# Lecture 4. Reading: Notes and Brennan

Save this PDF as:

Size: px
Start display at page:

Download "Lecture 4. Reading: Notes and Brennan"

## Transcription

1 Lecture 4 Postulates of Quantum Mechanics, Operators and Mathematical Basics of Quantum Mechanics Reading: Notes and Brennan.3-.6 ECE Dr. Alan Doolittle

2 Postulates of Quantum Mechanics Classical Mechanics describes the dynamical state variables of a particle as, y, z, p, etc... Quantum Mechanics takes a different approach. QM describes the state of any particle by an abstract Wave Function, (, y,z,t, we will describe in more detail later. We will introduce Five Postulates of Quantum Mechanics and one Governing Equation, known as the Schrödinger Equation. Postulates are hypotheses that can not be proven. If no discrepancies are found in nature, then a postulate becomes an aiom or a true statement that can not be proven. Newton s st and nd Laws and Mawell s equations are aioms you are probably already familiar with. There are also classical static variable such as mass, electronic charge, etc... that do not change during physical processes. ECE Dr. Alan Doolittle

3 Postulates of Quantum Mechanics Postulate The Wave Function, (, y,z,t, fully characterizes a quantum mechanical particle including it s position, movement and temporal prerties. (, y,z,t replaces the dynamical variables used in classical mechanics and fully describes a quantum mechanical particle. The two formalisms for the wave function: When the time dependence is included in the wavefunction, this is called the Schrödinger formulation. We will introduce a term later called an erator which is static in the Schrödinger formulation. In the Heisenberg formulation, the wavefunction is static (invariant in time and the erator has a time dependence. Unless otherwise stated, we will use the Schrödinger formulation ECE Dr. Alan Doolittle

4 Postulates of Quantum Mechanics Postulate The Probability Density Function of a quantum mechanical particle is: (, y,z,t (, y,z,t The probability of finding a particle in the volume between v and v+dv is: (, y,z,t (, y,z,t dv In D, the probability of finding a particle in the interval between & +d is: (, t (, t d Since (, t (, t d is a probability, then the sum of all probabilities must equal one. Thus, 4. (, t (, t d or (, y, z, t (, y, z, t dv This is the Normalization Condition and is very useful in solving problems. If fulfills 4., then is said to be a normalized wavefunction. ECE Dr. Alan Doolittle

5 Postulates of Quantum Mechanics Postulate (cont d (,y,z,t, the particle wave function, is related to the probability of finding a particle at time t in a volume ddydz. Specifically, this probability is: ( [, y, z, t ddydz or ddydz] But since is a probability, or in D (, y, z, t [ d] ddydz Solymar and Walsh figure 3. ECE Dr. Alan Doolittle

6 Postulates of Quantum Mechanics Postulate 3 Every classically obtained dynamical variable can be replaced by an erator that acts on the wave function. An erator is merely the mathematical rule used to describe a certain mathematical eration. For eample, the derivative erator is defined as d/d. The wave function is said to be the erators erand, i.e. what is being acted on. Classical Dynamical Variable Position Potential Energy V( f( Momentum p p Kinetic Energy m QM Operator Representation V( f( h i h m The following table lists the correspondence of the classical dynamical variables and their corresponding quantum mechanical erator: Total Energy (Kinetic + Potential Total Energy (Time Version f(p E E Total Total h m h f i + V( h i t ECE Dr. Alan Doolittle

7 Postulates of Quantum Mechanics Postulate 4 For each dynamical variable, ξ, there eists an epectation value, <ξ> that can be calculated from the wave function and the corresponding erator ξ (see table under postulate 3 for that dynamical variable, ξ. Assuming a normalized, 4. ξ (, t ξ(, t d or ξ y, z, t ξ (, y, z, t dv The epectation value, denoted by the braces <>, is also known as the average value or ensemble average. Brennan goes through a formal derivation on pages 8-9 of the tet and points out that the state of a QM particle is unknown before the measurement. The measurement forces the QM particle to be compartmentalized into a known state. However, multiple measurements can result in multiple state observances, each with a different observable. Given the probability of each state, the epectation value is a weighted average of all possible results weighted by each results probability. (, ECE Dr. Alan Doolittle

8 ECE Dr. Alan Doolittle Postulates of Quantum Mechanics Postulate 4 Eample: Lets assume the wave function of a QM particle is of an observable is given by: We can normalize (see postulate to get the constant A: Then we can calculate the epectation value of, <> as: Note: That the probability of observing the QM particle is 0 at 0 but multiple measurements will average to a net weighted average measurement of 0. otherwise A t 0, ( ( d d or, (, ( 3 A A A d A d t t -

9 Postulates of Quantum Mechanics Postulate 4 As the previous case illustrates, sometimes the average value of an observable is not all we need to know. Particularly, since the integrand of our <> equation was odd, the integrand was 0. (Looking at whether a function is even or odd can often save calculation effort. We may like to also know the standard deviation or the variance (standard deviation squared of the observable around the average value. In statistics, the standard deviation is: σ And since <>0, all we need is: Similarly, we could also calculate other higher order moments of distributions such as skewness, curtosis, etc... σ 3 0 (, t (, t d or d - σ σ ECE Dr. Alan Doolittle

10 Postulates of Quantum Mechanics Postulate 4 Real world, measurable observables need to be real values (i.e. not imaginary / comple. What makes for a real epectation value? To answer this, we need to consider some prerties of erators and devel (work toward the concept of a Hermitian Operator. Eigenfunctions and Eigenvalues: If the effect of an erator acting on an erand (wave function in our case is such that the erand is only modified by a scalar constant, the erand is said to be an Eigenfunction of that erator and the scalar is said to be an Eigenvalue. Eample: Operator ξ (, t λ (, t s Eigenfunction of Operator ξ Operand Eigenvalue ECE Dr. Alan Doolittle

