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8/3/2019 Chem 373- Lecture 1: Classical Mechanics and the Schrdinger Equation
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Welcome to the Chem 373
Sixth Edition
+ Lab Manual
http://www.cobalt.chem.ucalgary.ca/ziegler/Lec.chm373/index.html
It is all on the web !!
8/3/2019 Chem 373- Lecture 1: Classical Mechanics and the Schrdinger Equation
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Lecture 1: Classical Mechanics and the Schrdinger Equation
This lecture covers the following parts of Atkins
1. Further information 4. Classical mechanics (pp
911- 914 )
2. 11.3 The Schrdinger Equation (pp 294)
Lecture-on-line
Introduction to Classical mechanics and the Schrdinger
equation (PowerPoint)
Introduction to Classical mechanics and the Schrdinger
equation (PDF)
Handout.Lecture1 (PDF)Taylor Expansion (MS-WORD)
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Tutorials on-line
The postulates of quantum mechanics(This is the writeup for
Dry-lab-II)( This lecture has covered (briefly) postulates1-2)(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 Schrdinger Equation
The Time Independent Schrdinger Equation
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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
Atkins ,**)
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8/3/2019 Chem 373- Lecture 1: Classical Mechanics and the Schrdinger Equation
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8/3/2019 Chem 373- Lecture 1: Classical Mechanics and the Schrdinger Equation
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Classical Mechanics
A particle in 3-D has the following attributes
X
Y
Z
1. Mass m
m
mass
r
Position
2. Positionrr
v = dr /dt
velocit y
3. Velocityrv
Rate of change of position with time
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Expression for total energy
ET Ekin Epot(rr )
The total energy of a particle with positionrr ,
mass m and velocity rv also has energy
Kinetic energy duto motion
Potential energydue to forces
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p
vv
small mass large velocity
v
v
large mass small velocity
or
Ek1
2mv
2
The kinetic energy can be written as :
r
p mvv
Or alternatively in terms of the
linear momentum:
as:
Ekp2
2m
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A particle moving in a potential energy field V is subject to a
force
V(x)
X
F=-dV/dx
Force in one dimension
Force in direction of
decreasing potential energy
The potential energy and force
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F dVdx
ex dVdy
eyPotential energy V
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vF (dV/dx)
rex (dV/dy)
vey (dV/dz)
vez
vF vV gradV
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8/3/2019 Chem 373- Lecture 1: Classical Mechanics and the Schrdinger Equation
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The expression for the total energy in terms ofthe potential energy and the kinetic energy
given in terms of the linear momentum
The Hamiltonian will take on a special
importance in the transformation from
classical physics to quantum mechanics
E Ekin
Epot
p2
2mV(
rr )
is called the Hamiltonian
Hp
2
2m
V(r
r )
The Classical Hamiltonian
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8/3/2019 Chem 373- Lecture 1: Classical Mechanics and the Schrdinger Equation
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Quantum Mechanics
The particle is moving in the potential V(x,y,z)
We consider a particle of mass m,
Linear momentumr
p mrv
and positionr
r
rr
rp =
X
Y
Z m
Positionmass
mvr
Linear Momentum
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rr
rp =
X
Y
Z m
Positionmass
mvr
Linear Momentum
The classical Hamiltonian is given by
H 12m
px2 py
2 pz2 V(x,y,z)
H1
2m
rp
rp V(
vr )
1
2m
p2
V(v
r )
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Quantum Mechanical Hamiltonian
The quantum mechanical Hamiltonian H is constructed by the
following transformations :
HClass H1
2mpx
2 py2 pz
2 V( x, y, z)
Classical Mechanics Quantum Mechanics
x px x x ; pxh
i x
y py y y ; pyh
i y
z pz z z ; pz hi z
Here h 'h-bar'=h
2is a modification of Plancks constanth
h 1.05457 1034 Js
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H
1
2m ( px2
py2
pz2
) V( x, y, z)
1
2m[(h
i x
h
i x) (
h
i y
h
i y) (
h
i z
h
i z)] V(x,y,z)
We have
h
i y
h
i y
h2
i2 y y
h2
2
y2
Thus
Hh
2
2m[
2
x2
2
y2
2
z2 ] V(x, y,z )
h2 2 2 2
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By introducing the Laplacian: 22
x2
2
y2
2
z2
we have
Hh
2m
2V(x,y,z)
It is now a postulate of quantum mechanics that :
the solutions (x,y,z) to the Schrdinger equation
H (x,y,z) E (x,y,z)
h2
2m
2(r
r ) V(r
r ) (r
r ) E (r
r )
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)
Hh
2m[
x2
y2
z2 ] V(x, y,z )
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What you should learn from this lecture
Definition of :
Linear momentum (pm),
kinetic energy(p2
2m);
Potential Energy
Relation between force F
and potential energy V (rF = -
rV)
The definition of the Hamiltonian (H)
as the sum of kinetic and potential energ
with the potential energy written in terms
of the linear momentum
For single particle: Hp2
2m
V(rr )
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You must know that : The quantum mechanical Hamiltonian His constructed from the classical Hamiltonian H by
the transformation
HClassH
1
2mpx
2 py2 pz
2 V( x, y, z)
Classical Mechanics Quantum Mechanics
x px x x ; pxh
i x
y py y y ; pyh
i y
z pz z z ; pzh
i z
Appendix A
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The position of the particle is a function of time.
