Transcript Document

Chemistry 330
The Quantum Mechanics of
Simple Systems
Properties of an Acceptable
Wavefunction
The wavefunction
must be




Continuous
Single-valued
No singularities
Continuous first
derivatives
The Curvature of a
Wavefunction
The average kinetic
energy of a particle
can be ‘determined’
by noting its
average curvature.
The Wavefunction
The wavefunction  is a
probability amplitude
Square modulus (*
or 2) is a probability
density.
The probability of
finding a particle in the
region dx located at x is
proportional to 2 dx.
Particle Wavefunctions
The wavefunction
for a particle at a
well-defined location
is a sharply spiked
function
Zero amplitude
everywhere except
at the particle's
position.
Postulates of quantum
Mechanics
There exists a wavefunction that is the
solution of the Schrödinger equation
x, y, z, t   wx, y, z, t   iux, y, z, t 


d


1



* - probability density function
Postulates #2
The expectation value of any
observable is defined as follows
A    Â d


<A> - the expectation value of
the operator A
Postulates #3
The wavefunction must satisfy the
relationship

Ĥ  i
0
t
Ĥ
- the Hamiltonian
operator
The Spatial and Temporal
Functions
Consider the complete wavefunction
((x,y,z,t)) to be a product of a


Spatial function – (x,y,z)
Temporal function – f(t)
The Hamiltonian and the
energy
The eigenvalues for the Hamiltonian
operator are the total energy of the
system
The temporal function describes the
variation of the potential energy with
time
f t   Ae
iEt

Commutators and
Expectations Values
Two operators that commute

Observables corresponding to those
operators can have precise values
simultaneously
Two operators that don’t commute

Observables corresponding to those
operators can’t be determined
simultaneously
Hermitian Operators
For an operator
 Ĥ


d    Ĥ d


An operator that satisfies this condition
is said to be Hermitian
Superposition and Expectation
Values
A wavefunction that is written as a
linear combination
  c11  c 22  ...  cnn
The probability of measuring a particular
eigenvalue  {cn}2
Superposition and
Wavefunctions
The wavefunction for a
particle with an illdefined location
Superposition of several
wavefunctions of
definite wavelength
An infinite number of
waves is needed to
construct the
wavefunction of a
perfectly localized
particle.
The General Approach
Solve the Schrödinger equation for the
physical description of the system
Obtain expectation values for the
observables
Obtain the probability density function
at various points in space
The Free Particle
The particle moves in the absence of an
external force
x
Choose V = 0
The Schrödinger Equation
The Schrödinger equation
d   2mE 
  2   0
2
dx
  
2
The wavefunctions
ix
1  Ae
 2  Be ix
The Particle in a ‘Box’
A particle in a onedimensional region
with impenetrable
walls.
Potential energy is
zero between x = 0
and x = L,
Rises sharply to
infinity at the walls.
The Schrödinger Equation
The Schrödinger equation
d   2mE 
  2   0
2
dx
  
2
The wavefunctions
 nx 
1  C sin

 L 
n – energy level
L – length of box
The Energy Expression
n
En 
2
2mL
2 2
nh

2
8mL
2
h
E o  En1 
2
8mL
E  En1  En  2n  1E o
The energy of the
particle depends
directly on the value
of n!
2
2
2
The Solutions for the Particle
in a ‘Box’
The first five
normalized
wavefunctions of a
particle in a box.
Successive
functions possess
one more half wave
and a shorter
wavelength.
The Particle in a ‘Box’
The allowed energy
levels for a particle
in a box. Note that
the energy levels
increase as n2, and
that their separation
increases as the
quantum number
increases.
The Probability Distributions
The first two
wavefunctions and
the corresponding
probability
distributions
The probability
distribution in terms
of the darkness of
shading.
Orthogonal wavefunctions
A graphical
illustration of
orthogonality for two
wavefunctions
The integral is equal
to the total area
beneath the graph
of the product, and
is zero.
Tunnelling
Suppose that the energy at the walls
dos not rise abruptly to infinity!!
V
Tunneling Probability
The probability that the particle will
tunnel through the wall

