Lecture 1
Refresher on Quantum Mechanics
1. Introduction
Want to describe “mechanics” of atomic-scale things, like electrons in atoms and molecules.
Why? These ultimately determine the shape, the energy, and all the properties of matter.
When do we need quantum mechanics?
de Broglie wavelength (1924)
λ = h p = h mv
h = 6.626 × 10−34 J s (Planck's constant)
Car
m = 1000 kg
v = 100 km/hr
Typical value on the highway
p = 2.8 × 10−4 kg m/s
λ = 2.4 × 10−38 m
Too small to detect. Classical
object!
Electron
9.1 × 10−31 kg
v = 0.01 c
Typical value in atom
p = 2.7 × 10−24 kg m/s
λ = 2.4 × 10−10 m
Comparable to size of atom.
Must account for wave
properties of an electron!
How to describe wave properties of an electron? Schrödinger equation (1926?)
Kinetic energy + Potential energy = Total Energy
Expressed as differential equation (Single particle, non-relativistic):
!
!2 2
!
! ! r,t +V r,t ! r,t = !i! ! r,t
2m
!t
( ) ( ) ( )
( )
Ψ(r, t ) : wavefunction
Steady-state, or time-independent:
!
!2 2
! ! r +V r ! r = E! r
2m
() () ()
( )
()
! r,t = ! r e
!iEt
()
!
E: energy
© Prof. W. F. Schneider
CBE 60547 – Computational Chemistry
University of Notre Dame
Spring 2012
1/5
Lecture 1
Refresher on Quantum Mechanics
2. Postulates of Non-relativistic Quantum Mechanics
Postulate I: The physical state of a system is completely described by its wavefunction Ψ. In general,
Ψ is a complex function of the spatial coordinates and time. Ψ is required to be:
1. single-valued
2. continuous and twice-differentiable
3. square-integrable ( ∫ Ψ ∗ Ψdτ is defined over all finite domains)
For bound systems Ψ can always be normalized such that
∫ Ψ Ψdτ = 1.
∗
Postulate II: To every physically observable quantity M there corresponds a Hermitian quantum
mechanical operator M̂ . The only observable values of M are the eigenvalues of M̂ .
Physical quantity
Operator
Expression
Position x, y, z
xˆ, yˆ , zˆ
x⋅, y⋅, z ⋅
Linear momentum px, …
pˆ x , …
Angular momentum lx, …
lˆx , …
⎛ ∂
∂ ⎞
−ih⎜ y − z ⎟ , …
∂y ⎠
⎝ ∂z
Kinetic energy T
Tˆ
h2 2
−
∇
2m
Potential energy V
Vˆ
V (r )
Total energy E
Ĥ
−ih
−
∂
,…
∂x
h2 2
∇ + V (r )
2m
Postulate III: If a particular observable M is measured many times on many identical systems in a state Ψ,
the average value of the result will be the expectation value of the operator M̂ :
M = ∫ Ψ* ( Mˆ Ψ )dτ
Postulate IV: The energy-invariant states of a system are solutions of the equation
"
Ĥ !(r,t) = i! !(r,t), Ĥ = T̂ + Vˆ
"t
If the system is in a time-independent stationary state, this reduces to the Schrödinger equation:
Hˆ Ψ (r) = E Ψ (r)
( ( ) ( )) are called
Postulate V: (The uncertainty principle.) Operators that do not commute Aˆ Bˆ Ψ ≠ Bˆ Aˆ Ψ
conjugate. Conjugate observables cannot be specified together to arbitrary accuracy. For
example, the error (standard deviation) in the measured position and momentum of a particle
must satisfy ΔxΔpx ≥ h 2 .
