Lecture1

1. Further information 4. Classical mechanics (pp 911- 914 )
2. 11.3 The Schrödinger Equation (pp 294)
Lecture-on-line
Introduction to Classical mechanics and the Schrödinger equation (PowerPoint)
/ dt2 )t to
t2
vv (t o )∆ t
(d2rv / dt2 )t to ∆t2
v r
(t
o
)
vr(to + ∆t)
FHvo(wtoev)er
fromv
Newtons law:
V gradV
Thus :
m(d2rv / dt 2 )t to
rv(to
t) = rv(to) + vv(to )
/ dt2)t
to
t2
t) = rv(to) + vv(to )
t + 1 (d2rv 2
/ dt2 )t to
t2
Appendix A
Newton' s Equation and determination of position..cont
rv(to
t) = rv(to) + vv(to )
t + 1 (d2rv 2
1 t - 2m (gradV)t=t 0
t2
Newton' s Equation and determination of position..cont
X
Rate of change of position with time
Expression for total energy
The total energy of a mass m and velocity
vpr aarltsioclheasweitnherpgoysition
rr
,
ET Ekin Epot(rr )
Atkins ,**)
Classical Mechanics A particle in 3-D has the following attributes
mass
Z
mr
Position
1. Mass m 2. Position rr
velocit y
3. Velocity
vr
Y
v = dr/dt
Kinetic energy due to motion
Potential energy due to forces
The kinetic energy can be written as :
Ek
1 mv2 2
Or alternatively in terms of the linear momentum:
2m
You must know that : The quantum mechanical Hamiltonian Hˆ is constructed from the classical Hamiltonian H by the transformation
HClass
Hˆ
1 2m
pˆ x2
pˆ
2 y
pˆ z2
V(xˆ, yˆ, zˆ)
Classical Mechanics Quantum Mechanics
x
px
xˆ
x ; pˆx
h ix
y
py
yˆ
y ; pˆy
h iy
z
pz
zˆ
z ; pˆz
h iz
Appendix A Newton' s Equation and determination of position..cont
Audio-Visuals on-line Quantum mechanics as the foundation of Chemistry (quick time movie ****, 6 MB) Why Quantum Mechanics (quick time movie from the Wilson page ****, 16 MB) Why Quantum Mechanics (PowerPoint version without animations) Slides from the text book (From the CD included in
1 [(h h ) (h h ) (h h )] V(x, y, z) 2m i x i x i y i y i z i z
We have
h
h
h2
i y i y i2 y y
2
h2 y2
Thus
Hˆ
h2 [ 2m
2
x2
2
y2
2
z2 ] V(x, y,z )
Hˆ
h2 [ 2m
2
x2
2
y2
2
z2 ] V(x, y,z )
The quantum mechanical Hamiltonian Hˆ is constructed by the
following transformations :
HClass
Hˆ
1 2m
pˆ x2
pˆ
2 y
pˆ z2
V(xˆ, yˆ, zˆ)
Classical Mechanics Quantum Mechanics
The definition of the Hamiltonian (H)
as the sum of kinetic and potential energy,
with the potential energy written in terms
of the linear momentum For single particle: H p2 V(rr)
What you should learn from this lecture
Definitionof : Linear momentum (pm), kinetic energy(p2 );
2m Potential Energy
Relation between force F and potential energy V F(r = - r V)
Hˆ (x, y, z) E (x, y, z)
h2 2 (rr) V(rr) (rr) E (rr) 2m
h2 [ 2m
2
x2
2
y2
2
y2 ] V(x, y,z)
E
Contains all kinetic information about a particle moving in the Potential V(x,y,z)
The position of the particle is a function of time.
Let us assume that the particle at
t to
rv(t t) = rv(t ) = ? has the position
and the velocity
vr(torr)(to()drr / dt )t
Introduction to Classical mechanics and the Schrödinger equation (PDF)
Handout.Lecture1 (PDF) Taylor Expansion (MS-WORD)
Tutorials on-line The postulates of quantum mechanics (This is the writeup for Dry-lab-II)( This lecture has covered (briefly) postulates 1-2)(You are not expected to understand even postulates 1 and 2 fully after this lecture) The Development of Classical Mechanics Experimental Background for Quantum mecahnics Early Development of Quantum mechanics The Schrödinger Equation The Time Independent Schrödinger Equation
mass
Z
m rr
Y
Position
pr = mrv
X
Linear Momentum
The classical Hamiltonian is given by
H
1 2m
px2
p2y
pz2
V(x, y,z)
H 1 pr pr V(rvV (rv )
Quantum Mechanical Hamiltonian
2
2
2
By introducing the Laplacian : 2 x2 y2 z2 we have
Hˆ
h 2 V(x, y, z)
2m
It is now a postulate of quantum mechanics that :
the solutions (x, y, z) to the Schrödinger equation
to
(d vr / dt )t to∆t (d2vr / dt2 )t to ∆t2
vr (t o )
vr(to + ∆t)
What is
o
1