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Ch 2. The Schrödinger
Equation (S.E)
- Due to the contribution of wave-particle duality, an
appropriate wave equation need to be solved for the
microscopic world.
- Erwin Schrödinger was the first to formulate such an
equation
- We need to be familiar with operators, eigenfunction,
wavefunction, eigenvalues that are used in S.E.
MS310 Quantum Physical Chemistry
2.1 What determines if a system needs to be described using Q.M?
When do we use a particle description(classical) of an atomic or
molecular system and when do we use a wave (quantum
mechanical) description?
 two criteria are used!
1) The magnitude of the wavelength of the particle relative to the
dimension of the problem, a hydrogen molecule 100 pm vs.
a baseball 1x 10-34 m
 no way to show wave character of baseball
2) The degree to which the allowed energy values form a continu
ous energy spectrum
Boltzmann distribution :
ni gi [ i  j ] / kT

e
nj g j
MS310 Quantum Physical Chemistry
There are two limits
1) large T or small εi – εj
: small energy gap(almost continuous energy),
classical behavior
2) (εi – εj)/kT >> 1
: large energy gap, quantum behavior
Also we can derive the internal energy by Boltzmann distribution
3
E  kT
2
k : the Boltzmann constant
MS310 Quantum Physical Chemistry
Relative population in the different energy levels
a) Sharp energy levels
b) ∆E ≈ kT : classical behavior(nearby the continuous energy)
c) ∆E >> kT : quantum behavior(discrete energy)
MS310 Quantum Physical Chemistry
2.2 Classical waves and the nondispersive wave equation
Example of waves
a) Plane waves
b) spherical waves
c) cylindrical waves
Wave front : surface over the maximum or minimum amplitude
Direction of propagation of waves : blue arrows, perpendicular
to the surface
MS310 Quantum Physical Chemistry
Mathematically, we can describe the wave by wave function.
Amplitude of wave : related to position and time
Position : propagation depends on wavelength λ
Time : propagation depends on period T
x
t
 x, t   A sin 2   
 T
Arbitrary condition : Ψ(0,0)=0
Positions where amplitude is zero :
x
t
n t
2     n  x     
 T
2 T
x or t increase →
wave moves in the positive x direction
MS310 Quantum Physical Chemistry
Graph of wave functions
We use another form, too.
 x , t   A sin( kx  t )
k : wave vector, k 
2

ω : angular frequency, ω=2πν
Choice of a zero in time or position : arbitrary(free)
→ We can rewrite the wave equation
 x, t   A sin( kx  t   )
In this case, Ψ(0,0) ≠ 0 and φ : initial phase
MS310 Quantum Physical Chemistry
If two or more waves in same region : interference
1) constructive interference : enhancement of amplitude
2) destructive interference : cancellation of amplitude
Phase difference in interference
1) constructive interference : 2nπ → same sign
2) destructive interference : (2n+1)π → opposite sign
Also, we can think about two waves same frequency,
amplitude and opposite directions
 ( x, t )  A sin( kx  t )
2 ( x, t )  A sin( kx  t )
( x, t )  1 ( x, t )  2 ( x, t )  A[sin( kx  t )  sin( kx  t )]
MS310 Quantum Physical Chemistry
Use this formula
sin(   )  sin  cos   cos  sin 
Final result is simple than original equation
( x , t )  2 A sin kx cos t   ( x ) cos t
position of nodes are same at all time!
→ ‘standing’ waves
standing waves represent the stationary state(no change by time)
MS310 Quantum Physical Chemistry
Both of traveling waves and stationary waves, amplitude
and distance is not independent
∂2 Ψ( x, t ) 1 ∂2 Ψ( x, t )
: classical nondispersive wave eq.
= 2
2
2
∂x
v
∂t
v : velocity of wave propagation
This equation is a start point of the Schrödinger Equation.
Ex) 2-2. show that traveling wave satisfies wave equation.
Solution) Ψ( x, t ) = A sin( kx - ωt + φ)
∂2 Ψ( x, t )
= - k 2 A sin( kx - ωt + φ)
2
∂x
1 ∂2 Ψ( x, t )
ω2
= - 2 A sin( kx - ωt + φ)
2
2
v
∂t
v
Equating two results, v=ω/k
MS310 Quantum Physical Chemistry
2.3 Waves represented by complex functions
We know the traveling wave equation
 x, t   A sin( kx  t   )  A cos(kx  t   