© Prof. W. F. Schneider
CBE 60547 – Computational Chemistry
University of Notre Dame
Spring 2012
2/5
Lecture 1
Refresher on Quantum Mechanics
3. Note on constants and units
Resource on physical constants: http://physics.nist.gov/cuu/Constants/
Resource for unit conversions: http://www.digitaldutch.com/unitconverter/
Unit converter available in Calc for Gnu emacs
Atomic units common for quantum mechanical calculations
Atomic unit
SI unit
Common unit
Charge
e=1
1.6021×10−19 C
Length
a0 = 1 (bohr)
5.29177×10−11 m
−31
0.529177 Å
Mass
me = 1
9.10938×10
Angular momentum
ħ=1
1.054 572×10−34 J s
Energy
Eh (hartree)
4.359744×10−18 J
Electrostatic force
kg
9
1/(4πε0) = 1
27.2114 eV
-2
8.987552×10 C N m
1.38065×10−23 J K−1
Boltzmann constant
2
8.31447 J/mol K
(see http://en.wikipedia.org/wiki/Atomic_units)
Energy units
1 eV = 1.60218×10−19 J = 96.485 kJ/mol = 8065.5 cm−1 = 11064 K kB
4. Example: Energy states of an electron in a box
z
3D box → 3 degrees of freedom
0,
⎧
V (r ) = ⎨
⎩∞,
L
0 < x, y , z < L
x, y, z ≤ 0, x, y, z ≥ L
Schrødinger eq
h2 ⎛ ∂ 2
∂2
∂ 2 ⎞
−
⎜ 2 + 2 + 2 ⎟ψ ( x, y, z ) = Eψ ( x, y, z )
2me ⎝ ∂x
∂y
∂y ⎠
ψ ( x, y, z ) = 0, x, y, z ≤ 0, x, y, z ≥ L
e−
L
Second-order linear partial differential equation
Boundary value (eigenvalue) problem
y
L
Separable
ψ ( x, y, z ) = X ( x )Y ( y ) Z ( z )
© Prof. W. F. Schneider
x
CBE 60547 – Computational Chemistry
University of Notre Dame
Spring 2012
3/5
Lecture 1
−
Refresher on Quantum Mechanics
h2 ⎛ 1 ∂ 2 X ( x)
1 ∂ 2Y ( y)
1 ∂ 2 Z ( z ) ⎞
+
+
⎜
⎟ = E
2me ⎝ X ( x) ∂x 2
Y ( y) ∂y 2
Z ( z ) ∂z 2 ⎠
0 < x, y, z < L
ftn x + ftn y + ftn z = constant → each term must be constant
−
h2 ∂ 2 X ( x)
= Ex X ( x)
2me ∂x 2
nx π x
,
L
n 2π 2 h2
Enx = x 2
2me L
X ( x) = sin
X (0) = X ( L ) = 0
function that twice differentiated returns itself
nx = 1, 2,3,...
Solutions called eignefunctions/wavefunctions and eigenvalues
Characterized by quantum number, one for each degree of freedom
Normalization – require that wavefunction square integrates to 1
L
C 2 ! sin 2
0
Xn =
x
nx ! x
2
dx " C 2 X n X n = 1 #C = ±
x
x
L
L
n !x
2
sin x , 0 < X < L
L
L
Dirac notation
Note increasing
nodes with
increasing energy
E ∝ n2
ΔE ∝ n
ΔE E ∝ 1 n
See Ho, JPC B
2005, 109, 20657.
© Prof. W. F. Schneider
CBE 60547 – Computational Chemistry
University of Notre Dame
Spring 2012
4/5
Lecture 1
Refresher on Quantum Mechanics
3 dimensional solution
3/2
n πy
n πx
n πz
⎛ 2 ⎞
ψ ( x, y, z ) = X ( x )Y ( y ) Z ( z ) = ⎜ ⎟ sin x sin y sin z
L
L
L
⎝ L ⎠
E = Ex + E y + Ez =
(n
2
x
)
+ n y2 + nz2 π 2 h2
2
2me L
nx , ny , nz = 1, 2, 3,K
One quantum number for each dof
20
(4,1,1)
(3,2,2)
(1,4,1)
(2,3,2)
(1,1,4)
(2,2,3)
(3,2,1)
(2,3,1)
(1,2,3)
(3,1,1)
(1,3,1)
(1,1,3)
(2,2,1)
(1,2,2)
(2,1,2)
(2,1,1)
(1,2,1)
(1,1,2)
15
⎛ 2 2
⎞
E ⎜ π h
2me L2 ⎟⎠
⎝
(3,1,2)
(2,1,3)
(1,3,2)
(2,2,2)
10
5
(1,1,1)
zero point energy
0
Degeneracy
Symmetry
Energy levels – depend on volume à pressure!!
© Prof. W. F. Schneider
CBE 60547 – Computational Chemistry
University of Notre Dame
Spring 2012
5/5