# State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness

Save this PDF as:

Size: px
Start display at page:

Download "State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness"

## Transcription

1 Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators-> normalzaton, orthogonalty completeness egenvalues and expectaton values of operators Tme ndependent Schroednger equaton and statonary states. Probablty current.

2 Schroednger equaton. Schroednger equaton s a wave equaton, whch lnks tme evoluton of the wave functon of the state to the Hamltonan of the state. For most of systems Hamltonan represents total energy of the system T+V= knetc +potental. Hamltonan s defned also classcally, and equatons of motons for classcal systems can be wrtten usng dervatves of the Hamltonan. Classcally there s no need for a concept of the wave functon of the state, as any state can be totally specfed by gvng momentum and postons of all partcles.

3 - Frst I remnd you about a flat wave. Ths s the wave functon descrbng a free partcle. - I wll show that the flat wave s a soluton of the free Schroednger equaton. - It useful to test operators and propertes of wave functons on a flat wave to understand what they really mean.

4 A non-relatvstc partcle has the followng Hamltonan-> energy: H=[ p2 2 m V r ] where knetc energy s E k = m v 2 2 = p2 2 m we know that a free partcle ( propagatng n a place wthout potental) can be descrbed by a flat wave or a combnaton of flat waves- wave packet m r,t = Ae r,t = 2 h we note that: r,t t = 2 h h p r Et where h =E, =h/ p, ħ= h h p e p r Et d 3 p 3 2 h h p E e p r Et d 3 p

5 Thus r,t t = 2 h 3 2 h p p2 2 m e Now remnd ourselves Laplace operator h p r Et d 3 p 2 = 2 x 2 2 y 2 2 z 2 and note that for example : 2 r,t = 2 ħ x 2 2 r,t = 2 ħ thus h t ħ 2 p p x 2 e ħ 2 p p 2 e =[ h2 2 m 2 ] ħ p r Et d 3 p ħ p r Et d 3 p Ths s Schroednger equaton for free partcle

6 We can also note that dfferentatng over x (for example): h r,t = 2 h x and defne momentum operators h t h t = H =[ p 2 2 m V r ] h t 3 2 h p p x e p r Et d 3 p p x h =[ h2 2 m 2 V r ] x, p y h y, p z h z =[ h2 2 m 2 ] h t = H =[ p 2 2 m ] Natural extenson to the stuaton wth potental ( non-free partcle)

7 Postulates of Quantum Mechancs A ) For every classcal observable a lnear operator. It can depend on momentum and poston operators. Momentum operator (x) s proportonal to dervatve over x F q 1, q 2,.. q n, p 1, p 2,.. p n : B) A system s fully descrbed by a wave-functon whch fulflls wave equaton, Schroednger for non-relatvstc Hamltonan C) expectaton value of an operator F q n =q n, p n = ħ h t = p H =[ 2 2 m V r ] corresponds to the expectaton ( mean value) of the measurement result of the varable F taken over a bg number of ndependent measurements (ths s more dffcult then t sounds) D) the only possble results of sngle measurements of the varable F are the egenvalues of the operator F F = f (example of spn ½, z projectons ) f = constant < F >= * F d q n (after a measurement the system collapses to the state wth well defned varable f)

8 What's new here? You have heard ths before on kvantefyskk og statstk mekankk The dfference s that now we are tryng to wrte our wave functon n a more general way. It can be a functon of space varables, or momentum or perhaps even more general varables ( and tme). In practce we wll start wth space representaton, then we wll dscuss momentum representaton, and then other- general Drac representaton of the sate and vector representaton ( matrx mechancs)

9 Ad C. < F >= * F d d =dq 1... d q n In general we ntegrate over the set varables the wave functon of the system s defned on, and normalzed on. In space representaton of quantum mechancs we dscuss rght now ths s space coordnates. For example an expectaton value of a poston of a partcle wll be: < r t >= * r,t r r,t d 3 r < r t >= r r,t 2 d 3 r proper normalzaton 1= r,t 2 d 3 r what about momentum? Ths s probablty densty to fnd the state n locaton r

10 Hermtan operators. The expectaton values of an operator representng any real varable must be real < F >= * F d = F * d =< F > * Defnton of hermtan operator: The operator F s hermtan f : 1* F 2 d = 2 F 1 * d If the operator s hermtan ts expectaton value s REAL- that s a requrement for operators assocated wth observables Defnton of adjoned operator to F 1 * F 2 d = 2 F 1 * d Hermtan operators are self-adjoned meanng : F = F example, check p_x operator.