Let us assume that the particle at t to
has the position
rr (to )
and the velocity
r
v(to ) (d
r
r/dt )t to
What isv
r (to t) =v
r (t1) = ?
vr (to t) =
vr ( to)+(d
vr / dt)t to t +
1
2(d2
vr / dt2)t to t
2
vr (to t) =
vr (to) +
vv(to ) t +
1
2(d2
vr / dt2 )t t
o
t2
By Taylor expansion around
rr (to )
or
Newton's Equation and determination of position..cont
vr (t o )
v
r (to+
t)
(d2
vr/dt2 )t t
o
t2
(d
vr/dt )t t
ot
ppe d
Appendix A
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vr (t o )
vr (to + t)
vv (t
o) t
(d
2vr /dt
2)t t
o
t2
vr (to t) =
vr (to) +
vv(to ) t +
1
2(d2
vr / dt2 )t to t
2
vF(to )
vV gradV m(d2 vr /dt2 )t to
However from Newtons law:
vr (t
o
t) =vr (t
o
) +vv(t
o
) t -1
2m(gradV)
t=t0t2
Thus :
Newton's Equation and determination of position..contpp
Newton's Equation and determination of position cont
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vr (t
o)
vr (to
+ t)
v
v(to )
t
-1
m(gradV)
t = t ot
vr (to t) =
vr (to) +
vv(to ) t -
1
2m(gradV)t=t0 t
2
Newton s Equation and determination of position..cont
Appendix A
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At the later time : t1
to
t we have
vr (t1 t) =vr (t1 )+(d
vr / dt)t t1 t +1
2(d
2vr /dt
2)t t1 t
2(1)
The last term on the right hand side of eq(1)
can again be determined from Newtons equation
vF(t1 )
vV gradV m(d
2vr /dt
2)t t1
as
(d
2vr /dt2
)t t11
m(gradV)t t1
Newton's Equation and determination of position..cont
pp
Appendix A
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We can determine the first term on the right side of
eq(1) By a Taylor expansion of the velocity
vr (t1 t) =
vr (t1 )+(d
vr / dt)t t1 t +
1
2m
(gradV)t t1 t2(1)
(dvr / dt)t t1 (d
vr / dt)t t0
1
2
(d2vr / dt2 )t t0 t
or
(d
vr / dt)t t1
vv(to )
1
2m(gradV)t to t
Where both: vv(to ) and
1
m(gradV)t to are known
Newton's Equation and determination of position..cont
Newton's Equation and determination of position cont
Appendix A
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The position of a particle is determined at all
times from the position and velocity at to
vv(t2 ) (d
vr / dt)t t2
vv(t1)
1
m(gradV) t t1 t
Newton's Equation and determination of position..cont
v
r (t2 t) =v
r (t2 ) +v
v(t2 ) t +1
2 (d2v
r /dt2
)t t2 t2
(d2
vr /dt2)t t2
1
m(gradV)t t2
At t2 t0 2 t what aboutvr (t2 t) ?
r (t2 )
r(t2 t)
v(t2 ) t-1
m(gradV)
t = t 2
t