)  A cos(kx  t   ' )
Where φ’ = φ - π/2
i
Use the Euler’s formula e  exp(i )  cos   i sin 
 x , t   Re(A exp i ( kx  t   ' ))
We write the wave equation in complex form
 x , t   A exp i ( kx  t   ' )
Why uses a complex form?
1) All quantities can obtain by this form
2) easier than real form with differentiation and integration.
MS310 Quantum Physical Chemistry
Important formulas
1) Complex number : a+ib(a,b:real)
a
or equivalent form, re i (r  a 2  b 2 and   cos -1 )
r
2) Complex conjugate of f(number of function) : f*
substituting i to –i
ex) (a+ib)* = a-ib, ( re i )*  re  i
3) Magnitude of f(number of function) : |f|
| f |
f*f
ex) | a  ib | a 2  b 2 , | re i | r
MS310 Quantum Physical Chemistry
2.4 The Schrödinger Equation
How we can obtain the Schrödinger Equation?
→ start with the classical wave equation and stationary wave
( x , t )   ( x ) cos t
Substitute the wave equation by stationary wave, we obtain
d 2 ψ( x ) ω 2
+ 2 ψ( x ) = 0
2
dx
v
Using the relations ω=2πν and νλ=v
d 2ψ ( x) 4π 2
+ 2 ψ ( x) = 0
dx 2
λ
Until now, it is classical wave.
MS310 Quantum Physical Chemistry
‘Introduce’ the quantum mechanics by de Broglie relation  
h
p
d 2ψ ( x) 4π 2p 2
+
ψ ( x) = 0
2
2
dx
h
Momentum is related by total energy E and potential energy V(x)
p2
 E  V ( x ), p  2m( E  V ( x )
2m
Substituting the wave equation by this relation
d 2ψ ( x) 8π 2m
+ 2 [E - V( x)]ψ ( x) = 0
2
dx
h
Use ℏ=h/2π, we obtain the time-independent Schrödinger Eq.
 2 d 2 ( x)

 V ( x) ( x)  E ( x)
2
2m dx
MS310 Quantum Physical Chemistry
How one obtain the time-dependent Schrödinger Equation?
Starting from the solution of classical wave equation
( x , t )  Ae
 x
 i  t  
 
 Ae
x

 2i  vt  


On the other hand
E
E  h  2   
2
p
  h  2  1 
p
p
 2
Then
( x, t )  Ae
cf.  
h
2
i
    Et  px 

Wave equivalent of a free particle of
energy E and momentum p moving on the x-direction
MS310 Quantum Physical Chemistry
Let’s try now,
      i    i 
p2
 2
2
2
      p  x   p     2   p    
x 2
x  x         

 


i 

    E    E  
 i
t



i t
t
2
p
We know that E 
 V ( x, t )
2m
 p2

 E( x , t )  
 V ( x , t )  
 2m

p2

 ( x , t )  V ( x , t ) ( x , t )
2m
Now replace E( x , t ), p ( x , t )
2
( x , t )
 2  2 ( x , t )
Time-dependent
i


V
(
x
,
t
)

