Quantum-Mechanical View of Atoms

Since we cannot say exactly where an electron is, the Bohr picture of the atom, with electrons in neat orbits, cannot be correct.

Quantum theory describes an electron probability distribution; this figure shows the distribution for the ground state of hydrogen:

Quantum particle in a box

xL

U(x)

Lxx

LxxU

or 0

0 0

We have standing wave: kxCx sin

1-dimensional box

LC

n

L

kL

nk

2

22

n=1,2… - is the “quantum number”

Energy and momentum are quantized:mL

hn

m

pE

L

nhhp nn 2

222

82

2

Example: What is the energy difference between the first excited state and the ground state of an electron in the “box” of size L=1nm?

eVJJ

kgm

sJE

mL

h

mL

h

mL

hEEE

1.1108.11011.98

63.63

1011.9108

1063.63

8

3

8

1

8

2

19192

3129

234

2

2

2

22

2

22

12

potential energy

3-dimensional box

We have 3 independent standing waves, and 3 independent quantum numbers.

The hydrogen atom

r

exU

2

04

1

•The electron is moving in 3-dimensional space. •Because of that, we can expect 3 independent external quantum numbers.•However, the potential energy is function of one coordinate, r. •Because of that, we can expect that the energy depends only from one of these 3 numbers.•In addition, the electron has one internal quantum number.

mL

hnnn

m

pppE zyxzyx

2

2222222

82

potential energy:

The hydrogen atom

1) Principal quantum number n gives the total energy:

There are four different quantum numbers needed to specify the state of an electron in an atom.

2) Orbital quantum number l gives the magnitude of the angular momentum. (l can take on integer values from 0 to n – 1)

1 ,...1 ,0 nl

3) The magnetic quantum number, ml, gives the “direction” of the electron’s angular momentum. ( ml can take on integer values from –l to +l )

lml ,...1 ,0

4) The spin quantum number, ms, which for an electron can take on the values +½ and -½.

The need for this quantum number was found by experiment; spin is an intrinsically quantum mechanical quantity, although it mathematically behaves as a form of angular momentum.

Angular momentum

This plot indicates the quantization of angular momentum direction for l = 2. The other two components of the angular momentum are undefined.

The angular momentum quantum numbers do not affect the energy level of the hydrogen atom, but they do change the spatial distribution of the electron cloud.

61221 llL

,lz mL 2,1,0 lm

Zeeman effectIn a magnetic field, the spectral lines are split into several very closely spaced lines. This splitting, known as the Zeeman effect, demonstrates that the atoms energy levels are split. This means that, in magnetic field, the energy of state depend not only on principal quantum number, n but also on the “magnetic quantum number” ml.

Fine structureA careful study of the spectral lines showed that each actually consist of several very closely spaced lines even in the absence of an eternal magnetic field. This splitting is called “fine structure”. It is related to the spin of electron.

Transitions between energy levels

“Allowed” transitions between energy levels occur between states whose value of l differ by one:

Other, “forbidden,” transitions also occur but with much lower probability.Photon has a spin angular momentum of 1ħ.

Complex Atoms

Complex atoms contain more than one electron, so the interaction between electrons must be accounted for in the energy levels.

A neutral atom has Z electrons, as well as Z protons in its nucleus. Z is called the atomic number.

Four quantum numbers: n, l, ml , ms can be used to describe an electron in atom.

The energy depends mainly on n and l.

This table summarizes the four quantum numbers

The Pauli exclusion principle:

No two electrons in an atom can occupy the same quantum state.

More generally: No two identical particles whose spin quantum number is a half-integer (1/2, 3/2,…), including electrons, protons and neutrons can occupy the same quantum state.

The quantum state of an electron in atom is specified by the four quantum numbers. According to the Pauli principle no two electrons can have the same set.

The Periodic Table of the Elements

Electrons are grouped into shells and subshells:

•Electrons with the same n are in the same shell.

•Electrons with the same n and l are in the same subshell.

•The exclusion principle limits the maximum number of electrons in each subshell to 2(2l + 1).

21

,...1,0

s

l

m

lm 12 lml

l

Example 1:

electrons) 2 (maximum statesdiffernt 2

,0 0, 1For 21 sl mmln

Example 2:

electrons) 8 (maximum statesdiffernt 8 :total

statesdiffernt 6 ;1,01for

statesdiffernt 2 ;00for

0,1; 2For

21

21

sl

sl

mml

mml

ln

Electron configurations are written by : •the value for n •the letter code for l•and the number of electrons in the subshell as a superscript

Electron configurations

Example: A neutral atom of a certaine element has configuration given by :

Example: The ground-state configuration of

sodium:

Sodium has 11 electrons (Z=11). Ten of them form a closed neon-like core. The remaining electron is the valence electron.

What is the atomic number of this element?.3433221 6262622 dspspss

Notations:

Each value of l is given its own letter symbol.

This table shows the configuration of the outer electrons only

Atoms with the same number of electrons in their outer shells have similar chemical behavior. They appear in the same column of the periodic table.

The outer columns – those with full, almost full, or almost empty outer shells – are the most distinctive.

The inner columns, with partly filled shells, have more similar chemical properties.

Example: The electron configuration of the neutral fluorine atom in its ground state is522 221 pss

Make a list of the four quantum numbers of each electron in the fluorine atom.

n l ml ms orbital designation112222222

Summary

• n, the principal quantum number, can have any integer value, and gives the energy of the level

• l, the orbital quantum number, can have values from 0 to n – 1

• ml, the magnetic quantum number, can have values from –l to +l

• ms, the spin quantum number, can be +½ or -½

• Energy levels depend on n and l, except in hydrogen. The other quantum numbers also result in small energy differences

• Pauli exclusion principle: no two electrons in the same atom can be in the same quantum state

• Electrons are grouped into shells and subshells

• Periodic table reflects shell structure

Atoms with the same number of electrons in their outer shells have similar chemical behavior. They appear in the same column of the periodic table.