11 Postulates of Quantum Mechanics Postulate 4 Eample: consider the case of an eponential function and the differential erator ξ d d e λ s d d (, t e λ λ s s e λ s Eigenfunction Eigenvalue of derivative Operand erator Eigenfunctions and Eigenvalues are very important in quantum mechanics and will be used etensively. KNOW THEM! RECOGNIZE THEM! LOVE THEM! ECE Dr. Alan Doolittle

12 Linear Operators: ξ Postulates of Quantum Mechanics Postulate 4 An erator is a linear erator if it satisfies the equation c(, t cξ (, t where c is a constant. is a linear erator where as the logarithmic erator log( is not. Unlike the case for classical dynamical values, linear QM erators generally do not commute. Consider: Classical Mechanics : p p Quantum Mechanics : h i d d h i d d (, t p? h (, t i h (, t i? (, t p d d (, t h d + i d (, t (, t ECE Dr. Alan Doolittle

13 Postulates of Quantum Mechanics Postulate 4 If two erators commute, then their physical observables can be known simultaneously. However, if two erators do not commute, there eists an uncertainty relationship between them that defines the relative simultaneous knowledge of their observables. More on this later... ECE Dr. Alan Doolittle

14 Postulates of Quantum Mechanics Postulate 4: A Hermitian Operator (erator has the prerty of Hermiticity results in an epectation value that is real, and thus, meaningful for real world measurements. Another word for a Hermitian Operator is a Self-Adjoint Operator. Let us work our way backwards for the D case: ξ If ξ is real and the dynamical variable is real (i.e. physically meaningful then, ξ ξ This says nothing about ξ Thus taking the conjugate of ξ ξ more generally an erator is said to be hermician if and (, tξ (, tξ (, tξ (, tξ ξ (, td ξ (, td (, td (, td as in general, ξ everything in the above equation, ξ (, tξ (, tξ ξ (, td (, td it satisfies : This equation is simply the case where ECE Dr. Alan Doolittle

15 Postulates of Quantum Mechanics Postulate 4 Consider two important prerties of a Hermitian Operator Eigenvalues of Hermitian Operators are real, and thus, measurable quantities. Proof: Hermiticity states, Consider first the left side, Now consider the right hand side, (, tξ Thus, λ (, tξ (, tξ s λ (, td (, td s λ (, td λ (, tξ s (, tλ s (, t which is only true if (, tλ (, td (, t(, td s s λ s (, td (, td (, td is real. ECE Dr. Alan Doolittle

16 Postulates of Quantum Mechanics Postulate 4 Consider two important prerties of a Hermitian Operator Eigenfunctions corresponding to different and unequal Eigenvalues of a Hermitian Operator are orthogonal. Orthogonal functions of this type are important in QM because we can find a set of functions that spans the entire QM space (known as a basis set without duplicating any information (i.e. having the one function project onto another. Formally, ξ and λ (, t λ This is true if (, t functions (, td ξ λ is hermician and, are (, t 0 orthogonal and ξ if they satisfy : (, t λ (, t See Brennan section.6 for details ECE Dr. Alan Doolittle

17 Probability current density is conserved. Postulates of Quantum Mechanics Postulate 5 If a particle is not being created or destroyed it s integrated probability always remains constant ( for a normalized wave function. However, if the particle is moving, we can define a Probability Current Density and a Probability Continuity equation that describes the particles movement through a Gaussian surface (analogous to electromagnetics. Brennan, section.5 derives this concept in more detail than I wish to discuss and arrives at the following: Define Probability Density ρ Define Probability Current Density (see Brennan section.5 ( r ( r j h mi ( ( r ( r ( r ( r ρ Probability Current Density Continuity Equation ρ j + t 0 ECE Dr. Alan Doolittle

18 Postulates of Quantum Mechanics Postulate 5 Basically this epression states that the wave function of a quantum mechanical particle is a smoothly varying function. In an isotric medium, mathematically, this is stated in a simplified form as: Lim o Lim o (, t and (, t ( Simply stated, the wave function and it s derivative are smoothly varying (no discontinuities. However, in a non-isotric medium (eamples at a heterojunction where the mass of an electron changes on either side of the junction or at an abrupt potential boundary the full continuity equation MUST be used. Be careful as the above isotric simplification is quoted as a Postulate in MANY QM tets but can get you into trouble (see homework problem in non-isotric mediums. When in doubt, use ρ h mi o (, t o, t ρ ( j + 0 ( r ( r j ( r ( r ( r ( r o t ECE Dr. Alan Doolittle

19 Postulates of Quantum Mechanics Eample using these Postulates Consider the eistence of a wave function (Postulate of the form: ( A ( + cos( 0 for for - π -π π π Figure after Fred Schubert with gratitude. ECE Dr. Alan Doolittle

20 Postulates of Quantum Mechanics Eample using these Postulates Consider the eistence of a wave function (Postulate of the form: ( A ( + cos( 0 for for Applying the normalization criteria from Postulate : A A ( + cos ( + cos π ( + cos + cos π A + sin + (, t (, t d A ( 3π A + 4 3π sin - π π -π π d d π π ( 3π ( + cos( 0 for for - π -π π π ECE Dr. Alan Doolittle

21 Postulates of Quantum Mechanics Eample using these Postulates Show that this wave function obeys the Probability Continuity Equation (Postulate 5 at the boundaries +/-π: ( ( + cos( for - π π j 3π 0 for -π Define Probability Density ρ ( + cos + cos ρ t π ( r ( r ρ 3π ρ ρ is independent of time so 0 t so ( ( sin 3π h Probability Current Density j mi h ( cos sin ( cos + + mi 3π 3π 3π Probability Current Density Continuity Equation j + ( ( ( ( ( sin 3 π 0 ECE Dr. Alan Doolittle

