# Quantum Mechanics

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• School of Physical Sciences

PH 502Wavemechanics & Quantum Physics

COURSE SCRIPT

Dr. Peter Blmler

Peter BlmlerLinks (blue arrows) do not work in this summary!!!

Peter Blmler

Peter Blmler

Peter Blmler

• Dr. P. Blmler: PH 502 Wave Mechanics Syllabus page: 1

Dr. P. Blmler, School of Physical Sciences, University of Canterbury, UK

Syllabus for Part II (Dr. Blmler)

7. The Foundations and Postulates of Quantum Mechanics:7.1 Philosophy and Concept of Axiomatic Physics

7.2 Operators and Eigenfunctions:7.2.1 Properties of Operators7.2.2 Eigenfunctions and Eigenvalues7.2.3 Linear Combinations7.2.4 Orthogonality, Normalisation and Orthonormality7.2.5 Hermiticity7.2.6 Commutators7.2.7 List of Important Operators

7.3 Postulates (Axioms) of Quantum Mechanics

7.3.1 Wavefunction7.3.2 Physical Systems7.3.3 Schrdingers Equation7.3.4 Probability and Expectation Value

7.4 Uncertainty Principle (principle of indeterminacy)

7.4 Dirac Notation

8. Confined Particles8.1 Recapitulation: 1D-Problem particle in a box

E.1 Excursus: Separation of variables

8.2 2D-Problem: Particle in a Square Well8.2.1 Example Quantum Corrals

8.3 3D-Problem particle in a real box, Degeneracy

9. The Harmonic Oscillator9.1 Classical Description

9.2 1D Harmonic Oscillator in QM9.2.1 Solution of Hermites Differential Equation9.2.2 Correspondence Principle

9.3 The 2D Harmonic Oscillator

9.4 The 3D Harmonic Oscillator

10. Rotational Motion10.1 Classical Rotation in a 2D Plane

10.2 Some Quantum-Mechanical Considerations

E2: Excursus: Co-ordinate Transforms of the Laplacian OperatorE2.1 Method 1: Explicit Solution for 2D Circular Co-ordinatesE2.2 Method 2: Curvilinear Co-ordinate Transforms

10.3 Exact Solution for the Rigid Rotator

10.4 Particle Rotating on a Sphere

10.5 Angular Momentum

11. The Hydrogenic Atom11.1 Motion in a Coulomb field

11.2 Solution of the Radial Differential Equation

11.3 The Complete Solution: Atomic Orbitals

11.4 Linear Combinations: Hybrid Orbitals

11.5 Quantum numbers, Orbital Shapes and Degeneracy11.5.1 The Periodic Table of Elements

11.6 Electron Spin11.5.2 Selection Rules and Atomic Spectra

• Dr. P. Blmler: PH 502 Wave Mechanics Useful physical constants page: 2

Dr. P. Blmler, School of Physical Sciences, University of Canterbury, UK

Useful constants

Planck constant: h 6.62607610-34 Js = 4.13566910-15 eV s

p=

2hh 1.05457310-34 Js = 6.58212210-16 eV s

Speed of light (in vac.): c 2.99792458108 m/s

Electron mass: me 9.10939010-31 kg

510.9991 keV/c2

Proton mass: mp 1.67262310-27 kg

938.2723 MeV/c2

Neutron mass: mn 1.67492910-27 kg

939.5656 MeV/c2

Fundamental charge: e 1.60217710-19 C

Compton wavelength:cm

h

e=l C 2.4263105810

-12 m

Boltzmann constant: kB 1.38065810-23 J/K

Vacuum permittivity 0e ( )Jm/C10854188.8 212-

04pe ( )Jm/C10112650.1 210-

combinations:

hc = 1.986410-25 Jm = 1239.8 eV nmch = 3.161510-26 Jm = 197.33 eV nm

• Dr. P. Blmler: PH 502 Wave Mechanics Greek Alphabet/Units page: 3

Dr. P. Blmler, School of Physical Sciences, University of Canterbury, UK

Units

The Greek alphabet

A a alpha N n nu

B b beta X x xi

D d delta O o omicron

G g gamma P p pi

E e epsilon R r rho

Z z zeta S s, V sigma

H h eta T t tau

Q q, J theta U u upsilon

I i iota F f, j phi

K k kappa C c chi

L l lambda Y y psi

M m mu W w omega

Prefix Exponent Symbol Prefix Exponent Symbol

deci -1 d deca 1 dacenti -2 c hecto 2 hmilli -3 m kilo 3 k

micro -6 m mega 6 Mnano -9 n giga 9 Gpico -12 p tera 12 T

femto -15 f peta 15 Patto -18 a exa 18 E

zepto -21 z zetta 21 Zyocto -24 y yotta 24 Y

1 = 10-10 m

• Dr. P. Blmler: PH 502 Wave Mechanics Course Summary / Spine page: 4

Dr. P. Blmler, School of Physical Sciences, University of Canterbury, UK

COURSE SPINE

What is that?

This section is designed to give you a coarse outline of the learning outcomes of each section. Youmight find this useful for exam preparations and revision.

