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Quantum Physics I

by: Quinn Larkin

Quantum Physics I PHY 471

Quinn Larkin
GPA 3.67


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This 33 page Class Notes was uploaded by Quinn Larkin on Saturday September 19, 2015. The Class Notes belongs to PHY 471 at Michigan State University taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/207645/phy-471-michigan-state-university in Physics 2 at Michigan State University.

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Date Created: 09/19/15
Physics 471 Math Review Fall 2004 1 Complex Numbers While the values of physical observables must be real numbers quantum mechanical am plitudes are in general complex numbers Consequently some facility with the properties of complex numbers is needed in any study of quantum mechanics A complex number 2 is usually represented as 2 22y where z and y are real numbers and the symbol 2 satis es 22 71 2 can be viewed as V71 Apart from the introduction of 2 the algebraic properties of complex numbers are the same as those of real numbers For example 22 x 2y2 22 222y 2y2 22 7 22 22zy Notice that 22 is another complex number and that unlike the square of a real number it is not positive There is a positive number naturally associated with a complex number 2 It is best constructed by de ning the samplers conjugate 2 of 2 To get the complex conjugate of 2 one simply changes 2 to 72 1e 2 x239y 272y With this de nition the product 22 is M2 E 22 x 72y2 2y 22 y2 2 0 2 xzz 22 is called the absolute value or modulus of 2 Complex numbers reside in a plane called the complex plane whose 2 axis is the real axis and whose y axis is the imaginary axis The point7 xy lt gt z 2y can also be speci ed by the polar coordinates p6 as 2 pcos02sin0 p 22y2 tant9 a Typically the polar form is rewritten using Euler7s formula cos02sin0 ew to obtain 1 z x y pew In the latter form p is just lzl the modulus of z and 6 is called the phase of z Exercises H Find the product of the complex numbers 2 3239 and 1 4239 D Determine the reciprocal of z z W in terms of z and y Hint If the reciprocal is C X 23971 then 2 1 Use this to determine X and 1 9 Show that lewl 1 using Euler7s formula 4 Express 2 2 2239 in polar form Cf If 04 and B are real use Euler7s formula to show that eiaei 6ia 39 i7r6 CT Find the Cartesian form of z 46 I What can you conclude about 2 if z 2 2 Differential Equations In quantum mechanics like classical mechanics the dynamics of particles are determined by a differential equation The quantum mechanical differential equation is called Schrodinger7s equation In one dimension it is a second order differential equation for the wave function As such its solution will in general involve nding two independent functions which satisfy the differential equation and imposing certain boundary7 conditions on them dictated by the physical circumstances For many interesting applications the Schrodinger equation for 7 can be reduced to the form We dz with k being a real number The form of the solution is determined by the sign of the k2 term If it is k2 then the equation has the same form as the classical harmonic oscillator and the general solution is i kzmz 0 7 Acoskz B sinkx where A and B are independent of z and determined by the boundary conditions For the 7k2 case the general solution is 7 A6 B64 In quantum mechanical applications the boundary conditions usually restrict not only the form of the solution oscillating or exponential but also the values of k that are permissible The validity of these solutions and the use of boundary conditions are illustrated in the following exercises Exercises 1 Show by direct differentiation that the form of 72 for the 7k2 case is a solution to the differential equation 2 Show that the k2 oscillatory solution can also be written 72 06 D671 3 For the oscillatory k2 solution7 suppose z is restricted to 0 S x S a By requiring 0 0 and a 07 what are the restrictions on A7 B7 and k 4 In some quantum mechanics problems7 there are several regions with different values of k Suppose for 0 S x 3 17 the Schrodinger equation is dsz 2 d H 7W 0 and7 for a S x S 007 it is dsz 2 dzz 7 k 72 0 Determine the form of 72 in the two regions if 0 0 and as x a 007 72 a 0 3 Matrices For quantum systems with a nite number of degrees of freedom7 the equation of motion can be written as a relation involving nite dimensional square matrices We will develop this notion is some detail in class7 but it is useful to recall some simple properties of matrices A square matrix A is an n gtlt n array usually written as 011 012 013 39 39 39 aln 021 022 023 39 39 39 a2n A 031 032 033 39 39 39 13m anl anZ an 39 39 39 ann In quantum mechanics7 the notion of the hermiticz39ty of a matrix is very important lts de nition involves complex conjugation7 discussed above7 and the transpose7 AT7 of the matrix A AT is the matrix obtained