11 Ad hermtan, prove of hermetcty condton < F >= * F d = F * d (alpha, any constant number- take real functons PSI1, PSI2 1 e a 2 * F 1 e a 2 d = 1 e a 2 F 1 e a F 2 * d 1 * F 1 d e a 1 * F 2 e a 2 * F 1 2 * F 2 d = = 1 F 1 * d e a 1 F 2 * e a 2 F 1 * 2 F 2 * d 1* F 2 d = 2 F 1 * d 2* F 1 d = 1 F 2 * d

12 Commutator of operators Commutators: are mportant! for example varables who's operators commute can be measured smultaneously for the system. [ A, B ]= A B B A Lets check [ x, p x ] = x p x p x x x p x p x x =x ħ x ħ x x = x ħ x ħ x x ħ [ x, p x ]= ħ =ħ x poston varable, and p_x do not commute. What does that mean for subsequent measurements of px and x?

13 Egenfunctons and egenvalues: The result of an operator workng on a functon s usually a dfferent functon. If there exst a set of functons that : F n = f n n we call them egenfunctons of the operator F, and fn are egenvalues (constants) The set of egenvalues ( or spectrum of the operator) can be dscrete, contnuos or mxed. Example of contnuos- energy operator (Hamltonan) for a free partcle, dscrete= hamltona for harmonc oscllator, mxed- hamltonan for hydrogen atom. p x = ħ x ħ x f = f f f =const exp fx ħ That's the form of egenfunctons of momentum operator. They are not quadratcaly ntegrable and have to be normalzed n a dfferent way.

14 Expectaton value vs. egenvalue of an operator. < F >= * F d The expectaton ( mean value) of the measurement of the varable F taken over a bg number of ndependent measurements ( n practce over large number of dentcally prepared states- an ensamble ). But the only possble results of measurements of the varable F are the egenvalues of the operator F F = f (example of spn ½, z projectons ) If the system s n the state descrbed by the egenfuncton of the operator,the expectaton value of the measurement s equal to the egenvalue. As the egenvalue s the only possble measurement results, that s what we wll always get! < F >= * F d = n* F n d = f n n * n = f n for the egenstate number n Ths proof s straghtforward for normalzed quadratc-ntegrable egenfunctons, can be also proved for dfferent type of normalzaton

15 Orthogonalty of egenfunctons For a hermetc operator, egenfunctons correspondng to dfferent egenvalues are orthogonal n* F m d = m F n * d f m n * m d = f n * m n * d f m f n n * m d =0 f f m f n then n * m d =0 Drac notaton: Scalar product n * m d = n m n* F m d = n F m n F m = F n m Hermtan operator n Drac notaton

16 We try to normalze our set of egenfunctons of an operator typcally choosng a constant to multply the functons wth. n * m d = nm 0 for n = m, orthogonalty 1 for n=m, normalzaton Normalzaton of egenfunctons wth contnuous spectrum of egenvalues has to be a bt dfferent- functons are not quadratcally ntegrable f * ' f d = f ' f =< f' f> f x =c exp fx ħ * f ' x f x d x= 1 2 ħ C must be c= 1 2 ħ Drac delta Example momentum egenfunctons, here just 1-dm e x f f ' ħ dx= f ' f to get the normalzaton rght

17 More about Drac delta functon: Drac delta functon : f x x x ' dx= f x ' f x x dx= f 0 propertes : possble representaton: 1 x dx=1 x =lm 0 x =lm 1 2 x = 1 2 exp xy dy exp xy y dy =lm 0 x 2 2

18 x = x 2 2 =0.01 =0.1 =0.5 x

19 Completeness We assume that every reasonable, quadratcally ntegrable functon can be expressed as a lnear combnaton of egenfunctons of a hermtan operator F g= c g >= For egenfunctons wth contnuous spectrum of egenvalues we can represent a functon g n the followng way : (egenfunctons have to form a complete bass set) g= c f f df g >= df c f f >= df <f g> f > f * ' f d = f ' f =< f' f> then we have : c f = f * g d =< f g> we must also have : f * r ' f r d f = r ' r c >= < g> > for orthonormal set of egenfunctons the coeffcents are a scalar product of the functon n queston and approprate egenfuncton n * g d = c n * d =c n c n = n * g d =<n g>