22 Postulates of Quantum Mechanics Eample using these Postulates Show that this wave function obeys the smoothness constraint (Postulate 5 for isotric medium at the boundaries +/-π: ( 3π ( + cos( Lim π (or - π 3π 0 0 for 0 for for - π π -π π ( + cos ( + cosπ and -π or π Lim sin π (or - π 3π 0 0 for -π or π 3π sinπ 3π Thus, the wave function and its derivative are both zero and continuous at the boundary (same for -π. ECE Dr. Alan Doolittle

23 Postulates of Quantum Mechanics Eample using these Postulates What is the epected position of the particle? (Postulate 3 ( 3π ( + cos( 0 for for - π -π π π 3π d 3π ( + cos ( + cos 3π π ( + cos + cos π d d Since the function in parenthesis is even and is odd, the product (integrand is odd between symmetric limits of +/-π. Thus, 0 Note: Postulate 4 is not demonstrated because is not an Eigenfunction of any erator. ECE Dr. Alan Doolittle

24 Momentum-Position Space and Transformations Position Space Fourier Transform Momentum Space ( Φ( p πh e ip / h dp Φ ( p ( πh e ip / h d If ( is normalized then so is Φ( p ( ( d Φ ( p Φ( p dp If a state is localized in position,, it is delocalized in momentum, p. This leads us to a fundamental quantum mechanical principle: You can not have infinite precision measurement of position and momentum simultaneously. If the momentum (and thus wavelength from ph/λ is known, the position of the particle is unknown and vice versa. For more detail, see Brennan p. 4-7 ECE Dr. Alan Doolittle

25 Momentum-Position Space and Transformations b ( e Fourier Transform φ( p be b h p The above wave function is a superposition of 7 simple plane waves and creates a net wave function that is localized in space. The red line is the amplitude envele (related to. These types of wave functions are useful in describing particles such as electrons. The wave function envele can be approimated with a Gaussian function in space. Using the Fourier relationship between space and momentum, this can be transformed into a Gaussian in momentum (see Brennan section.3. The widths of the two Gaussians are inversely related. ECE Dr. Alan Doolittle

26 Momentum-Position Space and Transformations b ( e Fourier Transform φ( p be b h p The widths of the two Gaussians are inversely related. Thus, accurate knowledge of position leads to inaccurate knowledge of momentum. ECE Dr. Alan Doolittle

27 Uncertainty And the Heisenberg Uncertainty Principle Due to the probabilistic nature of Quantum Mechanics, uncertainty in measurements is an inherent prerty of quantum mechanical systems. Heisenberg described this in 97. For any two Hermitian erators that do not commute (i.e. B OP A OP A OP B OP there observables A and B can not be simultaneously known (see Brennan section.6 for proof. Thus, there eists an uncertainty relationship between observables whose Hermitian erators do not commute. Note: if erators A and B do commute (i.e. ABBA then the observables associated with erators A and B can be known simultaneously. ECE Dr. Alan Doolittle

28 Uncertainty And the Heisenberg Uncertainty Principle Consider the QM variable, ξ, with uncertainty (standard deviation ξ defined by the variance (square of the standard deviation or the mean deviation, ( ξ <ξ ><ξ> ξ is the standard/mean deviation for the variable ξ from its epectation value <ξ> There are two derivations of the Heisenberg Uncertainty Principle in QM tets. The one in Brennan is more precise, but consider the following derivation for it s insight into the nature of the uncertainty. Brennan s derivation will follow. ECE Dr. Alan Doolittle

29 Uncertainty And the Heisenberg Uncertainty Principle Insightful Derivation: Consider a wave function with a Gaussian distribution defined by: The normalization constant can be determined as, Taking the Fourier transformation of this function into the momentum space we get, ip / h Φ( p ( πh e d Φ Φ ( p πh ( p ( 4π 4 ( A ( ξ A e π ( 4π ( ξ 4 ( 4π 4 ( ξ ( ξ π h ( ξ π h ( ξ ( ξ e ( ξ e e h p ip / h ( ξ d The standard deviation in momentum space is inversely related to the standard deviation in position space. ECE Dr. Alan Doolittle

30 Uncertainty And the Heisenberg Uncertainty Principle Insightful Derivation: The standard deviation in momentum space is inversely related to the standard deviation in position space. Defining: Position Space σ ( ξ and σ p h ( ξ / ( 4π 4 σ X ( σ X ( e Φ( p ( σ X π ( 4π 4 ( σ X e π σ X σ ( and σ ( p p Φ ( p Momentum Space ( p 4π 4 σ p ( σ σ p π ( p 4π 4 ( σ σ p e π e p p σ ( p h σ p ( p ECE Dr. Alan Doolittle

31 Uncertainty And the Heisenberg Uncertainty Principle Precise Derivation: The solution for the Gaussian distribution is eact only for that distribution. However, the assumption of a Gaussian distribution overestimates the uncertainty in general. Brennan (p derives the precise General Heisenberg Uncertainty Relationship using: A If two erators do not commute (i.e. AB BA in general a relationship can be written such that AB-BAiC B Given (A the following prerty proven true by Merzbacher 970: ( A ( B 0.5C Thus, given the position erator,, and momentum erator, p (ħ/i(d/d, Letting this erate on wave function h i p ( p p d d p h i d d ic ic ic ECE Dr. Alan Doolittle