From the PDF-File you can always link to

Blue colour provides links to the particular section!

The following symbols mean:

Recap of material from previous courses

Mathematical detail, method or solution

Illustration in a MAPLE program

Workshop or exercise

Recap pervious knowledge!

Classical wave mechanics (PH 301)

Qualitative description of quantum mechanics (PH 301)

Partial differential equations (PH 501)

• Dr. P. Blmler: PH 502 Wave Mechanics Course Summary / Spine page: 5

Dr. P. Blmler, School of Physical Sciences, University of Canterbury, UK

7. The Foundations and Postulatesof Quantum Mechanics:

7.2 Operators and Eigenfunctions

Definitions and examples

To know:1. Properties of operators, notation2. Eigenfunctions and eigenvalues, degeneracy, physical meaning3. Linear combination4. Orthogonality, normalisation and orthonormality5. Hermiticity, real eigenvalues6. Commutators

7. Important operators: UKHLLLpppzyx ,,,,,,,, zyxzyx,,,

Exercises and workshops

7.3/7.4 The Postulates of Quantum Mechanics:

Definitions and examples

To know:1. Wavefunction, probability interpretation, normalisation2. Hermitian operators and physical observables3. Expectation values and deductions4. Schdinger equations5. Uncertainty principle (as in (7.4.1))

Exercises and workshops

• Dr. P. Blmler: PH 502 Wave Mechanics 7.2 Operators page: 6

Dr. P. Blmler, School of Physical Sciences, University of Canterbury, UK

7.2. Operators and Eigenfunctions:

What is an operator?

An operator is a symbol that tells you to do something to whatever follows the symbol.

More mathematically it can be understood as a symbol that represents a rule to transform a followingfunction into another function.

The notation in this script for operators is A and an expression like fA will be referred to as A

operating on f . For rigorousity we will very often use symbols like A representing a vector-operator. The underline shall simply remind you that this operator may act differently on individualco-ordinates (or vector components).

Comparison: Scalars, Functions, Vectors, Matrices, Operators

Scalars and Functions:

A linear function converts a scalar into another:

b = f(a)

Vectors and Matrices:

A matrix is a linear vector function, thus it converts a vector into another:

aMb =

example:

++++

++

=

===

=

333232131

323222121

313212111

3

2

1

333231

232221

131211

3

2

1

amamam

amamam

amamam

a

a

a

mmm

mmm

mmm

am

b

b

b

kikaMb

Functions and Operators:

An operator is a function of higher order, thus it generally converts a function into another function:

In quantum mechanics, physical observables (e. g. energy, position, linear and angular momentum)are represented mathematically by operators. For example the total energy (kinetic energy K plus

potential energy U) is represented by an operator UKH += , where H is called Hamilton-operator or simply Hamiltonian. It is defined as:

)()( xaxb M=

• Dr. P. Blmler: PH 502 Wave Mechanics 7.2 Operators page: 7

Dr. P. Blmler, School of Physical Sciences, University of Canterbury, UK

Um

+-= 22

2

hHWe will later derive this formula. The link to experimentally assessable values of the energy (its

average value) of a system described by H and in a state described by a wavefunction y(r) is given

by its expectation value H .

rrr d)()(* yy= HH

7.2.1 Properties of Operators:

Most of the properties of operators are obvious. Nevertheless, they are summarised forcompleteness:

Sum and Difference: of two operators A and B :

( )( ) fff

fff

BABA

BABA

-=-

+=+

(7.2.1)

The product of two operators is defined by:

[ ]ff BABA = (7.2.2)which means: First operate with Bon f then operate with A on the result of fB .

Equality: Two operators are equal if ff BA = holds for all functions f. (7.2.3)

Identity: The identity operator I does nothing (multiplies with 1 or adds 0) (7.2.4)

The nth power of an operator nA is defined as n successive applications of A , e.g.

[ ]fff AAAAA 2 == (7.2.5)The exponential of an operator )exp(A is defined by a Taylor expansion:

K++++==!3

!2

)exp(

32 AAAIA Ae (7.2.6)

The associative law holds for operators:

• Dr. P. Blmler: PH 502 Wave Mechanics 7.2 Operators page: 8

Dr. P. Blmler, School of Physical Sciences, University of Canterbury, UK

( ) ( ) ff CBACBA = (7.2.7)The commutative law does not generally hold for operators!

ff ABBA (7.2.8)

see commutators!

Examples:

1) Addition of a constant: cxfxf += )()(A

2) Derivative (wrt x): )()(dd

)( xfxfx

xf ==D (could be named prime operator or dot

operator for derivatives wrt time)

3) For D operating on f(x) =exp(ax):

( ) ( ) ( ) f(x)aaxaaxx

axf(x) ==== expexpdd

exp DD

4) For 2D operating on f(x) =exp(ax2):

( )[ ] ( ) ( )[ ]( )( ) ( ) ( )( ) ( )f(x)axaaxaxaxaaxx

xa

axxaaxx

axf(x)

22222

2222

12expexp2expdd

2

exp2expddexp

+

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