20 Completeness, contnuaton: Lets now consder that our functons are normalzed n the normal space, so f * ' f d = f * ' r f r d 3 r= f ' x f x f ' y f y f ' z f z etc.. g r = c r g r = g r ' we must n analogy have for contnuous spectrum: c n = n * r ' g r ' d 3 r ' * r ' r d 3 r ' f * r ' f r d f = r ' r * r ' r = r r ' 3-dm Drac delta

21 What s the nterpretaton of expanson coeffcents cn? We see that modulus squared of an expanson coeffcent wll correspond to a probablty to measure certan egenvalue: Lets take spectrum of egenfunctons of gven operator F and expand a functon of state (g) nto t : g= c and check what s the expectaton value of the operator F for the state descrbed by g < F >= g * F g d = n < F >= n < F >= n c n * n * f c = n c n * c f,n = c n * n * F c 2 f c c n * c f n * d but we know that from nterpretaton of expectaton value we must have < F >= P f where P_ s the probablty that the value f_ wll be measured.

22 P f df = c f 2 df = f * g d 2 df where F f = f f Thus we have. The probablty that measurng observable assocated wth F on a state descrbed by a wave functon g wll gve a result f_ s the followng: P = c 2 = * g d 2 where F = f For contnuos spectrum of F we can prove n analogy that: F = f c f 2 df The probablty that measurng observable assocated wth F on a state descrbed by a wave functon g wll gve a result beetween f and f+df s

23 Statonary states: If the Hamltonan does not contan tme explctly we can try to separate the solutons n to tme dependent part and coordnates dependent part. We obtan that there s new constant nvolved proportonal to the tme dervatve of the tme dependent functon dvded by the functon tself. We call t ENERGY. We obtan tme-ndependent Shroednger equaton for the part whch does not depend on tme. t, r r T t dt t /dt ħ =const E T t H r =E r Et/ħ T =C e h T t t r =[ h2 2 m 2 r V r r ]T t What statonary means n practce? Expectaton values of operators do not depend on tme. ( for normal operators whch do not contan tme dervatves ) Prove t!

24 Probablty current : Probablty nterpretaton for the partcle wave functon: modulus square of t s a probablty to fnd a partcle n a gven place (probablty densty). THUS: Integral over space has to gve 1. However locally spacal probablty densty can change wth tme. We can defne the probablty current, useful when dscussng movement of partcles r,t = * = * * = t t t t ħ 2 m * 2 2 * = change of probablty densty wth tme (at a gven place) s related to the out-flow of the current. t = ħ 2 m * * = j j= ħ 2 m * * =R * ħ m ħ 2 m * * h t h2 =[ 2 m 2 V r ] h * h2 =[ t 2 m 2 V r ] *

### QUANTUM MECHANICS, BRAS AND KETS

PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented

More information

### 5. Simultaneous eigenstates: Consider two operators that commute: Â η = a η (13.29)

5. Smultaneous egenstates: Consder two operators that commute: [ Â, ˆB ] = 0 (13.28) Let Â satsfy the followng egenvalue equaton: Multplyng both sdes by ˆB Â η = a η (13.29) ˆB [ Â η ] = ˆB [a η ] = a

More information

### The material in this lecture covers the following in Atkins. 11.5 The informtion of a wavefunction (d) superpositions and expectation values

Lecture 7: Expectaton Values The materal n ths lecture covers the followng n Atkns. 11.5 The nformton of a wavefuncton (d) superpostons and expectaton values Lecture on-lne Expectaton Values (PDF) Expectaton

More information

### 8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

### v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

More information

### Chapter 3 Group Theory p. 1 - Remark: This is only a brief summary of most important results of groups theory with respect

Chapter 3 Group Theory p. - 3. Compact Course: Groups Theory emark: Ths s only a bref summary of most mportant results of groups theory wth respect to the applcatons dscussed n the followng chapters. For

More information

### where the coordinates are related to those in the old frame as follows.

Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product

More information

### A. Te densty matrx and densty operator In general, te many-body wave functon (q 1 ; :::; q 3N ; t) s far too large to calculate for a macroscopc syste