32 Uncertainty And the Heisenberg Uncertainty Principle Precise Derivation (cont d: A If two erators do not commute (i.e. AB BA in general a relationship can be written such that AB-BAiC B Given (A the following prerty proven true by Merzbacher 970: ( A ( B 0.5C p p Letting this erate on wave function ( p p ic h d h d ic i d i d h d h d ( ic i d i d h d h h d ic i d i i d h ic i h C ic ECE Dr. Alan Doolittle

33 Uncertainty And the Heisenberg Uncertainty Principle Precise Derivation (cont d Thus, since ( A ( B 0.5C, with in this case, A and B p, p For this case, the uncertainies associated with is, and p p ( ( p ih h is p must be related to h ( ( p h Note the etra factor of ½. ECE Dr. Alan Doolittle

34 Uncertainty And the Heisenberg Uncertainty Principle Energy Time Derivation: Using: Group Velocity, v group / t ω/ k and the de Broglie relation pħ k and the Planck Relationship Eħ ω: Using Group Velocity Using de Broglie relation Using Planck Relationship ( ( p t ϖ k t ϖ k ( p ( h k ( t( ϖ ( t E h h h h ( t( E h ECE Dr. Alan Doolittle

35 Operators in Momentum Space Postulate 3 (additional information Equivalent momentum space erators for each real space counterpart. Classical Dynamical Variable QM Operator Real Space Representation QM Operator MomentumSpace Representation Position h i p Potential Energy V( V( V h i p f( f( f h i p Momentum p h i p Kinetic Energy p m h m p m f(p h f i f ( p Total Energy (Kinetic + Potential E Total h m + V( p + V h i p m Total Energy (Time Version E Total h i t h i t ECE Dr. Alan Doolittle

36 Dirac Notation There eists a short hand notation known as Dirac notation that simplifies writing of QM equations and is valid in either position or momentum space. ξ or in momentum space, ξ Φ (, tξ (p, tξ (, td Φ(p, tdp Note: The left side, <, is known as a Bra while the right side, >, is known as a ket (derived from the Bracket notation. The ket is by definition, the comple conjugate of the argument. ξ (, tξ (, td Dirac notation is valid in either position or momentum space so the variables,, y, z, t, and p can be tionally left out. ECE Dr. Alan Doolittle

37 Dirac Notation The erator acts on the function on the right side: ξ ξ (, tξ (, td If you need the erator to act on the function to the left, write it eplicitly. ξ (, tξ (, td Note that in this eample, we have commuted the left hand side (erator and with the right hand side (. This is okay. While not all erators commute, any function times another function does commute (i.e. once the erator conjugate has acted on the, the result is merely a function which always commutes. Thus, it is the erator that may or may not commute but the functions that result from an erator acting on a wave function always commute. ECE Dr. Alan Doolittle

38 Dirac Notation The Dirac form of Hermiticity is: (, tξ (, td ξ ξ or ξ ξ (, tξ (, td ECE Dr. Alan Doolittle

39 ECE Dr. Alan Doolittle The Dirac Delta Function is: Dirac Delta Function ( ( ( ( for 0 for o o 0 lim d e o o o o o δ δ δ π σ δ σ σ Some useful prerties of the delta function are: ( ( ( ( ( ( ( ( ( o o o o o o o o f d d d d d f f d f f f a a ( ( ( ( for 0 ( ( o δ δ δ δ δ δ δ δ δ o 0

40 Dirac and Kronecker Delta Function The Dirac Delta Function is: δ ( o for o 0 otherwise The Kronecker Delta Function is: δ ij 0 if i j if i j ECE Dr. Alan Doolittle

### Harmonic Oscillator Physics

Physics 34 Lecture 9 Harmonic Oscillator Physics Lecture 9 Physics 34 Quantum Mechanics I Friday, February th, 00 For the harmonic oscillator potential in the time-independent Schrödinger equation: d ψx

More information

### 1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as

Chapter 3 (Lecture 4-5) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series

More information

### Hermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1)

CHAPTER 4 PRINCIPLES OF QUANTUM MECHANICS In this Chapter we will continue to develop the mathematical formalism of quantum mechanics, using heuristic arguments as necessary. This will lead to a system

More information

### Chemistry 431. NC State University. Lecture 3. The Schrödinger Equation The Particle in a Box (part 1) Orthogonality Postulates of Quantum Mechanics

Chemistry 431 Lecture 3 The Schrödinger Equation The Particle in a Box (part 1) Orthogonality Postulates of Quantum Mechanics NC State University Derivation of the Schrödinger Equation The Schrödinger

More information

### 1 The Fourier Transform

Physics 326: Quantum Mechanics I Prof. Michael S. Vogeley The Fourier Transform and Free Particle Wave Functions 1 The Fourier Transform 1.1 Fourier transform of a periodic function A function f(x) that

More information

### A. The wavefunction itself Ψ is represented as a so-called 'ket' Ψ>.

Quantum Mechanical Operators and Commutation C I. Bra-Ket Notation It is conventional to represent integrals that occur in quantum mechanics in a notation that is independent of the number of coordinates

More information

### Introduces the bra and ket notation and gives some examples of its use.

Chapter 7 ket and bra notation Introduces the bra and ket notation and gives some examples of its use. When you change the description of the world from the inutitive and everyday classical mechanics to

More information

### From Einstein to Klein-Gordon Quantum Mechanics and Relativity

From Einstein to Klein-Gordon Quantum Mechanics and Relativity Aline Ribeiro Department of Mathematics University of Toronto March 24, 2002 Abstract We study the development from Einstein s relativistic

More information

### Mathematical Formulation of the Superposition Principle

Mathematical Formulation of the Superposition Principle Superposition add states together, get new states. Math quantity associated with states must also have this property. Vectors have this property.

