16 Chapter 1 The Schrodinger Equation 1.7 Complex Numbers We have seen that the wave function can be complex,so we now review some properties of complex numbers. A complex number z is a number of the form =x+iy.wherei=V (1.25) a negative quantity).If y 01n1. )then z is a real numbe 0.then z is an imagin 683 a er, is an imaginary nu er,an cial ely r Re(): and y are c =m(z)】 y parts The lex nu ary nart on the vertical axis.This diagra m immediately suggests defining two quantities that charac- terize the complex number:the distance r of the point z from the origin is called the absolute value or modulus of zand is denoted by the angle that the radius vector to the point z makes with the positive horizontal axis is called the phase or argument of z.We have l=r=(2+y)ir tano=y/x (1.26 x=rcos. y=rsin So we may write=x+iy as z=r cos ir sin0=ree 1.27) since(Prob.4.3) ei=cos0 isin 1.28) conjugate of the complex numberis defined as 来三x-y=re0 1.29 FIGURE1.3 (a)Plotof a complex numberz=x+iy. (b)Plot of the number -2+ b
16 Chapter 1 | The Schrödinger Equation 1.7 Complex Numbers We have seen that the wave function can be complex, so we now review some properties of complex numbers. A complex number z is a number of the form z = x + iy, where i K 2-1 (1.25) and where x and y are real numbers (numbers that do not involve the square root of a negative quantity). If y = 0 in (1.25), then z is a real number. If y 0, then z is an imaginary number. If x = 0 and y 0, then z is a pure imaginary number. For example, 6.83 is a real number, 5.4 - 3i is an imaginary number, and 0.60i is a pure imaginary number. Real and pure imaginary numbers are special cases of complex numbers. In (1.25), x and y are called the real and imaginary parts of z, respectively: x = Re(z); y = Im1z2. The complex number z can be represented as a point in the complex plane (Fig. 1.3), where the real part of z is plotted on the horizontal axis and the imaginary part on the vertical axis. This diagram immediately suggests defining two quantities that characterize the complex number z: the distance r of the point z from the origin is called the absolute value or modulus of z and is denoted by z ; the angle u that the radius vector to the point z makes with the positive horizontal axis is called the phase or argument of z. We have 0z0 = r = 1x2 + y221>2 , tan u = y>x (1.26) x = r cos u, y = r sin u So we may write z = x + iy as z = r cos u + ir sin u = reiu (1.27) since (Prob. 4.3) eiu = cos u + i sin u (1.28) The angle u in these equations is in radians. If z = x + iy, the complex conjugate z* of the complex number z is defined as z* K x - iy = re-iu (1.29) y r x (a) (b) u Figure 1.3 (a) Plot of a complex number z 5 x 1 iy. (b) Plot of the number 22 1 i
1.8Units 17 Ifis areal number.its imaginary part Thus is real if and only ifTaking =+iy)x-iy)=x2+iyx -iyx -i2y2 2z*=x2+y2=r2=22 1.30 1.3 22「2 It is easy to prove.either from the definition of complex conjugate or from(131),that (12)-效 (1.32 Likewise. (/)=/统(a+)°=着+嘉，(a-2)=济-克1.33 For the absolute values of products and quotients,it follows from(1.31)that lazal lallzal. 1.34 Therefore,if is a complex wave function,we have 2川=2=* (1.35) We now obtain a formula for the nth roots of the number 1.We may take the phase n being a posi- =1.Thus is an nth root of unity.There are n different complex nth roots of unity.and taking n successive values of the integer k gives us all of them: =e2，k=0,1,2.,n-1 (1.36 differs by ar (),we get the of the two square roots of:for =3.the thre and so on. 1.8 Units This book uses SI units.In the International System (SD).the units of length,mass.and time are the meter(m),kilogram(kg),and second(s).Force is measured in newtons (N)and energy in joules (J).Coulomb's law for the magnitude of the force between two charges and 2 separated by a distance r in vacuum is written in SI units as F= (1.37)
1.8 Units | 17 If z is a real number, its imaginary part is zero. Thus z is real if and only if z = z*. Taking the complex conjugate twice, we get z back again, 1z*2* = z. Forming the product of z and its complex conjugate and using i 2 = -1, we have zz* = (x + iy)(x - iy) = x 2 + iyx - iyx - i 2 y 2 zz* = x 2 + y 2 = r 2 = 0z0 2 (1.30) For the product and quotient of two complex numbers z1 = r1eiu1 and z2 = r2eiu2 , we have z1z2 = r1r2ei1u1+u22 , z1 z2 = r1 r2 ei1u1-u22 (1.31) It is easy to prove, either from the definition of complex conjugate or from (1.31), that 1z1z22* = z1 * z* 2 (1.32) Likewise, 1z1>z22* = z1 *>z2 *, 1z1 + z22* = z* 1 + z2 *, 1z1 - z22* = z1 * - z2 * (1.33) For the absolute values of products and quotients, it follows from (1.31) that 0z1z2 0 = 0z1 0 0z2 0 , ` z1 z2 ` = 0z1 0 0z2 0 (1.34) Therefore, if c