G25.2651: Statstcal Mecancs Notes for Lecture 13 I. PRINCIPLES OF QUANTUM STATISTICAL MECHANICS Te problem of quantum statstcal mecancs s te quantum mecancal treatment of an N-partcle system. Suppose te

More information

### BERNSTEIN POLYNOMIALS

On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

### Recurrence. 1 Definitions and main statements

Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

### Linear Algebra for Quantum Mechanics

prevous ndex next Lnear Algebra for Quantum Mechancs Mchael Fowler 0/4/08 Introducton We ve seen that n quantum mechancs, the state of an electron n some potental s gven by a ψ x t, and physcal varables

More information

### 2.4 Bivariate distributions

page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

More information

### The quantum mechanics based on a general kinetic energy

The quantum mechancs based on a general knetc energy Yuchuan We * Internatonal Center of Quantum Mechancs, Three Gorges Unversty, Chna, 4400 Department of adaton Oncology, Wake Forest Unversty, NC, 7157

More information

### What is Candidate Sampling

What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

More information

### Moment of a force about a point and about an axis

3. STATICS O RIGID BODIES In the precedng chapter t was assumed that each of the bodes consdered could be treated as a sngle partcle. Such a vew, however, s not always possble, and a body, n general, should

More information

### Graph Theory and Cayley s Formula

Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll

More information

### LECTURE II. Hamilton-Jacobi theory and Stäckel systems. Maciej B laszak. Poznań University, Poland

LECTURE II Hamlton-Jacob theory and Stäckel systems Macej B laszak Poznań Unversty, Poland Macej B laszak (Poznań Unversty, Poland) LECTURE II 1 / 17 Separablty by Hamlton and Jacob Consder Louvlle ntegrable

More information

### benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

### b) The mean of the fitted (predicted) values of Y is equal to the mean of the Y values: c) The residuals of the regression line sum up to zero: = ei

Mathematcal Propertes of the Least Squares Regresson The least squares regresson lne obeys certan mathematcal propertes whch are useful to know n practce. The followng propertes can be establshed algebracally:

More information

### A Note on the Decomposition of a Random Sample Size

A Note on the Decomposton of a Random Sample Sze Klaus Th. Hess Insttut für Mathematsche Stochastk Technsche Unverstät Dresden Abstract Ths note addresses some results of Hess 2000) on the decomposton

More information

### Section 2 Introduction to Statistical Mechanics

Secton 2 Introducton to Statstcal Mechancs 2.1 Introducng entropy 2.1.1 Boltzmann s formula A very mportant thermodynamc concept s that of entropy S. Entropy s a functon of state, lke the nternal energy.

More information

### 5.74 Introductory Quantum Mechanics II

MIT OpenCourseWare http://ocw.mt.edu 5.74 Introductory Quantum Mechancs II Sprng 9 For nformaton about ctng these materals or our Terms of Use, vst: http://ocw.mt.edu/terms. 4-1 4.1. INTERACTION OF LIGHT

More information

### APPLICATIONS OF VARIATIONAL PRINCIPLES TO DYNAMICS AND CONSERVATION LAWS IN PHYSICS

APPLICATIONS OF VAIATIONAL PINCIPLES TO DYNAMICS AND CONSEVATION LAWS IN PHYSICS DANIEL J OLDE Abstract. Much of physcs can be condensed and smplfed usng the prncple of least acton from the calculus of

More information

### NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582

NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 7. Root Dynamcs 7.2 Intro to Root Dynamcs We now look at the forces requred to cause moton of the root.e. dynamcs!!

More information

### Consider a 1-D stationary state diffusion-type equation, which we will call the generalized diffusion equation from now on:

Chapter 1 Boundary value problems Numercal lnear algebra technques can be used for many physcal problems. In ths chapter we wll gve some examples of how these technques can be used to solve certan boundary

More information

### The Mathematical Derivation of Least Squares

Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the age-old queston: When the hell

More information

### x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

More information

### Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

### Solution of Algebraic and Transcendental Equations

CHAPTER Soluton of Algerac and Transcendental Equatons. INTRODUCTION One of the most common prolem encountered n engneerng analyss s that gven a functon f (, fnd the values of for whch f ( = 0. The soluton

More information

### Laws of Electromagnetism

There are four laws of electromagnetsm: Laws of Electromagnetsm The law of Bot-Savart Ampere's law Force law Faraday's law magnetc feld generated by currents n wres the effect of a current on a loop of