More information

### - develop a theory that describes the wave properties of particles correctly

Quantum Mechanics Bohr's model: BUT: In 1925-26: by 1930s: - one of the first ones to use idea of matter waves to solve a problem - gives good explanation of spectrum of single electron atoms, like hydrogen

More information

### Bra-ket notation - Wikipedia, the free encyclopedia

Page 1 Bra-ket notation FromWikipedia,thefreeencyclopedia Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. It can also be used to denote abstract

More information

### a) Calculate the zero-point vibrational energy for this molecule for a harmonic potential.

Problem set 4 Engel P7., P7.6-0. Problem 7. The force constant for a H 8 F molecule is 966 N m -. a) Calculate the zero-point vibrational energy for this molecule for a harmonic potential. The following

More information

### Physics 53. Wave Motion 1

Physics 53 Wave Motion 1 It's just a job. Grass grows, waves pound the sand, I beat people up. Muhammad Ali Overview To transport energy, momentum or angular momentum from one place to another, one can

More information

### 4. The Infinite Square Well

4. The Infinite Square Well Copyright c 215 216, Daniel V. Schroeder In the previous lesson I emphasized the free particle, for which V (x) =, because its energy eigenfunctions are so simple: they re the

More information

### From Fourier Series to Fourier Integral

From Fourier Series to Fourier Integral Fourier series for periodic functions Consider the space of doubly differentiable functions of one variable x defined within the interval x [ L/2, L/2]. In this

More information

### CHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often

7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.

More information

### Three Pictures of Quantum Mechanics. Thomas R. Shafer April 17, 2009

Three Pictures of Quantum Mechanics Thomas R. Shafer April 17, 2009 Outline of the Talk Brief review of (or introduction to) quantum mechanics. 3 different viewpoints on calculation. Schrödinger, Heisenberg,

More information

### Second postulate of Quantum mechanics: If a system is in a quantum state represented by a wavefunction ψ, then 2

. POSTULATES OF QUANTUM MECHANICS. Introducing the state function Quantum physicists are interested in all kinds of physical systems (photons, conduction electrons in metals and semiconductors, atoms,

More information

### 2 A Differential Equations Primer

A Differential Equations Primer Homer: Keep your head down, follow through. [Bart putts and misses] Okay, that didn't work. This time, move your head and don't follow through. From: The Simpsons.1 Introduction

More information

### Lecture 18: Quantum Mechanics. Reading: Zumdahl 12.5, 12.6 Outline. Problems (Chapter 12 Zumdahl 5 th Ed.)

Lecture 18: Quantum Mechanics Reading: Zumdahl 1.5, 1.6 Outline Basic concepts of quantum mechanics and molecular structure A model system: particle in a box. Demos how Q.M. actually obtains a wave function.

More information

### (x) = lim. x 0 x. (2.1)

Differentiation. Derivative of function Let us fi an arbitrarily chosen point in the domain of the function y = f(). Increasing this fied value by we obtain the value of independent variable +. The value

More information

### The Sixfold Path to understanding bra ket Notation

The Sixfold Path to understanding bra ket Notation Start by seeing the three things that bra ket notation can represent. Then build understanding by looking in detail at each of those three in a progressive

More information

### The Schrödinger Equation

The Schrödinger Equation When we talked about the axioms of quantum mechanics, we gave a reduced list. We did not talk about how to determine the eigenfunctions for a given situation, or the time development

More information

### Time dependence in quantum mechanics Notes on Quantum Mechanics

Time dependence in quantum mechanics Notes on Quantum Mechanics http://quantum.bu.edu/notes/quantummechanics/timedependence.pdf Last updated Thursday, November 20, 2003 13:22:37-05:00 Copyright 2003 Dan

More information

### Continuous Random Variables

Continuous Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Continuous Random Variables 2 11 Introduction 2 12 Probability Density Functions 3 13 Transformations 5 2 Mean, Variance and Quantiles

More information

### Mixed states and pure states

Mixed states and pure states (Dated: April 9, 2009) These are brief notes on the abstract formalism of quantum mechanics. They will introduce the concepts of pure and mixed quantum states. Some statements

More information

### For the case of an N-dimensional spinor the vector α is associated to the onedimensional . N

1 CHAPTER 1 Review of basic Quantum Mechanics concepts Introduction. Hermitian operators. Physical meaning of the eigenvectors and eigenvalues of Hermitian operators. Representations and their use. on-hermitian

More information

### CHAPTER 5 THE HARMONIC OSCILLATOR

CHAPTER 5 THE HARMONIC OSCILLATOR The harmonic oscillator is a model which has several important applications in both classical and quantum mechanics. It serves as a prototype in the mathematical treatment

More information

### Lecture-XXIV. Quantum Mechanics Expectation values and uncertainty

Lecture-XXIV Quantum Mechanics Expectation values and uncertainty Expectation values We are looking for expectation values of position and momentum knowing the state of the particle, i,e., the wave function

More information

### Notes on wavefunctions III: time dependence and the Schrödinger equation

Notes on wavefunctions III: time dependence and the Schrödinger equation We now understand that the wavefunctions for traveling particles with momentum p look like wavepackets with wavelength λ = h/p,

More information

### ANALYSIS AND APPLICATIONS OF LAPLACE /FOURIER TRANSFORMATIONS IN ELECTRIC CIRCUIT

www.arpapress.com/volumes/vol12issue2/ijrras_12_2_22.pdf ANALYSIS AND APPLICATIONS OF LAPLACE /FOURIER TRANSFORMATIONS IN ELECTRIC CIRCUIT M. C. Anumaka Department of Electrical Electronics Engineering,

More information

### Fourier Series and Sturm-Liouville Eigenvalue Problems

Fourier Series and Sturm-Liouville Eigenvalue Problems 2009 Outline Functions Fourier Series Representation Half-range Expansion Convergence of Fourier Series Parseval s Theorem and Mean Square Error Complex

More information

### 2. Introduction to quantum mechanics

2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian

More information

### The Schrödinger Equation. Erwin Schrödinger Nobel Prize in Physics 1933

The Schrödinger Equation Erwin Schrödinger 1887-1961 Nobel Prize in Physics 1933 The Schrödinger Wave Equation The Schrödinger wave equation in its time-dependent form for a particle of energy E moving

More information

### Quick Reference Guide to Linear Algebra in Quantum Mechanics

Quick Reference Guide to Linear Algebra in Quantum Mechanics Scott N. Walck September 2, 2014 Contents 1 Complex Numbers 2 1.1 Introduction............................ 2 1.2 Real Numbers...........................