Transmission probability

 
L

e e
T  1 
161   

 L 2
  2mE
1
2

1
The Particle in a 2D ‘Box’
A two-dimensional
square well.
Potential energy is
zero between x = 0
and x = L1 and y= 0
and y = L2,
Rises sharply to
infinity at the walls.
The Schrödinger Equation
The Schrödinger equation
 d  d    2mE 



  2   0
2
2 
dx    
 dx
2
2
The wavefunctions
 nx x 
2

Xx  
sin
L1
 L1 
 ny x 
2

Yy  
sin
L2
 L2 
The Energy Expression
The energy of the
particle depends
directly on the
values of nx and ny
and ny !
n x , y  h
2
Enx ,y  
2
8mL x ,y 
E  Ex  Ey
2
The Quantum Mechanical
Harmonic Oscillator
Examine a particle
undergoing
harmonic motion.
F  kx
1 2
V  kx
2
The Schrödinger Equation
The Schrödinger equation
 d  1 2
  kx   E
2
2m dx
2

2
2
The wavefunctions
 v  NvHv y  e
y
2
2
2

x
 
y  ;  


 mk 
Hv- Hermite Polynomials
v – vibrational quantum number
1
4
The Energy Expression




k
Ev  
v 1
2
m
v – vibrational
quantum number
 h v  1
2
The force constant –
k
E o  1 h
2
Particle mass – m
The energy of the
oscillator depends



The First Wavefunction of the
QM Oscillator
The normalized
wavefunction and
probability
distribution (shown
also by shading) for
the lowest energy
state of a harmonic
oscillator.
Harmonic Oscillator
The normalized
wavefunction and
probability
distribution (shown
also by shading) for
the first excited state
of a harmonic
oscillator.
The Wavefunctions
The first five
normalized
wavefunctions of the
QM harmonic
oscillator
The Probability Distributions
The probability
distributions for the
first five states of a
harmonic oscillator
Note – regions of
highest probability
move towards the
turning points of the
classical motion as v
increases.
Angular Momentum of A
Particle Confined to a Plane
Represented by a
vector of length |ml|
units along the zaxis
Orientation that
indicates the
direction of motion
of the particle.
Quantization of Rotation
Examine a particle
undergoing rotation
in a plane

ml 
E
2
2I
Jz  ml
The Schrödinger Equation
The Schrödinger equation
 1  
 E
2
2
2m r 
2
2
  

E

2
2I 
2
2
The Wavefunctions
The wavefunctions are dependent on
the quantum number ml
m l 
1 i ml
e
2
The Momenta and their
Operators
The angular momentum operators are
written as follows

l̂z  i

Eigenvalue - Jz
2

1 

1
 
2
2
L̂   
sin 

2
2 
 sin   
 sin  
Eigenvalue – J 2
Wavefunctions of the Particle
on a Ring
The real parts of the
wavefunctions of a
particle on a ring.
For shorter
wavelengths, the
magnitude of the
angular momentum
around the z-axis
grows in steps of ħ.
3-D Rotation
Suppose we allow
the particle to move
on the surface of a
sphere.
Two angles


 - the azimuthal
angle
 - the colatitude
The Schrödinger Equation
The Schrödinger equation

2
   V  E
2m
2
2 – the Laplacian Operator
The Wavefunctions
The wavefunctions are dependent on
the angles  and .
The Schrödinger equation is simplified
by the separation of variables
technique.
,    
The Solutions
The solutions to
the SE for this
systems are the
spherical
harmonics
l
ml
Yl,ml
0
0
1/(4)1/2
1
0
3/(4)1/2 Cos
1
±
1
-
1/2 Sin
3/(8)
+
e±i
The Probability Distributions
A representation of
the wavefunctions of
a particle on the
surface of a sphere.
Space Quantization
Represent the vectors for the angular
momenta as a series of cones!
The Stern-Gerlach Experiment
a) The experimental
arrangement for the
Stern-Gerlach
experiment: the
magnet provides an
inhomogeneous field.
b) The classically
expected result.
c) The observed outcome
using silver atoms.
The Spin Functions of
Electrons
An electron spin (s = 1/2)
can take only two
orientations with
respect to a specified
axis.
 electron (top) electron with ms = +1/2;
 electron (bottom) is
an electron with ms = 1/ .
2