More information

### Support Vector Machines

Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

### Communication Networks II Contents

8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

More information

### Harvard University Division of Engineering and Applied Sciences. Fall Lecture 3: The Systems Approach - Electrical Systems

Harvard Unversty Dvson of Engneerng and Appled Scences ES 45/25 - INTRODUCTION TO SYSTEMS ANALYSIS WITH PHYSIOLOGICAL APPLICATIONS Fall 2000 Lecture 3: The Systems Approach - Electrcal Systems In the last

More information

### 6. EIGENVALUES AND EIGENVECTORS 3 = 3 2

EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a non-zero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :

More information

### Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

More information

### Lesson 2 Chapter Two Three Phase Uncontrolled Rectifier

Lesson 2 Chapter Two Three Phase Uncontrolled Rectfer. Operatng prncple of three phase half wave uncontrolled rectfer The half wave uncontrolled converter s the smplest of all three phase rectfer topologes.

More information

### THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

More information

### Introduction to Statistical Physics (2SP)

Introducton to Statstcal Physcs (2SP) Rchard Sear March 5, 20 Contents What s the entropy (aka the uncertanty)? 2. One macroscopc state s the result of many many mcroscopc states.......... 2.2 States wth

More information

### PERRON FROBENIUS THEOREM

PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()

More information

### Least Squares Fitting of Data

Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng

More information

### The mathematical representation of physical objects and relativistic Quantum Mechanics.

The mathematcal representaton of physcal obects and relatvstc Quantum Mechancs. Enrque Ordaz Romay 1 Facultad de Cencas Físcas, Unversdad Complutense de Madrd Abstract The mathematcal representaton of

More information

### Feature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College

Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure

More information

### Logistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification

Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson

More information

### EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN

EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson - 3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson - 6 Hrs.) Voltage

More information

### Questions that we may have about the variables

Antono Olmos, 01 Multple Regresson Problem: we want to determne the effect of Desre for control, Famly support, Number of frends, and Score on the BDI test on Perceved Support of Latno women. Dependent

More information

### THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

More information

### Lecture 3. 1 Largest singular value The Behavior of Algorithms in Practice 2/14/2

18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest

More information

### HÜCKEL MOLECULAR ORBITAL THEORY

1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ

More information

### DILL CH_10 Maxwell Boltzmann law

DILL CH_1 Maxwell oltzmann law The basc concepts Entropy and temperature The Maxwell oltzmann dstrbuton Speed dstrbuton of the molecules M dstrbuton of harmonc oscllators Ensten's theory of lattce heat

More information

### Chapter 7 Symmetry and Spectroscopy Molecular Vibrations p. 1 -

Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p - 7 Symmetry and Spectroscopy Molecular Vbratons 7 Bases for molecular vbratons We nvestgate a molecule consstng of N atoms, whch has 3N degrees

More information

### Multiple stage amplifiers

Multple stage amplfers Ams: Examne a few common 2-transstor amplfers: -- Dfferental amplfers -- Cascode amplfers -- Darlngton pars -- current mrrors Introduce formal methods for exactly analysng multple

More information

### PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

### Extending Probabilistic Dynamic Epistemic Logic

Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

More information

### DEFINING %COMPLETE IN MICROSOFT PROJECT

CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

### ChE 4520/5520: Mass Transport. Objective/Introduction. Outline. Gerardine G. Botte

ChE 450/550: Mass Transport Gerardne G. Botte Objectve/Introducton In prevous chapters we neglected transport lmtatons In ths chapter we wll learn how to evaluate the effect of transport lmtatons We wll

More information

### The OC Curve of Attribute Acceptance Plans

The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

### + + + - - This circuit than can be reduced to a planar circuit

MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

More information

### Do Hidden Variables. Improve Quantum Mechanics?

Radboud Unverstet Njmegen Do Hdden Varables Improve Quantum Mechancs? Bachelor Thess Author: Denns Hendrkx Begeleder: Prof. dr. Klaas Landsman Abstract Snce the dawn of quantum mechancs physcst have contemplated

More information

### THE TITANIC SHIPWRECK: WHO WAS

THE TITANIC SHIPWRECK: WHO WAS MOST LIKELY TO SURVIVE? A STATISTICAL ANALYSIS Ths paper examnes the probablty of survvng the Ttanc shpwreck usng lmted dependent varable regresson analyss. Ths appled analyss

More information

### Formula of Total Probability, Bayes Rule, and Applications

1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.