More information

### 1 Lecture 3: Operators in Quantum Mechanics

1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators

More information

### Basic Quantum Mechanics

Basic Quantum Mechanics Postulates of QM - The state of a system with n position variables q, q, qn is specified by a state (or wave) function Ψ(q, q, qn) - To every observable (physical magnitude) there

More information

### CHEM3023: Spins, Atoms and Molecules

CHEM3023: Spins, Atoms and Molecules Lecture 2 Bra-ket notation and molecular Hamiltonians C.-K. Skylaris Learning outcomes Be able to manipulate quantum chemistry expressions using bra-ket notation Be

More information

### Conceptual Approaches to the Principles of Least Action

Conceptual Approaches to the Principles of Least Action Karlstad University Analytical Mechanics FYGB08 January 3, 015 Author: Isabella Danielsson Supervisors: Jürgen Fuchs Igor Buchberger Abstract We

More information

### Quantum Mechanics: Postulates

Quantum Mechanics: Postulates 5th April 2010 I. Physical meaning of the Wavefunction Postulate 1: The wavefunction attempts to describe a quantum mechanical entity (photon, electron, x-ray, etc.) through

More information

### Homework One Solutions. Keith Fratus

Homework One Solutions Keith Fratus June 8, 011 1 Problem One 1.1 Part a In this problem, we ll assume the fact that the sum of two complex numbers is another complex number, and also that the product

More information

### 2. Wavefunctions. An example wavefunction

2. Wavefunctions Copyright c 2015 2016, Daniel V. Schroeder To create a precise theory of the wave properties of particles and of measurement probabilities, we introduce the concept of a wavefunction:

More information

### Lecture 22 Relevant sections in text: 3.1, 3.2. Rotations in quantum mechanics

Lecture Relevant sections in text: 3.1, 3. Rotations in quantum mechanics Now we will discuss what the preceding considerations have to do with quantum mechanics. In quantum mechanics transformations in

More information

### Linear Algebra In Dirac Notation

Chapter 3 Linear Algebra In Dirac Notation 3.1 Hilbert Space and Inner Product In Ch. 2 it was noted that quantum wave functions form a linear space in the sense that multiplying a function by a complex

More information

### Chapter 3. Forces, Momentum & Stress. 3.1 Newtonian mechanics: a very brief résumé

Chapter 3 Forces, Momentum & Stress 3.1 Newtonian mechanics: a very brief résumé In classical Newtonian particle mechanics, particles (lumps of matter) only experience acceleration when acted on by external

More information

### An introduction to generalized vector spaces and Fourier analysis. by M. Croft

1 An introduction to generalized vector spaces and Fourier analysis. by M. Croft FOURIER ANALYSIS : Introduction Reading: Brophy p. 58-63 This lab is u lab on Fourier analysis and consists of VI parts.

More information

### MATH 461: Fourier Series and Boundary Value Problems

MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter

More information

### PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Exam Solutions Dec. 13, 2004

PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Exam Solutions Dec. 1, 2004 No materials allowed. If you can t remember a formula, ask and I might help. If you can t do one part of a problem,

More information

### Lecture 34: Principal Axes of Inertia

Lecture 34: Principal Axes of Inertia We ve spent the last few lectures deriving the general expressions for L and T rot in terms of the inertia tensor Both expressions would be a great deal simpler if

More information

### Special Relativity and Electromagnetism Yannis PAPAPHILIPPOU CERN

Special Relativity and Electromagnetism Yannis PAPAPHILIPPOU CERN United States Particle Accelerator School, University of California - Santa Cruz, Santa Rosa, CA 14 th 18 th January 2008 1 Outline Notions

More information

### The Essentials of Quantum Mechanics

The Essentials of Quantum Mechanics Prof. Mark Alford v7, 2008-Oct-22 In classical mechanics, a particle has an exact, sharply defined position and an exact, sharply defined momentum at all times. Quantum

More information

### Lecture 2. Observables

Lecture 2 Observables 13 14 LECTURE 2. OBSERVABLES 2.1 Observing observables We have seen at the end of the previous lecture that each dynamical variable is associated to a linear operator Ô, and its expectation

More information

### Solved Problems on Quantum Mechanics in One Dimension

Solved Problems on Quantum Mechanics in One Dimension Charles Asman, Adam Monahan and Malcolm McMillan Department of Physics and Astronomy University of British Columbia, Vancouver, British Columbia, Canada

More information

### Rutgers - Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006

Rutgers - Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006 QA J is an angular momentum vector with components J x, J y, J z. A quantum mechanical state is an eigenfunction of J 2 J

More information

### EXERCISES FOR CHAPTER 6: Taylor and Maclaurin Series

EXERCISES FOR CHAPTER 6: Taylor and Maclaurin Series. Find the first 4 terms of the Taylor series for the following functions: (a) ln centered at a=, (b) centered at a=, (c) sin centered at a = 4. (a)

