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

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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