More information

### Hedging Interest-Rate Risk with Duration

FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton

More information

### INTRODUCTION. governed by a differential equation Need systematic approaches to generate FE equations

WEIGHTED RESIDUA METHOD INTRODUCTION Drect stffness method s lmted for smple D problems PMPE s lmted to potental problems FEM can be appled to many engneerng problems that are governed by a dfferental

More information

### Mean Molecular Weight

Mean Molecular Weght The thermodynamc relatons between P, ρ, and T, as well as the calculaton of stellar opacty requres knowledge of the system s mean molecular weght defned as the mass per unt mole of

More information

### Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008

Rsk-based Fatgue Estmate of Deep Water Rsers -- Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn

More information

### greatest common divisor

4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

More information

### Algebraic Quantum Mechanics, Algebraic Spinors and Hilbert Space.

Algebrac Quantum Mechancs, Algebrac Spnors and Hlbert Space. B. J. Hley. Theoretcal Physcs Research Unt, Brkbeck, Malet Street, London WCE 7HX. b.hley@bbk.ac.uk Abstract. The orthogonal Clfford algebra

More information

### On entropy for mixtures of discrete and continuous variables

On entropy for mxtures of dscrete and contnuous varables Chandra Nar Balaj Prabhakar Devavrat Shah Abstract Let X be a dscrete random varable wth support S and f : S S be a bjecton. hen t s wellknown that

More information

### 1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)

6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes

More information

### Face Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching)

Face Recognton Problem Face Verfcaton Problem Face Verfcaton (1:1 matchng) Querymage face query Face Recognton (1:N matchng) database Applcaton: Access Control www.vsage.com www.vsoncs.com Bometrc Authentcaton

More information

### VLSI Technology Dr. Nandita Dasgupta Department of Electrical Engineering Indian Institute of Technology, Madras

VLI Technology Dr. Nandta Dasgupta Department of Electrcal Engneerng Indan Insttute of Technology, Madras Lecture - 11 Oxdaton I netcs of Oxdaton o, the unt process step that we are gong to dscuss today

More information

### Aryabhata s Root Extraction Methods. Abhishek Parakh Louisiana State University Aug 31 st 2006

Aryabhata s Root Extracton Methods Abhshek Parakh Lousana State Unversty Aug 1 st 1 Introducton Ths artcle presents an analyss of the root extracton algorthms of Aryabhata gven n hs book Āryabhatīya [1,

More information

### 1. Measuring association using correlation and regression

How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a

More information

### 1 Example 1: Axis-aligned rectangles

COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

More information

### Implementation of Deutsch's Algorithm Using Mathcad

Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"

More information

### Comment on Rotten Kids, Purity, and Perfection

Comment Comment on Rotten Kds, Purty, and Perfecton Perre-André Chappor Unversty of Chcago Iván Wernng Unversty of Chcago and Unversdad Torcuato d Tella After readng Cornes and Slva (999), one gets the

More information

### DETERMINATION THERMODYNAMIC PROPERTIES OF WATER AND STEAM

DETERMINATION THERMODYNAMIC PROPERTIES OF WATER AND STEAM S. Sngr, J. Spal Afflaton Abstract Ths work presents functons of thermodynamc parameters developed n MATLAB. There are functons to determne: saturaton

More information

### We are now ready to answer the question: What are the possible cardinalities for finite fields?

Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the

More information

### copyright 1997 Bruce A. McCarl and Thomas H. Spreen.

Appendx I: Usng Summaton Notaton Wth GAMS... AI-1 AI.1 Summaton Mechancs... AI-1 AI.1.1 Sum of an Item.... AI-1 AI.1.2 Multple Sums... AI-2 AI.1.3 Sum of Two Items... AI-2 AI.2 Summaton Notaton Rules...

More information

### LOOP ANALYSIS. The second systematic technique to determine all currents and voltages in a circuit

LOOP ANALYSS The second systematic technique to determine all currents and voltages in a circuit T S DUAL TO NODE ANALYSS - T FRST DETERMNES ALL CURRENTS N A CRCUT AND THEN T USES OHM S LAW TO COMPUTE

More information

### Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

### Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks

Bulletn of Mathematcal Bology (21 DOI 1.17/s11538-1-9517-4 ORIGINAL ARTICLE Product-Form Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz

More information

### Solution : (a) FALSE. Let C be a binary one-error correcting code of length 9. Then it follows from the Sphere packing bound that.