More information

### ON A COMPATIBILITY RELATION BETWEEN THE MAXWELL-BOLTZMANN AND THE QUANTUM DESCRIPTION OF A GAS PARTICLE

International Journal of Pure and Applied Mathematics Volume 109 No. 2 2016, 469-475 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v109i2.20

More information

### CHAPTER 12 MOLECULAR SYMMETRY

CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical

More information

### PROVING STATEMENTS IN LINEAR ALGEBRA

Mathematics V2010y Linear Algebra Spring 2007 PROVING STATEMENTS IN LINEAR ALGEBRA Linear algebra is different from calculus: you cannot understand it properly without some simple proofs. Knowing statements

More information

### An introduction to the electric conduction in crystals

An introduction to the electric conduction in crystals Andreas Wacker, Matematisk Fysik, Lunds Universitet Andreas.Wacker@fysik.lu.se November 23, 2010 1 Velocity of band electrons The electronic states

More information

### Linear Algebra Concepts

University of Central Florida School of Electrical Engineering Computer Science EGN-3420 - Engineering Analysis. Fall 2009 - dcm Linear Algebra Concepts Vector Spaces To define the concept of a vector

More information

### CHAPTER 16: Quantum Mechanics and the Hydrogen Atom

CHAPTER 16: Quantum Mechanics and the Hydrogen Atom Waves and Light Paradoxes in Classical Physics Planck, Einstein, and Bohr Waves, Particles, and the Schrödinger equation The Hydrogen Atom Questions

More information

### Introduction to quantum mechanics

Introduction to quantum mechanics Lecture 3 MTX9100 Nanomaterjalid OUTLINE -What is electron particle or wave? - How large is a potential well? -What happens at nanoscale? What is inside? Matter Molecule

More information

### Physics 53. Wave Motion 2. If at first you don't succeed, try, try again. Then quit. No use being a damn fool about it.

Physics 53 Wave Motion 2 If at first you don't succeed, try, try again. Then quit. No use being a damn fool about it. W.C. Fields Waves in two or three dimensions Our description so far has been confined

More information

### Harmonic Oscillator and Coherent States

Chapter 5 Harmonic Oscillator and Coherent States 5. Harmonic Oscillator In this chapter we will study the features of one of the most important potentials in physics, it s the harmonic oscillator potential

More information

### Physics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives

Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring

More information

### Lecture 2: Essential quantum mechanics

Department of Physical Sciences, University of Helsinki http://theory.physics.helsinki.fi/ kvanttilaskenta/ p. 1/46 Quantum information and computing Lecture 2: Essential quantum mechanics Jani-Petri Martikainen

More information

### H = = + H (2) And thus these elements are zero. Now we can try to do the same for time reversal. Remember the

1 INTRODUCTION 1 1. Introduction In the discussion of random matrix theory and information theory, we basically explained the statistical aspect of ensembles of random matrices, The minimal information

More information

### 8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology. Problem Set 5

8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Tuesday March 5 Problem Set 5 Due Tuesday March 12 at 11.00AM Assigned Reading: E&R 6 9, App-I Li. 7 1 4 Ga. 4 7, 6 1,2

More information

### We can represent the eigenstates for angular momentum of a spin-1/2 particle along each of the three spatial axes with column vectors: 1 +y =

Chapter 0 Pauli Spin Matrices We can represent the eigenstates for angular momentum of a spin-/ particle along each of the three spatial axes with column vectors: +z z [ ] 0 [ ] 0 +y y [ ] / i/ [ ] i/

More information

### Matter Waves. Solutions of Selected Problems

Chapter 5 Matter Waves. Solutions of Selected Problems 5. Problem 5. (In the text book) For an electron to be confined to a nucleus, its de Broglie wavelength would have to be less than 0 4 m. (a) What

More information

### INTRODUCTION TO FOURIER TRANSFORMS FOR PHYSICISTS

INTRODUCTION TO FOURIER TRANSFORMS FOR PHYSICISTS JAMES G. O BRIEN As we will see in the next section, the Fourier transform is developed from the Fourier integral, so it shares many properties of the

More information

### Quantum interference with slits

1 Quantum interference with slits Thomas V Marcella Department of Physics and Applied Physics, University of Massachusetts Lowell, Lowell, MA 01854-2881, USA E-mail: thomasmarcella@verizon.net In the experiments

More information

### Introduction to Fourier Series

Introduction to Fourier Series We ve seen one eample so far of series of functions. The Taylor Series of a function is a series of polynomials and can be used to approimate a function at a point. Another

More information

### Chapter 18 4/14/11. Superposition Principle. Superposition and Interference. Superposition Example. Superposition and Standing Waves

Superposition Principle Chapter 18 Superposition and Standing Waves If two or more traveling waves are moving through a medium, the resultant value of the wave function at any point is the algebraic sum

More information

### Quiz 1(1): Experimental Physics Lab-I Solution

Quiz 1(1): Experimental Physics Lab-I Solution 1. Suppose you measure three independent variables as, x = 20.2 ± 1.5, y = 8.5 ± 0.5, θ = (30 ± 2). Compute the following quantity, q = x(y sin θ). Find the

More information

### The Quantum Harmonic Oscillator Stephen Webb

The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems

More information

### 3 Trigonometric Fourier Series

3 Trigonometric Fourier Series Ordinary language is totally unsuited for epressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical

More information

### Fourier Transforms The Fourier Transform Properties of the Fourier Transform Some Special Fourier Transform Pairs 27

24 Contents Fourier Transforms 24.1 The Fourier Transform 2 24.2 Properties of the Fourier Transform 14 24.3 Some Special Fourier Transform Pairs 27 Learning outcomes In this Workbook you will learn about