by interchanging the row and columns of A7 and the hermitian conjugate of A7 Al7 is the complex conjugate of this matrix Explicitly7 A is an 021 031 am a a a a 12 22 32 n2 Ar 113 123 133 an aln aZn 19m 39 39 39 ann 3 Physics 471 Inclass Discussion Questions I am grateful to Michael Dubson of the University of Colorado for the vast majority of these questions QMll In Classical Mechanics can this equation be FdP derived net d t A Yes B No Answer No Newton S 2nd Law is a hypothesis to be compared with experiment It works amazingly well and is the foundation of Classical Mechanics net a d L QM12 Can this equation be derived A Yes B No Answer Yes it can be derived from Newton s 2Ild Law QMl3 In Quantum Mechanics can this equation be 712 82 1 3T derived 2m 6x2 V05 2 4715 A Yes B No Answer No the Schrodinger Equation is a hypothesis that must be compared with experiment It works amazingly well and is one of the foundations of Quantum Mechanics QM14 What is the modulus amplitude of 6XpiTE4 e1754 A 1 B 2 C J5 Di E 11 Answer A See the Math Review Sheet QMlS The probability density P 2 is plotted for a normalized wavefunction PX What is the probability that a position measurement will result in a measured value between 2 and 5 I Pl2 A23 BO3 04 CO4 03 D05 02 EO6 01 I I I I I 1 2 3 4 5 6 x Answer The area under the curve between X 2 and X 5 is 06 Since the P is normalized the total area is l QMl6 For a large number N of independent measurements of a random variable X which statement is true A 78gt Z ltXgt2 always B X2 2 ltXgt2 0r X2gt lt ltXgt2 depending on the probability distribution Answer A is true because 38gt ltxgt2 Z X ltxgt2gt Z 0 QMl7 Two traveling waves 1 and 2 are described by the equations y1 X t 2 sin2X t y2 X t 4 sinX 2t All the numbers are in the appropriate SI mks units Which wave has the higher speed A l B 2 C Both have the same speed The wavelength 9 of wave 1 is most nearly A lm B 2m C 3m D 4m The period of wave 2 is most nearly Als B2s C3s D4s Answers Traveling waves are represented by any function of X vt where v is the velocity In this problem the waves are sinusoidal and are represented as SlIllXDt where v oak The wavenumber is k 275 and the angular frequency is 0 275T So the first wave has k 2 m391 and 0 l s39l while the second wave has klm391andoa 2 s39l The answers are therefore B C C QM18 Choose the correct answer Ter2 dx 2 2T pee Mob 0 Txe zxzdsz 00 C None of the above Answer B because the integrand is odd in X QM19 Choose the correct answer 6 gt0 A ie Mx aydx 2e xa2dx C None of the above Answer C because the Gaussian is centered about xa hence the integrand is neither even nor odd in X Consider the two normalized wavefunctions shown in pictures 1 and 2 l 2 ILPI2 P2 QMllO Which of the following statements is true A ltXgt is the same for both wavefunctions B ltXgt is larger in 1 than in 2 C ltXgt is smaller in 1 than in 2 D I have no idea without doing an integral QMll 1 Which of the following statements is true A ltAX2gt is close to the same for l and 2 B ltAX2gt is much larger in 1 than in 2 C ltAX2gt is much smaller in 1 than in 2 D I have no idea without doing an integral Answers A for both questions If you don t believe this calculate ltAX2gt for a distribution like the one in 2 ILPI2 I 2 QMl12 Imagine that the wavefunctions shown above each represent a particle moving in one dimension at a speci c time In each case we measure the position of the particle Then we repeat the whole experiment many times with the same initial wavefunction and record our ndings in our laboratory notebook Choose the most accurate statement A In both experiments the probability of nding the particle with X lt a is 50 B The results of the two experiments will be very similar C The results of the two experiments will be quite different D Statements A and B are both true E Statements A and C are both true Answer E In the rst experiment the position measurements will all hover near two values In the second experiment they will be distributed uniformly over the range shown QMll3 Let y1Xt and y2Xt both be solutions of the same wave equation that is 631i mdthDLyh 6 vzmz 6 vza same V in both eqns Is the function ysum X t y1Xt y2Xt a solution of the 53m1 13km wave equation thatis is it true that 5X2 y at2 A Yes always B No never C sometimes depending on the functions yl and y2 Answer A because the differential equation is linear in y QM114 Given wX Asink X Bcosk X 7 the boundary condition V0 0 implies what AAO BBO Ck0 Dkn7 n123 ENone ofthese Answer B QM115 The stationary states Pn form an orthonormal set meaning Tn dX 6mn What is the value of I IPm chq njdx A Z 0n B cn cm C cm D CmZ 0n E None of these Answer C because chgmn cm 71 QMll7 Consider the third stationary state of the in nite square well W3 xlgsml37 What is the expectation value of the position ltXgt in that state A Zero B a2 C a D Can t answer because the state isn t normalized E None of