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 1-dimensional box U(x) potential energy x L We have standing wave: n=1,2… - is the “quantum number” Energy and momentum are quantized: 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?

The hydrogen atom 3-dimensional box We have 3 independent standing waves, and 3 independent quantum numbers. The hydrogen atom potential energy: 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.

The hydrogen atom There are four different quantum numbers needed to specify the state of an electron in an atom. 1) Principal quantum number n gives the total energy: 2) Orbital quantum number l gives the magnitude of the angular momentum. (l can take on integer values from 0 to n – 1) 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 ) 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.

Transitions between energy levels Zeeman effect In 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 structure A 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). Example 1: Example 2:

Electron configurations Notations: Each value of l is given its own letter symbol. 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 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. Example: A neutral atom of a certaine element has configuration given by : What is the atomic number of this element?

This table shows the configuration of the outer electrons only

n l ml ms orbital designation 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 is Make a list of the four quantum numbers of each electron in the fluorine atom. n l ml ms orbital designation 1 2

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