Chapter 28
Quantum Mechanics of Atoms
Chapter 28
• Quantum Mechanics – A New Theory
• The Wave Function and Its Interpretation; the Double­Slit Experiment
• The Heisenberg Uncertainty Principle
• Philosophy: Probability versus Determinism
• Quantum­Mechanical View of Atoms
• The Hydrogen Atom; Quantum Numbers
• Complex Atoms; the Exclusion Principle
• The Periodic Table of Elements
Quantum Mechanics – A ‘New’ Theory
Quantum mechanics incorporates wave­particle duality. It successfully explains energy states in atoms and molecules, the relative brightness of spectral lines, and many other phenomena.
It is widely accepted as being the fundamental theory underlying all physical processes.
Quantum mechanics is essential to understanding atoms and molecules, but can also have effects on larger scales.
The Wave Function and Its Interpretation; the Double­Slit Experiment
An electromagnetic wave has oscillating electric and magnetic fields. This leads directly to interference and diffraction – the fields add as vectors at any point (superposition principle).
But, What is oscillating in a matter wave?
The Wave Function and Its Interpretation; the Double­Slit Experiment
This role is played by the wave function, Ψ. The square of the wave function at any given point is proportional to the number of electrons expected to be found there.
For a single electron, the wave function is related to the probability of finding the electron at that point.
Added up over all of space this has to be exactly 1.
The Wave Function and Its Interpretation; the Double­Slit Experiment
For example: the interference pattern is observed after many electrons have gone through the slits.
If we send the electrons through one at a time, we cannot predict the path any single electron will take, but we can predict the overall distribution ­ this is just the probability of finding a given electron a that point
The Heisenberg Uncertainty Principle
Quantum mechanics tells us there are limits to measurement – not because of the limits of our instruments, but inherently.
This is due to the wave­particle duality, and to an interaction between the observing equipment and the object being observed.
The Heisenberg Uncertainty Principle
Imagine trying to see an electron with a powerful microscope. At least one photon must scatter off the electron and enter the microscope, but in doing so it will transfer some of its momentum to the electron.
The Heisenberg Uncertainty Principle
The uncertainty in the momentum of the electron is taken to be the momentum of the photon – it could transfer anywhere from none to all of its momentum. In addition, the position can only be measured to about one wavelength of the photon.
The Heisenberg Uncertainty Principle
Combining, we find the combination of uncertainties:
This is called the Heisenberg uncertainty principle.
It tells us that the position and momentum cannot simultaneously be measured with a precision better than this relationship.
The Heisenberg Uncertainty Principle
This relation also applies between the uncertainty in time and the uncertainty in energy:
This says that if an energy state only lasts for a limited time, its energy will be uncertain. It also says that conservation of energy can be violated if the time is short enough. Philosophic Implications; Probability versus Determinism
Newtonian mechanics describes a deterministic world. If you know the forces on an object and its initial velocity, you can predict where it will go ­
Exactly, if you know the initial conditions exactly !
Quantum mechanics is very different – you can predict what large numbers of electrons will produce, but have no idea what any individual one will do. Quantum­Mechanical 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: it has spherical symmetry.
Quantum Mechanics of the Hydrogen Atom; Quantum Numbers
There are four different quantum numbers needed to specify the state of an electron in an atom.
• Principal quantum number n gives the total energy:
This is exactly the same as found by Bohr.
Quantum Mechanics of the Hydrogen Atom; Quantum Numbers
2. Orbital quantum number l gives 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, and can take on integer values from –l to +l.
Quantum Mechanics of the Hydrogen Atom; Quantum Numbers
This plot indicates the quantization of angular momentum direction for l = 2. The other two components of the angular momentum are undefined ­ meaning L (as a vector) can have a direction anywhere on a conical surface, only the angle to the axis can be defined.
Quantum Mechanics of the Hydrogen Atom; Quantum Numbers
1. 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. [ There is also s =1/2 which is the electron’s “spin angular momentum” which only becomes important when one has multiple electrons ]
Quantum Mechanics of the Hydrogen Atom; Quantum Numbers
This table summarizes the four quantum numbers that are important for an electron in an atom..
Other notation: s­state l=0 1s, 2s, etc
p­state l=1 2p, 3p, etc d­state l=2 3d, 4d, etc Quantum Mechanics of the Hydrogen Atom; Quantum Numbers
The angular momentum quantum numbers do not affect the energy level much, but they do change the spatial distribution of the electron cloud.
