Lecture 4. Reading: Notes and Brennan


 Milton Weaver
 10 months ago
 Views:
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 89 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 SelfAdjoint 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 nonisotric 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 nonisotric 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 MomentumPosition 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. 47 ECE Dr. Alan Doolittle
25 MomentumPosition 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 MomentumPosition 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 ABBAiC 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 ABBAiC 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 timeindependent Schrödinger equation: d ψx
More information1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as
Chapter 3 (Lecture 45) 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 informationHermitian 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 informationChemistry 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 information1 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 informationA. The wavefunction itself Ψ is represented as a socalled 'ket' Ψ>.
Quantum Mechanical Operators and Commutation C I. BraKet Notation It is conventional to represent integrals that occur in quantum mechanics in a notation that is independent of the number of coordinates
More informationIntroduces 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 informationFrom Einstein to KleinGordon Quantum Mechanics and Relativity
From Einstein to KleinGordon 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 informationMathematical 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 192526: 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 informationBraket notation  Wikipedia, the free encyclopedia
Page 1 Braket notation FromWikipedia,thefreeencyclopedia Braket notation is the standard notation for describing quantum states in the theory of quantum mechanics. It can also be used to denote abstract
More informationa) Calculate the zeropoint vibrational energy for this molecule for a harmonic potential.
Problem set 4 Engel P7., P7.60. Problem 7. The force constant for a H 8 F molecule is 966 N m . a) Calculate the zeropoint vibrational energy for this molecule for a harmonic potential. The following
More informationPhysics 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 information4. 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 informationFrom 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 informationCHAPTER 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 informationThree 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 informationSecond 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 information2 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 informationLecture 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 informationThe 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 informationThe 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 informationTime 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 13:22:3705:00 Copyright 2003 Dan
More informationContinuous 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 informationMixed 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 informationFor the case of an Ndimensional 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. onhermitian
More informationCHAPTER 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 informationLectureXXIV. Quantum Mechanics Expectation values and uncertainty
LectureXXIV 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 informationNotes 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 informationANALYSIS 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 informationFourier Series and SturmLiouville Eigenvalue Problems
Fourier Series and SturmLiouville Eigenvalue Problems 2009 Outline Functions Fourier Series Representation Halfrange Expansion Convergence of Fourier Series Parseval s Theorem and Mean Square Error Complex
More information2. 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 informationThe Schrödinger Equation. Erwin Schrödinger Nobel Prize in Physics 1933
The Schrödinger Equation Erwin Schrödinger 18871961 Nobel Prize in Physics 1933 The Schrödinger Wave Equation The Schrödinger wave equation in its timedependent form for a particle of energy E moving
More informationQuick 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 information1 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 informationBasic 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 informationCHEM3023: Spins, Atoms and Molecules
CHEM3023: Spins, Atoms and Molecules Lecture 2 Braket notation and molecular Hamiltonians C.K. Skylaris Learning outcomes Be able to manipulate quantum chemistry expressions using braket notation Be
More informationConceptual 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 informationQuantum 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, xray, etc.) through
More informationHomework 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 information2. 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 informationLecture 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 informationLinear 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 informationChapter 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 informationAn 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. 5863 This lab is u lab on Fourier analysis and consists of VI parts.
More informationMATH 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 informationPHY4604 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 informationLecture 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 informationSpecial 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 informationThe Essentials of Quantum Mechanics
The Essentials of Quantum Mechanics Prof. Mark Alford v7, 2008Oct22 In classical mechanics, a particle has an exact, sharply defined position and an exact, sharply defined momentum at all times. Quantum
More informationLecture 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 informationSolved 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 informationRutgers  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 informationEXERCISES 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 informationON A COMPATIBILITY RELATION BETWEEN THE MAXWELLBOLTZMANN AND THE QUANTUM DESCRIPTION OF A GAS PARTICLE
International Journal of Pure and Applied Mathematics Volume 109 No. 2 2016, 469475 ISSN: 13118080 (printed version); ISSN: 13143395 (online version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v109i2.20
More informationCHAPTER 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 informationPROVING 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 informationAn 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 informationLinear Algebra Concepts
University of Central Florida School of Electrical Engineering Computer Science EGN3420  Engineering Analysis. Fall 2009  dcm Linear Algebra Concepts Vector Spaces To define the concept of a vector
More informationCHAPTER 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 informationIntroduction 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 informationPhysics 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 informationHarmonic 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 informationPhysics 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 informationLecture 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 JaniPetri Martikainen
More informationH = = + 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 information8.04: 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, AppI Li. 7 1 4 Ga. 4 7, 6 1,2
More informationWe can represent the eigenstates for angular momentum of a spin1/2 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 informationMatter 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 informationINTRODUCTION 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 informationQuantum interference with slits
1 Quantum interference with slits Thomas V Marcella Department of Physics and Applied Physics, University of Massachusetts Lowell, Lowell, MA 018542881, USA Email: thomasmarcella@verizon.net In the experiments
More informationIntroduction 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 informationChapter 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 informationQuiz 1(1): Experimental Physics LabI Solution
Quiz 1(1): Experimental Physics LabI 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 informationThe 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 information3 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 informationFourier 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 informationState 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 informationMatrix 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 informationQuantum 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 informationPHY 481/581 Intro NanoMaterials Science and engineering: Some Basics of Quantum Mechanics
PHY 481/581 Intro NanoMaterials Science and engineering: Some Basics of Quantum Mechanics http://creativecommons.org/licenses/byncsa/2.5/ 1 Basics of Quantum Mechanics  Why Quantum Physics?  Experimental
More informationz 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 informationAssessment 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 information1. 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 informationGenerally 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 informationF ij = Gm im j r i r j 3 ( r j r i ).
Physics 3550, Fall 2012 Newton s Third Law. Multiparticle 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 informationCHAPTER 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 informationThis 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 information2008 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 informationPath 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 informationConcepts 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 informationFourierSeries. 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 informationPower 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 informationContents Quantum states and observables
Contents 2 Quantum states and observables 3 2.1 Quantum states of spin1/2 particles................ 3 2.1.1 State vectors of spin degrees of freedom.......... 3 2.1.2 Pauli operators........................
More informationParticle 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 informationThe 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 informationIntroduction 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, SE16 91 Stockholm, Sweden Abstract In the present notes I try to give a better
More information202 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 informationTHE DOUBLE SLIT INTERFERENCE OF THE PHOTONSOLITONS
Bulg. J. Phys. 29 (2002 97 108 THE DOUBLE SLIT INTERFERENCE OF THE PHOTONSOLITONS P. KAMENOV, B. SLAVOV Faculty of Physics, University of Sofia, 1126 Sofia, Bulgaria Abstract. It is shown that a solitary
More information