chemistry 330 the mathematics behind quantum mechanics
TRANSCRIPT
Chemistry 330
The Mathematics Behind Quantum Mechanics
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Coordinate Systems Function of a coordinate system
locate a point (P) in space Describe a curve or a surface in space
Types of co-ordinate systems Cartesian Spherical Polar Cylindrical Elliptical
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Cartesian Coordinates The familiar x, y, z, axis system Point P - distances along the three
mutually perpendicular axes (x,y,z).
z
x
y
P(x,y,z)
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Spherical Coordinates Point P is based on a distance r
and two angles ( and ).
z
x
y
P(r,, )r
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The Transformation To convert spherical polar to
Cartesian coordinates
cossinrx
sinsinry
cosrz2222 zyxr
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Cylindrical Coordinates Point P is based on two distances
and an angle ().
z
x
y
P(r,,z)r
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The Transformation To convert cylindrical to Cartesian
coordinates
cosrx
sinry
zz 2222 zyxr
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Differential Volume Elements Obtain d for the various
coordinate systems Cartesian coordinates
z-
y-
x-
dxdydzd
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Differential Volume Elements Spherical polar coordinates
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0
r0
ddrdrd 2 sin
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Differential Volume Elements Cylindrical coordinates
z-
20
r0
dzrdrdd
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Vectors and Vector Spaces Vector – used to represent a physical
quantity Magnitude (a scalar quantity – aka length) Direction
Normally represent a vector quantity as follows
r
or r
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Components of a Vector A unit vector – vector with a
length of 1 unit. Three unit vectors in Cartesian
space z
x
yi j
k
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Vector Magnitude Magnitude of the vector is defined
in terms of its projection along the three axes!!
krjrirr zyxˆˆˆ
Magnitude ofr
21
2z
2y
2x rrrr
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Vectors (cont’d) Any vector can be written in terms
of its components - projection Vectors can be added or subtracted
Graphically Analytically
Note – vector addition or subtraction yields another vector
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Vector Multiplication Scalar Product – yields a number
cos2121 rrrr
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Vector Multiplication Cross Product – yields another
vector
kabba
jcaacibccbrr
2121
2121212121
kcjbiar
kcjbiar
2222
1111
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The Complex Number System Let’s assume we wanted to take the
square root of the following number.
1i
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Define the imaginary unit
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Imaginary Versus Complex Numbers A pure imaginary number = bi
b is a real number A complex number
C = a + bi Both a and b are real numbers
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The Complex Plane Plot a complex number on a
‘modified x-y’ graph. Z = x + yi
y
x
Z = x + yi
Z = x - yi
- R
I
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The Complex Conjugate Suppose we had a complex number
C = a + bi The complex conjugate of c
C*= a – bi Note
(C C*) = (a2 + b2) A real, non-negative number!!
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Other Related Quantities For the complex number
Z = x +yi
Magnitude 21
22 yxr
Phase
xy1tan
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Complex Numbers and Polar Coordinates The location of any point in the complex
plane can be given in polar coordinates
y
x
Z = x + yi
R
I
r
X = r cos
y = r sin z =r cos+ i sin = r e i
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Differential Equations Equations that contain derivatives
of unknown functions There are various types of
differential equations (or DE’s) First order ordinary DE – relates the
derivative to a function of x and y. Higher order DE’s contain higher
order derivatives
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Partial DE’s In 3D space, the relationship
between the variables x, y, and z, takes the form of a surface.
x
y
yz
xz
yxfz
,Function
Derivatives
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Partial DE’s (cont’d) For a function U(x,y,z)
A partial DE may have the following form
0z
Uy
Ux
U2
2
2
2
2
2
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Other Definitions Order of a DE
Order of the derivatives in it. Degree of the DE
The number of the highest exponent of any derivative
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Methods of Solving DE’s Find the form of the function
U(x,y,z)that satisfies the DE Many methods available (see math
367) Separation of variable is the most
often used method in quantum chemistry
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Operators An operator changes one function
into another according to a rule. d/dx (4x2) = 8x
The operator – the d/dx The function f(x) is the operand
Operators may be combined by Addition Multiplication
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Operators (cont’d) Operators are said to commute iff
the following occurs
yxfMKyxfKM ,ˆˆ,ˆˆ
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The Commutator Two operators will commute if the
commutator of the operators is 0!
yxfMKyxfKM
K and M of commutator the
,ˆˆ,ˆˆ
ˆˆ
If = 0, the operators commute!!
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Operators (the Final Cut) The gradient operator ( - del)
kz
jy
ix
ˆˆˆ
The Laplacian operator (2 – del squared)
2
2
2
2
2
22
zyx
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The Laplacian in Spherical Coordinates The Laplacian operator is very
important in quantum mechanics. In spherical coordinates
2
2
22
22
22
r1
r1
rr
rr1
sin
sinsin
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Eigenvalues and Eigenfunctions Suppose an operator operates on a
function with the following result
yxPfyxfP ,,ˆ
P is an eigenvalue of the operator
f(x,y) is an eigenfunction of the operator
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Eigenfunctions (cont’d) Operators often have more than one set
of eigenfunctions associated with a particular eigenvalue!!
These eigenfunctions are degenerate
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Linear Operators Linear operators are of the form
xfPcxfPc
xfcxfcP
2211
2211
ˆˆ
ˆ
Differential and integral operators are linear operators
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Symmetric and Anti-symmetric Functions For a general function f(x), we
change the sign of the independent variable If the function changes sign – odd If the sign of the function stays the
same – even Designate as
Symmetric – even Antisymmetric – odd
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Integrating Even and Odd Functions Integrate a function over a symmetric
interval (e.g., -x t +x)
x
x
0xf
x
0
x
x
xf2xf
if f(x) is odd
if f(x) is even
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Mathematical Series Taylor Series
!
...
!!
nax
af
2ax
af1
axafafxf
nn
2lll
When a = 0, this is known as a McLaren Series!
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Periodic Functions Sin(x) and cos(x) are example of
periodic functions! Real period functions are generally
expressed as a Fourier series
....sinsin
...coscos
x2bxb
x2axaa21xf
21
21o
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Normalization A function is said to be normalized
iff the following is true
12 xfxfN
N – normalization constant
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Orthogonal Functions Two functions (f(x) and g(x) are said to
be orthogonal iff the following is true
0xg xf
Orthogonal – right angles!!
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The Kronecker Delta (fg)
If our functions f(x) and g(x) are normalized than the following condition applies
fgxg xf f(x) = g(x), fg = 1
f(x) g(x), fg = 0