Download Key Concepts for Exam #2

Key Concepts for Exam #1
Correspondence Principle
Quantum mechanics approaches classical mechanics in the limit of large quantum
numbers.
Photoelectric Effect
If the frequency of incident light is above the threshold frequency, then as the intensity of
light increases, the kinetic energy of ejected electrons remains constant and the number of
electrons increases. In addition, as the frequency of light increases, the kinetic energy of ejected
electrons increases and the number of electrons remains constant.
If the frequency of the light is below the threshold frequency, then each individual photon
lacks enough energy to remove an electron from the metal and no electrons are released from the
metal.
Blackbody Radiation
c
T max  2 where the constant c 2  1.44 10 2 m
5
Bohr Atom: Spectral Lines
 1
1
1 
 R 2  2 

 n1 n2 
where Rydberg's constant, R, is 1.097  107 m1
DeBroglie Waves
h

p
where h is Planck's constant, and p is the momentum of the particle
Well-behaved Wave Functions
1. The function must be single-valued; i.e. at any point in space, the function  2 must have
only one numerical value.
2. The function must be finite and continuous at all points in space. The first and second
derivatives of the function must be finite and continuous.
3. The function  2 must have a finite integral over all space.
all space 
*
d  a finite number
Harmonic Oscillators: Hermite Polynomials
The Schrodinger equation for the harmonic oscillator is
 2 d 2 1 2
ˆ
H  
 kx   E
2 dx 2 2
The energy eigenvalues for the harmonic oscillator are given by
1 h k 
1

E  v 
  v  h
2  2 m 
2

where Planck's constant, h,is 6.626  1034 J s; k is the force constant;  is the frequency of the
oscillator, and v = 0, 1, 2, 3, …
1 k
2 
where k is the force constant,  is the reduced mass, and  is the frequency of the oscillator.

Kronnecker delta  nm 
 nm    *n  m d
The eigenfunctions  n and  m are orthogonal when  nm  0 . The other possible value
of the Kronnecker delta is 1, and it occurs when n = m. The Kronnecker delta can only equal zero
or one; no other values are possible.
Expectation Values
 * ˆ  d

 ˆ 
*
  d
or
 ˆ    * ˆ  d , where  is normalized
Orthogonality
Two functions f and g are orthogonal to each other if
f
*
gdq   g * fdq  0 where f
*
is the
complex conjugate of f and g * is the complex conjugate of g.
Particle In a Box
The potential energy, V, is zero inside the box, and infinite outside the box. The wave function
must be zero outside the box. Therefore, the wave function must equal zero at x = a and x = 0
(where a is the length of the box).
h2n2
The energy eigenvalue for a one dimensional box of length a is E 
where n  1,2,3,...
8ma 2
The energy eigenvalue for a three dimensional box of sides a, b, and c is
2
2
h 2  n x n y n z2 
E
  where n x , n y , n z  1,2,3,...
 
8m  a 2 b 2 c 2 


Linear Momentum
d  d
pˆ x  i

dx i dx
Angular Momentum
The angular momentum operators have the following eigenvalue equations
Lˆ2Yl ,m  l l  1Yl ,m where l  0,1,2,...
where m  0,1,...,l
Lˆ Y  m Y
z l ,m
l l ,m
Spin Operators

Sˆ z  s    s  m s  s
2
Sˆ 2  s  ss  1 2  s
l
where m s  
where s 
1
2
1
(for electrons)
2
Other Concepts
 Complex conjugates, probability density
 Operators, commutators, eigenvalues, eigenfunctions
 How to normalize wave functions
 Orbitals and quantum numbers: m, l , ml , ms
 Heisenberg's Uncertainty Principle
 Pauli Exclusion Principle
 Legendrian Operator, Laplacian Operator
 Diatomic Rigid Rotor
 Plots of radial distribution functions (e.g. Rn2,l r 2 vs. r)



Bohr magneton
Aufbau Principle for Atomic Electron Configurations
Closed vs. open shell of electrons
Relevant pages in Mortimer are Chapters 14, 15, 16, and Sections 19.1 and 19.2. Please note that
the sections from Chapter 16 that address symmetrization, Russell-Saunders coupling, and Slater
determinants will NOT be covered by Exam #1.
Note: This list is only a guide to help you study. It is NOT comprehensive, and the exam may
cover any topics discussed in class.