+\frac{1}{2}$ and $-\frac{1}{2}$
		$+1/2$
		$-1/2$

What are the numbers of electrons that can fill the different subshells ($l$)?

$s$ has one P.O. Box, or $m_{l}$, therefore it can only hold 2 electrons
(one with $+1/2$ and one with $-1/2$)

$p$ has a total of three $m_{l}$, therefore they can hold up to 6 electrons

$d$ has a total of five $m_{l}$, therefore they can hold up to 10 electrons

$f$ has a total of seven $m_{l}$, therefore they can hold up to 14 electrons

What is Hund's rule?

Subshells with more than one $m_{l}$, i.e. $p$, $d$ and $f$, will fill so that they are all half-filled before ``doubling up''.

Half-filled subshells will fill with similar spins, either $+\frac{1}{2}$ or $-\frac{1}{2}$; this is called paramagnetic. When all $m_{l}$ are filled with two $e^{-}$ (one $+\frac{1}{2}$ and one $-\frac{1}{2}$), this is termed diamagnetic.

Figure 28.4: Application of Hund's rule: Paramagnetic (left) and diamagnetic (right) $e^{-}$ filling.