the above Answer B QMl18 Consider the third stationary state of the in nite wX square well W3 xlgsml37 What is the expectation value of the momentum ltpgt in that state A A positive number B Zero C A negative number D Purely imaginary E None of the above Answer B QMll9 Consider the third 00 stationary state of the in nite square well a a a What is OK in that state Remember q ltxltxgt2gt ltAx2gt A Zero B Nonzero but smaller than a2 C A value close to a2 D A value larger than a2 E I have no idea without doing an integral Answer B Any measurement of the position will give a value within a2 of the average value QMlZO The Uncertainty Principle is often stated as apex 2 712 q ltxltxgt2gt ltAx2gt sz ltpltpgt2gt ltAp2gt What does the Uncertainty Principle mean A Physicists aren t really sure what predictions Quantum Mechanics makes about experiments B Quantum Mechanics is not an exact theory because it only gives approximate predictions about the results of experiments C If we restrict a particle to a narrow region of space and then measure its momentum there will be large spread in the values we get from the momentum measurement D The Uncertainty Principle would not apply if we knew the exact initial wavefunction Pxt0 for the particle E None of the above Answer C QMl21 Consider the third 00 stationary state of the in nite square well W3 x JZSm3 xj a a What is GP in that state Remember opz ltpltpgt2gt ltAp2gt A Zero B Nonzero but smaller than ha C A value close to 716 D A value larger than ha E I have no idea without doing an integral Answer D because we said before that 6X lt a2 and the Uncertainty Principle must be obeyed QMl22 We want to expand the square wave shown in a Fourier Series A ft fx b0 flan sin2 m ii cos27ml 711 T nl T Which of the following statements about the Fourier coef cients is true A Only the an s for odd n are nonzero B Only the an s for even 11 are nonzero C Only the his for odd n are nonzero D Only the his for even 11 are nonzero E None of the above statements are true Answer A Draw pictures of the even and odd sines and cosines and see how they match up with the square wave QMl23 PI and Pz are two stationary states HIPI Z EIIPI and HIPZ Z EZIPZ They are nondegenerate meaning E1 7 B Is P III2 III1 also a stationary state A Yes always B No never C Possibly yes depending on eigenvalues Answer B Plug 1 into the timeindependent Schrodinger equation and see if it is a solution It won t be QM124 A11 stationary states have the form 7103 2 2 INKE Z WX39e tso that l PXatl WXl 1s t1me independent Consider the sum of two nondegenerate stationary states LIJsumXt L1J1 LP2 W1X eimlt 1032t 2 Is this wavefunction stationary that is is IIPsum X 0 time independent A Yes always B No never CDepends on the w39s and on the 0339s Answer B See Problem 1 on Homework 3 QM125 Consider the function fX which is a sin wave of length L x sinkx lt X lt O elsewhere ANAL VVV VVV A L 7 y L Which statement is closest to the truth A fX has a single wellde ned wavelength B fX is made up of a range of wavelengths Answer B Think in terms of the Fourier transform QM126 Which of the graphs correctly shows parts of the wavefunction P that satis es Wquot kzw At W AK A w B V A X A X V C w h Ah D A V gt gt A X V X QM127 Which of the graphs correctly shows parts of the wavefunction P that satis es Wquot KZW Answers 26 A 27 B QMl28 Consider a particle of mass m in a 1D infinite square well of width 3701 The initial wavefunction I113530 looks like the back of an armadillo We measure the energy of the particle What are the possible outcomes of the measurement A Are you joking Not enough information is given h27z2 B 2m37a2 72272112 C 2m37a2 for integern Answer C The possible outcomes are the energies of the stationary states x2 b QM129 Consider the function fx e What can you say about the integral 00 ik x j fx 6 dx 00 It is A zero B nonzero and pure real C nonzero and pure imaginary D nonzero and complex Answer B QM l 3 0 Consider the wave function WW 0 Z A eXpik X 03 0 k gt 0 What does the probability density I IPX7 t 392 look like A P2 A constant in X and t X A LP2 B like this at an instant in time moving to the right a X llPlz C like this at an instant in time not moving right or left but oscillating up and down like a standing wave D None of these something entirely different Answer A QMl3 1 How does the energy E of the 11 3 state of an in nite square well of width 6 compare with the energy of the n3 state of an in nite well with a larger width The larger well has A lower energy B higher energy C the same energy Answer A QMl32 Compared to the in nite square with the same width 6 the ground state energy of a nite square well is A the same B higher C lower m Igt W a gt 11 Answer C The kinetic energy in the well is lower on the right because the wavelength is longer Look at the rst term in the Schrodinger equation


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