2s 2pz {2px} +/− i{2py}
Quantum Mechanics of the Hydrogen Atom; Quantum Numbers
“Allowed” transitions between energy levels occur between states whose value of l differ by one:
Other, “forbidden,” transitions can also occur for some conditions but with much lower probability.
Energy Splittings in a Magnetic Field
When B = 0
All the ml states Have the same energy ( n fixed ) When B present, it defines the z­axis
And ∆E = MB ml
(Zeeman Splitting)
∆l = +/−1, ∆m = 0,+/−1
B=0
Complex Atoms; the Exclusion Principle
Complex atoms contain more than one electron, so the interaction between electrons must be accounted for in the energy levels. This means that the energy depends on both n and l.
A neutral atom has Z electrons, and Z protons in its nucleus. Z is the ‘atomic number’.
Complex Atoms; the Exclusion Principle
In order to understand the electron distributions in atoms, another principle is needed. This is the Pauli exclusion principle:
No two electrons in an atom can occupy the same quantum state.
The ‘quantum state’ is specified by the four quantum numbers; no two electrons can have the same set of quantum numbers.
Complex Atoms; the Exclusion Principle
This chart shows the occupied – and some unoccupied – states in He, Li, and Na.
The Periodic Table of the Elements
We now have enough information to understand the basic organization of the periodic table. 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 l sub­shell to 2(2l + 1).
The Periodic Table of the Elements
Each value of l is given its own letter symbol. Maximum number of electrons is the number of ml values time number of ms values or
2(2l+1) in an l ­ level
The Periodic Table of the Elements
Electron ‘configurations’ are written by giving the value for n, the letter code for l, and the number of electrons in the sub­shell as a superscript.
For example, here is the ground­state electronic configuration of sodium (Z = 11) :
The Periodic Table of the Elements
This table shows the configuration of the outer electrons only.
Full shells
Z = 2,10,18
etc
are very stable and rather inert chemically
Unfilled inner shells, like
3dn will form magnetic ions
The Periodic Table of the Elements
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. Those with one e− or one empty spot tend to ‘ionize’ easily to end up with full shells, and quite often form ionic bonds with each other.
The inner columns, with partly filled outer shells, have more similar chemical properties. These atoms tend to form more covalent chemical bonds.
X­Ray Spectra and Atomic Number
Inner electrons can be ejected by high­energy electrons. The resulting sharp lines in the X­ray spectrum are characteristic of the element.
This example is for molybdenum.
X­Ray Spectra and Atomic Number
Measurement of these spectra allows determination of inner energy levels, as well as Z, as the wavelength of the shortest X­rays is inversely proportional to Z2.
Fluorescence and Phosphorescence
If an electron is excited to a higher energy state, it may emit two or more photons of longer wavelength as it returns to the lower level.
Flourescence – UV absorption, visible emission
Phosphorescence ­ metastable (long lived) middle state
Lasers
1. Lots of atoms need to be excited into a higher stste to cause population inversion.
2. The higher state must be a metastable state, so that once the population is inverted, it can stay that way. The laser transitions then will occur through stimulated emission rather than spontaneously.
Lasers
The laser beam is narrow, only spreading due to diffraction, which is determined by the size of the end mirror. Laser light is “coherent” – or in phase.
An inverted population can be created by exciting electrons to a state from which they decay to a metastable state. This is called ‘optical pumping’.
Summary of Chapter 28
• Quantum mechanics is the basic theory at the atomic level; it is statistical rather than deterministic
• Heisenberg uncertainty principle:
• Electron state in atom is specified by four numbers, n, l, ml, and ms
Summary of Chapter 28
• 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 in atoms depend on n and l, except for H. The other quantum numbers also result in small energy differences, mostly in magnetic fields.
Summary of Chapter 28
• Pauli exclusion principle: no two electrons in the same atom can be in the same quantum state; this dictates the structure of the Periodic Table given the rules for the allowed quantum numbers.
• Electrons are grouped into shells and sub­shells
• Periodic table reflects this shell structure
• Order of energies: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, (6d, 5f) with some mixing, initially to get half full or full inner shells by ‘stealing’ an outer s­electron