Introduction to Quantum Mechanics -...

Introduction to Quantum

Mechanics

and Quantum Numbers

The Quantum Mechanical Model

quantum mechanics: the application of

quantum theory to explain the

properties of matter, particularly

electrons in atoms

Schrödinger’s Standing Wave

Erwin Schrödinger and Louis de Broglie

found that an electron bound to a

nucleus in an atom resembled a

standing wave, so they began research

on a description of the atom based on

wave behaviour instead of particle

behaviour.

Schrödinger and de Broglie took the

idea of standing waves and applied it to

the electron in a hydrogen atom.

In their model, the electron is a circular

standing wave around the nucleus. This

circular standing wave consists of

wavelengths that are multiples of whole

numbers (n = 1, 2, 3, 4, ...).

Only certain circular orbits have a

circumference into which a whole

number of wavelengths can fit.

Any other orbits of the standing electron

wave are not allowed because they

would cause the standing wave to

cancel out or collapse.

Orbitals and Probability Distributions

Schrödinger’s work on quantum

mechanics led to his development of a

mathematical equation, called the

Schrödinger wave equation, that could

be used to calculate electron energy

levels.

Orbital: the region around the nucleus

where an electron has a high probability

of being found

http://hyperphysics.phy-

astr.gsu.edu/%E2%80%8Chbase/quant

um/schr2.html

Werner Heisenberg came up with a

statistical approach for locating

electrons.

To measure the location and speed of

an object, you must be able to observe

it.

Heisenberg’s Uncertainty Principle: the

idea that it is impossible to know the

exact position and speed of an electron

at a given time

The best we can do is to describe the

probability of finding an electron in a

specific location.

wave function: the mathematical

probability of finding an electron in a

certain region of space

Quantum mechanics does not describe

how an electron moves or even if it

moves. It only tells us the statistical

probability of finding the electron in a

given location in an atom. The area or

region where we are likely to find an

electron is an orbital.

Using wave functions, physicists have

created a three-dimensional electron

probability density, which is a plot that

indicates regions around the nucleus

with the greatest probability of finding

an electron.

The electron probability density plot for

a hydrogen electron in the ground state

(lowest energy state) is spherical and is

called the 1s orbital.

The greatest probability of finding the

electron occurs at a distance rmax from

the nucleus. This distance is the same

as the distance Bohr calculated for the

radius of the first circular orbit of

hydrogen’s electron.

The two main ideas of the quantum

mechanical model of the atom are that

electrons can be in different orbitals by

absorbing or emitting quanta of energy,

and that the location of electrons is

given by a probability distribution.

Quantum Numbers

There are 4 quantum numbers

(numbers that describe the quantum

mechanical properties of orbitals; from

the solutions to Schrödinger’s wave

equation)

The Principal Quantum

Number (n)

The integer, n, that Bohr used to label

the orbits and energies describes a

main shell of electrons, and is referred

to today as the principal quantum

number. Bohr’s theory used only one

quantum number, which is the main

reason that it worked well for hydrogen

but not for other atoms.

The Secondary Quantum

Number, (l)

Arnold Sommerfeld (1915) boldly

employed elliptical orbits to extend the

Bohr theory and successfully explain

that the main lines of the bright-line

spectrum for hydrogen were actually

composed of more than one line.

He introduced the secondary quantum

number, l, to describe additional

electron energy sublevels, or

subshells, that formed part of a main

energy level.

Using the analogy of a staircase for an

energy level, this means that one of

Bohr’s main energy “steps” is actually a

group of several little “steps”.

Notice that the number of sublevels equals

the value of the principal quantum number.

The Magnetic Quantum

Number, ml

The scientific work of analyzing atomic

spectra was still not complete. If a gas

discharge tube is placed near a strong

magnet, some single lines split into new

lines that were not initially present. This

observation was first made by Pieter

Zeeman in 1897 and is called the

normal Zeeman Effect.

He observed, for example, triplets

where only one line existed without the

magnetic field. The Zeeman effect was

explained using another quantum

number, the magnetic quantum

number, ml , added by Arnold

Sommerfeld and Peter Debye (1916).

Their explanation was that orbits could

exist at various angles. The idea is that

if orbits are oriented in space in different

planes, the energies of the orbits are

different when the atom is near a strong

magnet.

Shapes and Orientations of Orbitals

The Spin Quantum Number,

ms

Paramagnetism is another kind of

magnetism of substances and is

recognized as a relatively weak

attraction to a strong magnet.

Paramagnetism refers to the magnetism

of individual atoms; ferromagnetism is

due to the magnetism of a collection of

atoms.

Samuel Goudsmit and George

Uhlenbeck, found that a fourth quantum

number was necessary to account for

the details of the emission spectra of

atoms due to paramagnetism.

Since they knew from classical physics

that a spinning charge produces a

magnetic moment, it seemed

reasonable to assume that the electron

could have two oppositely directed “spin

states”

In 1925, Wolfgang Pauli, suggested that

each electron spins on its axis. For an

electron, the two spins are equal in

magnitude but opposite in direction, and

these are the only choices; i.e., the spin

is quantized to two and only two values.

This fourth quantum number is called

the spin quantum number,ms, and is

given values of either +1/2 or -1/2.

Qualitatively, we refer to the spin as

either clockwise or counterclockwise or

as up or down.

Pauli’s Exclusion Principle

In a given atom, no two

electrons can have the same

set of four quantum numbers

(n, l, ml, and ms).

Summary of Quantum Numbers