MATH 29T Exam : Part I Solutons. TRUE/FALSE? Prove your answer! (a) (5 pts) There exsts a bnary one-error correctng code of length 9 wth 52 codewords. (b) (5 pts) There exsts a ternary one-error correctng

More information

### The example below solves a system in the unknowns α and β:

The Fnd Functon The functon Fnd returns a soluton to a system of equatons gven by a solve block. You can use Fnd to solve a lnear system, as wth lsolve, or to solve nonlnear systems. The example below

More information

### 1 Approximation Algorithms

CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

More information

### PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB.

PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. INDEX 1. Load data usng the Edtor wndow and m-fle 2. Learnng to save results from the Edtor wndow. 3. Computng the Sharpe Rato 4. Obtanng the Treynor Rato

More information

### 1. The scalar-valued function of a second order tensor φ ( T) , m an integer, is isotropic. T, m an integer is an isotropic function.

ecton 4. 4. ppendx to Chapter 4 4.. Isotropc Functons he scalar- vector- and tensor-valued functons φ a and of the scalar varable φ vector varable v and second-order tensor varable B are sotropc functons

More information

### Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

### SCALAR A physical quantity that is completely characterized by a real number (or by its numerical value) is called a scalar. In other words, a scalar

SCALAR A phscal quantt that s completel charactered b a real number (or b ts numercal value) s called a scalar. In other words, a scalar possesses onl a magntude. Mass, denst, volume, temperature, tme,

More information

### Introduction to Regression

Introducton to Regresson Regresson a means of predctng a dependent varable based one or more ndependent varables. -Ths s done by fttng a lne or surface to the data ponts that mnmzes the total error. -

More information

### CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES

CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable

More information

### Scalar and Vector Quantization

Scalar and Vector Quantzaton Máro A. T. Fgueredo, Departamento de Engenhara Electrotécnca e de Computadores, Insttuto Superor Técnco, Lsboa, Portugal maro.fgueredo@st.utl.pt November 2008 Quantzaton s

More information

### Binary Dependent Variables. In some cases the outcome of interest rather than one of the right hand side variables is discrete rather than continuous

Bnary Dependent Varables In some cases the outcome of nterest rather than one of the rght hand sde varables s dscrete rather than contnuous The smplest example of ths s when the Y varable s bnary so that

More information

### Quantization Effects in Digital Filters

Quantzaton Effects n Dgtal Flters Dstrbuton of Truncaton Errors In two's complement representaton an exact number would have nfntely many bts (n general). When we lmt the number of bts to some fnte value

More information

### A random variable is a variable whose value depends on the outcome of a random event/experiment.

Random varables and Probablty dstrbutons A random varable s a varable whose value depends on the outcome of a random event/experment. For example, the score on the roll of a de, the heght of a randomly

More information

### Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

More information

### The eigenvalue derivatives of linear damped systems

Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by Yeong-Jeu Sun Department of Electrcal Engneerng I-Shou Unversty Kaohsung, Tawan 840, R.O.C e-mal: yjsun@su.edu.tw

More information

### ME 563 HOMEWORK # 1 (Solutions) Fall 2010

ME 563 HOMEWORK # 1 (Solutons) Fall 2010 PROBLEM 1: (40%) Derve the equatons of moton for the three systems gven usng Newton-Euler technques (A, B, and C) and energy/power methods (A and B only). System

More information

### The covariance is the two variable analog to the variance. The formula for the covariance between two variables is

Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.

More information

### Chapter 7. Random-Variate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 Random-Variate Generation

Chapter 7 Random-Varate Generaton 7. Contents Inverse-transform Technque Acceptance-Rejecton Technque Specal Propertes 7. Purpose & Overvew Develop understandng of generatng samples from a specfed dstrbuton

More information

### Vision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION

Vson Mouse Saurabh Sarkar a* a Unversty of Cncnnat, Cncnnat, USA ABSTRACT The report dscusses a vson based approach towards trackng of eyes and fngers. The report descrbes the process of locatng the possble

More information