CHEM H90 Lecture Notes - Lecture 7: Uncertainty Principle, Wave Function
Lecture #7
• more about quantization
o only discrete energy values are allowed
o particle is not in one place with a certain speed, but is everywhere with a certain
probability density
o stays in this state (called eigenstate) until something happens
o impossible to know where, but can find probabilities from Schrodinger equation
(square the wavefunction)
• Born interpretation
o figured out that square of wavefunction is the probability of finding an electron
• Heisenberg uncertainty principle
o cannot know both speed and position of a quantum particle simultaneously
• degeneracy
o in more than 1 dimension, can have several states with the same energy
o due to symmetry of the potential
o e.g., 3D cubic box of side L
▪ Eijk=h2(i2+j2+k2)/(8mL2), i,j,k = 1,2,3..
▪ one quantum number for each dimension
▪ ground state energy is 3h2/(8mL2)
▪ first excited states are (2,1,1)
• symmetry and degeneracy
o because atom is spherical and space id 3D, find each energy level is degenerate, i.e.,
more than 1 allowed level
o for given n, can have ang mom/ up to n-1
o each/ has 2/+1 states
o in each level, electron can either be spin up or spin down
• important effects of quantum mechanics on electrons
o energy is discrete, not continuous (quantized)
o electron is a wavefunction, whose square gives probability density of finding it
(Born)
o impossible to simultaneously measure positon and speed (Heisenberg)
o can have several wavefunction with same energy (degeneracy) due to symmetry
• what we should have learned
o classical mechanics fails for very small particles
o light can behave as particles (photons)
o electrons can behave as waves
o bound systems have discrete energies, i.e., their energies are quantized
o symmetry leads to degeneracy (more than 1 state per energy level)
• pattern emerges
o for each value of l, there are 2/+1 orbitals
o for each value of n, l ranged from 0 to n-1
o for each n, there are n2 orbitals
o for each n, electrons have 2n2 places to go
o if n is 1, 2n2 is 2
o if n is 2, 2n2 is 8
• a very deep thing
find more resources at oneclass.com
find more resources at oneclass.com