(
x
,
t
)
Schrödinger Eq.
t
2m x 2
MS310 Quantum Physical Chemistry
One of our focus is stationary system.
In this case, both of time-dependent and time-independent are satisfied.
( x , t )
i
 E( x, t )
t
For stationary state,  ( x , t )   ( x ) f ( t )
Substitute the equation, we obtain
E
i t
df ( t )
df ( t )
E
i
 Ef ( t ),
  i f ( t ), f ( t )  e 
dt
dt

i
E
t

Finally, ( x , t )   ( x )e
→ same form as classical standing wave
MS310 Quantum Physical Chemistry
2.5 Solving the Schrödinger Equation
Key concept : operators, observables, eigenfunctions and
eigenvalues
Operators in a classical mechanics
Ex) Velocity in Newton’s second law
d2x
m 2  F ( x, t )
dt
t
1 2
v ( t 2 )  v ( t1 )   F ( x , t )dt
m t1
1) Integrate the force acting on the particle over the interval
2) Multiply by the inverse of the mass
3) Add the quantity to the velocity at time t1
MS310 Quantum Physical Chemistry
How about a operators in Quantum Mechanics?
→ every measurable quantities(observables) have their
operator each.
(ex : energy, momentum, position)
Notation : caret, Ô
Differential equations : set of solutions
Operator Ô has a set of eigenfunctions and eigenvalues
Ô n  an n
Ψn : eigenfunctions, an : eigenvalues
Ex) hydrogen atom
eigenfunctions : each orbitals(1s, 2s, 2px, …)
eigenvalues : each orbiral energies
MS310 Quantum Physical Chemistry
We see the time-independent Schrödinger equation.
2  2
{
 V ( x )} n ( x )  E n n ( x )
2
2m x
2  2

 V ( x ) : energy operator(H amiltonian ) Hˆ
2
2m x
ˆ  E 
This equation can be written by H
n
n n
Important physical implication : measurement process in Q.M.
MS310 Quantum Physical Chemistry
2.6 Eigenfunctions of Q.M. operator are orthogonal
Orthogonality in 3-dimensional vector space :
x•y = x•z = y•z = 0
Similar, orthogonality in functional space is defined by

*

i
 ( x ) j ( x )dx  0, unless i  j

 i ( x ), j ( x ) : eigenfunct ions of quantum operator
If I = j, the integral has a nonzero value
Functions can be normalized and form an orthonormal set.

*

 i ( x ) i ( x )dx  1, i  1,2,..., n

MS310 Quantum Physical Chemistry
In 3-dimension, normalization must be 3-dimension.
Closed-shell atoms are spherical symmetric and we normalized
the wave functions in spherical coordinate
Volume element in spherical
coordinate : r2 sin θ dr dθ dφ,
not a dr dθ dφ
MS310 Quantum Physical Chemistry
2.7 Eigenfunctions of Q.M. operator form a complete set
completeness in 3-dimensional vector space :
Any vector in 3-dimensional can be represented by linear
combination of vector x, y, and z
Similar, completeness in functional space :
Wave function can be expanded in the eigenfunctions of any
Q.M. operator

f ( x )   bn n ( x )
n 1
It is same formalism as a Fourier series.
We choose a periodic function in [-b,b]
Fourier series is represented by
nx
nx
f ( x )  d 0   [cn sin(
)  d n cos(
)]
b
b
n 1

MS310 Quantum Physical Chemistry
If f(x) even : cn=0, dn calculated by orthogonality
mx
mx
mx
f
(
x
)
cos(
)
dx

(cos(
))
(
d

d
cos
(
))dx

0
n
b

b
b
b
m
b
b
b
b
  (cos(
b
mx
mx
))d m cos(
)dx  bd m
b
b
1
m x
d m   f ( x ) cos(
)dx , m  0
b b
b
b
Generally
b
1
d0 
f ( x )dx

2b  b
And this approximation is nearly exact.
m
f ( x )  d 0   d n cos(
n 1
nx
)
b
MS310 Quantum Physical Chemistry
Accuracy of Fourier series
a) Yellow line : real function

x2
x 2
x
2b2
f ( x )  {2( )  1}e
, 1   1
b
b
b) red line : fourier series approximation, n=2,3,4 and 6
MS310 Quantum Physical Chemistry
Summary
- The time-dependent and time-independent
Schrödinger equations play the role in solving
quantum mechanical problems.
- Operators, eigenfunctions, wave functions and
eigenvalues are key concepts to solve quantum
mechanical wave equations.
MS310 Quantum Physical Chemistry