More information

### State of Stress at Point

State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

More information

### Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,

LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x

More information

### Quantum Mechanics and Representation Theory

Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30

More information

### PHY 481/581 Intro Nano-Materials Science and engineering: Some Basics of Quantum Mechanics

PHY 481/581 Intro Nano-Materials Science and engineering: Some Basics of Quantum Mechanics http://creativecommons.org/licenses/by-nc-sa/2.5/ 1 Basics of Quantum Mechanics - Why Quantum Physics? - Experimental

More information

### z 0 and y even had the form

Gaussian Integers The concepts of divisibility, primality and factoring are actually more general than the discussion so far. For the moment, we have been working in the integers, which we denote by Z

More information

### Assessment Plan for Learning Outcomes for BA/BS in Physics

Department of Physics and Astronomy Goals and Learning Outcomes 1. Students know basic physics principles [BS, BA, MS] 1.1 Students can demonstrate an understanding of Newton s laws 1.2 Students can demonstrate

More information

### 1. The quantum mechanical state of a hydrogen atom is described by the following superposition: ψ = (2ψ 1,0,0 3ψ 2,0,0 ψ 3,2,2 )

CHEM 352: Examples for chapter 2. 1. The quantum mechanical state of a hydrogen atom is described by the following superposition: ψ = 1 14 2ψ 1,, 3ψ 2,, ψ 3,2,2 ) where ψ n,l,m are eigenfunctions of the

More information

### Generally Covariant Quantum Mechanics

Chapter 15 Generally Covariant Quantum Mechanics by Myron W. Evans, Alpha Foundation s Institutute for Advance Study (AIAS). (emyrone@oal.com, www.aias.us, www.atomicprecision.com) Dedicated to the Late

More information

### F ij = Gm im j r i r j 3 ( r j r i ).

Physics 3550, Fall 2012 Newton s Third Law. Multi-particle systems. Relevant Sections in Text: 1.5, 3.1, 3.2, 3.3 Newton s Third Law. You ve all heard this one. Actioni contrariam semper et qualem esse

More information

### CHAPTER III - MARKOV CHAINS

CHAPTER III - MARKOV CHAINS JOSEPH G. CONLON 1. General Theory of Markov Chains We have already discussed the standard random walk on the integers Z. A Markov Chain can be viewed as a generalization of

More information

### This is the 39th lecture and our topic for today is FIR Digital Filter Design by Windowing.

Digital Signal Processing Prof: S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 39 FIR Digital Filter Design by Windowing This is the 39th lecture and

More information

### 2008 AP Calculus AB Multiple Choice Exam

008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus

More information

### Path Integrals in Quantum Field Theory A Friendly Introduction

Section 2 Classical Mechanics and Classical Field Theory Path Integrals in Quantum Field Theory A Friendly Introduction Chris Elliott October, 203 Aims of this Talk In this talk I hope to demystify (at

More information

### Concepts for specific heat

Concepts for specific heat Andreas Wacker, Matematisk Fysik, Lunds Universitet Andreas.Wacker@fysik.lu.se November 8, 1 1 Introduction In this notes I want to briefly eplain general results for the internal

More information

### FourierSeries. In 1807, the French mathematician and physicist Joseph Fourier submitted a paper C H A P T E R

C H A P T E R FourierSeries In 87, the French mathematician and physicist Joseph Fourier submitted a paper on heat conduction to the Academy of Sciences of Paris. In this paper Fourier made the claim that

More information

### Power Series. Chapter Introduction

Chapter 10 Power Series In discussing power series it is good to recall a nursery rhyme: There was a little girl Who had a little curl Right in the middle of her forehead When she was good She was very,

More information

### Contents Quantum states and observables

Contents 2 Quantum states and observables 3 2.1 Quantum states of spin-1/2 particles................ 3 2.1.1 State vectors of spin degrees of freedom.......... 3 2.1.2 Pauli operators........................

More information

### Particle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 7 : Symmetries and the Quark Model. Introduction/Aims

Particle Physics Michaelmas Term 2011 Prof Mark Thomson Handout 7 : Symmetries and the Quark Model Prof. M.A. Thomson Michaelmas 2011 206 Introduction/Aims Symmetries play a central role in particle physics;

More information

### The Central Equation

The Central Equation Mervyn Roy May 6, 015 1 Derivation of the central equation The single particle Schrödinger equation is, ( H E nk ψ nk = 0, ) ( + v s(r) E nk ψ nk = 0. (1) We can solve Eq. (1) at given

More information

### Introduction to Green s Functions: Lecture notes 1

October 18, 26 Introduction to Green s Functions: Lecture notes 1 Edwin Langmann Mathematical Physics, KTH Physics, AlbaNova, SE-16 91 Stockholm, Sweden Abstract In the present notes I try to give a better

More information

### 202 6TheMechanicsofQuantumMechanics. ) ψ (r) =± φ (r) ψ (r) (6.139) (t)

0 6TheMechanicsofQuantumMechanics It is easily shown that ˆ has only two (real) eigenvalues, ±1. That is, if ˆ ψ (r) = λψ (r), ˆ ψ (r) = ˆ ψ ( r) = ψ (r) = λ ˆ ψ ( r) = λ ψ (r) (6.137) so λ =±1. Thus,

More information

### THE DOUBLE SLIT INTERFERENCE OF THE PHOTON-SOLITONS

Bulg. J. Phys. 29 (2002 97 108 THE DOUBLE SLIT INTERFERENCE OF THE PHOTON-SOLITONS P. KAMENOV, B. SLAVOV Faculty of Physics, University of Sofia, 1126 Sofia, Bulgaria Abstract. It is shown that a solitary

More information