Experiments that lead to the demise of classical mechanics for molecules, atoms, and light!
1. blackbody radiation
2. photoelectric effect
3. Davisson and Germer electron scattering
4. Atomic spectra, and Bohr's model
Introduction of quantization and wave properties
De Broglie waves
Classical Waves:
1. equations of motion for "things" bobbing around in the sea [wave motion].
2. Travelling waves, and the Classical Wave Equation
3. Superposition of waves
Start with the classical wave equation
Introduce lambda=h/p, and E = h mu [de Broglie and Einstein] into this equation
We showed we could get the SCHRODINGER WAVE EQUATION
This is a quantum mechanical EQUATION OF MOTION
Momentum is represented by an operator, and so is energy E.
The Schrodinger equation is a differential equation: first order in time, second order in space.
Introduce stationary states for classical waves
Show that if psi(x,y,z,t) = psi(x,y,z) exp[-iEt/hbar], then we get
the time-independent Schrodinger Eqn : H psi = E psi
This is an eigenvalue-eigenvector equation
Given only H, the Hamiltonian operator, we can solve for psi and E
Many chemical problems are TIME INDEPENDENT so we can use the Time-independent Scrodinger EQN.
What do wavefunctions mean?
Born interpretation of what psi . psi* dx is: probability densities
Postulates:
I. physical properties represented by psi
II. solutions obtained from time-dependent or time-independent Schrodinger Equations.
III. dynamic variables represented as operators.
IV. Operators derived from classical expressions using the definition for the "p" operator and "x" operator
V. The eigenvalues represent the possible values for measuring the quantity represented by the operator
VI. Expectation values are measured as an average over many measurments, like
Properties of Operators: hermitian, eigenfunctions are orthogonal and properties of
degenerate eigenfunctions.
Exactly solvable 1-D problems
Particle in a box
Energy is quantized: E = n^2h^2 / 8mL^2
The stationary states are bounded by the end of the box
The stationary states are sinusoidal functions with NODES
Electronic states of conjugated chains are approximately like electrons in a 1-D box.
Transitions between the energy levels [E1, E2] can be done using light of a particular frequency, mu:
h mu = E1 - E2
Separation of variables: particle in a 3-D box
degeneracy
Free-particle
Description of tunneling
One electron atoms
Definition of the reduced mass, and reduction of the two body problem [nucleus+electron]
to a one-body problem.
Introduction to spherical polar coordinates to make this problem SEPARABLE
Write the Laplacian operator, and the Hamiltonian in spherical polar coordinates
Show that the equation Hpsi = E psi is separable in these coordinates.
Get three equations to solve: phi equation, theta equation and RADIAL equation (r).
The phi equation is easily solved, and theta solutions are the Associated Legendre functions
The radial wavefunction is solved using an eqaution solved by Laguerre.
The total wavefunction is a PRODUCT of R(r) Theta (theta) Phi (phi)
the angular functions Phi * Theta are called SPHERICAL HARMONICS
You can get these functions in NORMALIZED FORM from tables.
There are 3 quantum numbers: n, l, and ml.
The energy only depends on n, so there are many degenerate states for each n!
The l quantum numbers can go from 0.... n-1. They can be denoted by letters: s,p,d.. etc
s-orbitals are spherically symmetric
l=1 are the p orbitals. l=1, ml= is the pz orbital. px and py are made from linear combinations
of the spherical harmonics Y(1,1) and Y(1,-1)
RADIAL DISTRIBUTION FUNCTIONS r^2 |R(r)|^2 give the probability of finding an
electron in a thin spherical shell
a distance r from the nucleus.
The quantum numbers l and ml are related to the ANGULAR MOMENTUM of an electron in
an ORBITAL (wavefunction).
The total angular momentum is l(l+1) hbar^2
the projection of the angular momentum in the z direction Lz = ml hbar.
Lx and Ly cannot be measured simultaneously with Lz
consequences: p1, p0 and p-1 are degenerate in energy.
they all have the same l value and hence same total angular momentum
they all differ in their angular momentum in the z-direction, so in a MAGNETIC FIELD in the
z direction, they differ in energy
Electrons have another quantum number! Cannot get it from the SCHRODINGER EQUATION
Its called spin; which has a quantum number S, and Sz = -1/2 or + 1/2.
Many electron atoms cannot be solved exactly. eg. He has a nuclear charge Z=2, and 2 electrons
the ELECTRON REPULSION makes this problem NOT SEPARABLE
Two methods can be used to solve He approximately:
1. the variational principle
2. perturbation methods
Both give energies close to experiment of -79eV.
Many electron wavefunctions need to be anti-symmetric [Pauli Principle]
The ZEROTH ORDER wavefunctions for many electron systems can be written as SLATER determinants
We generated the zeroth order wavefunctions for He in the ground state [1s1s]
and
the first excited state [1s2s]
This excited state can be a triplet [S=1] or singlet [S=0] state
the triplet is lower in energy
Total angular momentum: J = L+S
Many electron states are given a shorthand notation -> term symbols!
Let's do some molecules!
Lecture 1
Lecture 2
Lecture 3
Lecture 4
Lecture 5
Lecture 6
Lecture 7
Lecture 8
Lecture 9
Lecture 10
Lecture 11
Lecture 12
Lecture 13
Lecture 14
Lecture 15
Lecture 16