is a complex wave function, we have 0 c2 0 = 0 c0 2 = c* c (1.35) We now obtain a formula for the nth roots of the number 1. We may take the phase of the number 1 to be 0 or 2p or 4p, and so on. Hence 1 = ei2pk , where k is any integer, zero, negative, or positive. Now consider the number v, where v K ei2pk>n , n being a positive integer. Using (1.31) n times, we see that vn = ei2pk = 1. Thus v is an nth root of unity. There are n different complex nth roots of unity, and taking n successive values of the integer k gives us all of them: v = ei2pk>n , k = 0, 1, 2, c, n - 1 (1.36) Any other value of k besides those in (1.36) gives a number whose phase differs by an integral multiple of 2p from one of the numbers in (1.36) and hence is not a different root. For n = 2 in (1.36), we get the two square roots of 1; for n = 3, the three cube roots of 1; and so on. 1.8 Units This book uses SI units. In the International System (SI), the units of length, mass, and time are the meter (m), kilogram (kg), and second (s). Force is measured in newtons (N) and energy in joules (J). Coulomb’s law for the magnitude of the force between two charges Q1 and Q2 separated by a distance r in vacuum is written in SI units as F = Q1Q2 4pe0r 2 (1.37)
18 Chapter 1 The Schrodinger Equation where the charges and 2 are in coulombs(C)and eo is a constant(called the permittivity of vacuumor the electric constant)whose value is8.854 X 10-12 C2N-m2 (see the Appendix for accurate values of physical constants). 1.9 Calculus =0 dc d(cf) df dx d(sin cx) =C cos cx. d(cosc）=-c sin cx,. dx -+ =费+ dx x dx dg dx of(x)dx=cf(x)dx [fx)+8(x)]dx /f(x)dx+ g(x)d =x +1 d= r=t sin cr dr =_cos cx cos cx dx sincx 厂网s=se-so）ha竖-） Summary The state of a quantum-mechanical system is described by a state function or wave function w.which is a function of the coordinates of the particles of the system and of the time The state function changes with time according to the time-dependent Schrodinger equa- tion,which for a one-particle,one-dimensional system is Eq.(1.13).For such a system,the quantity(x,)dx gives the probability that a measurement of the particle's position
18 Chapter 1 | The Schrödinger Equation where the charges Q1 and Q2 are in coulombs (C) and e0 is a constant (called the permittivity of vacuum or the electric constant) whose value is 8.854 * 10-12 C2 N-1 m-2 (see the Appendix for accurate values of physical constants). 1.9 Calculus Calculus is heavily used in quantum chemistry, and the following formulas, in which c, n, and b are constants and f and g are functions of x, should be memorized. dc dx = 0, d1cf2 dx = c df dx, dxn dx = nxn-1 decx dx = cecx d1sin cx2 dx = c cos cx, d1cos cx2 dx = -c sin cx, d ln cx dx = 1 x d1f + g2 dx = df dx + dg dx, d1fg2 dx = f dg dx + g df dx d1f>g2 dx = d1fg-12 dx = -fg-2 dg dx + g-1 df dx d dx f1g1x22 = df dg dg dx An example of the last formula is d3sin1cx224>dx = 2cx cos1cx 22. Here, g1x2 = cx 2 and f = sin. Lcf1x) dx = c Lf1x) dx, L3f1x2 + g1x)4 dx = Lf1x) dx + Lg1x) dx L dx = x, L xn dx = xn+1 n + 1 for n - 1, L 1 x dx = ln x L ecx dx = ecx c , L sin cx dx = - cos cx c , L cos cx dx = sin cx c L c b f1x2 dx = g1c2 - g1b2 where dg dx = f1x2 Summary The state of a quantum-mechanical system is described by a state function or wave function , which is a function of the coordinates of the particles of the system and of the time. The state function changes with time according to the time-dependent Schrödinger equation, which for a one-particle, one-dimensional system is Eq. (1.13). For such a system, the quantity 0 1x, t2 0 2 dx gives the probability that a measurement of the particle’s position
ling to s pote not dep on then the can exi f a numbe e of one-par one re the tin t wave function(x)is a s dependent S Sch. ion.19) dinger cqua Problems Answers to numerical problems are given at the end of the book. Sec. 12 141.516171.819 general Pobs.1.1-1.61.7-1.81.9-1.111.12-1.191.20-1.291.30-1.311.321.33 1.1 True or false?(a)All photons have the same energy.(b)As the frequency of light increases its wavelength decre s.(c)If violet light with 1.2 (b)An Nd:YAG n whose wavelength is 1064n mit ulse of 1064-nm radiation of a er5×10w and durations.Find the number of photons emitted in hs pulse.(Recall tha 1W=1J/s) 1.3 Calculate the energy of one mole of UV photons of wavelength 300 nm and compare it with a typical singl a energy of 400 k 1.4 The work function of very pure Na is 2.75 eV,where 1ev 1.602J.(a)Calculate th that in Na.(c)The work function of sodium that has not been very carefully purified is substantially less than 2.75 eV.bec use of adsorbed sulfur and other subst nces derived from atmospher hen impure N Is exposed to .will the ma t。 mation toPlanck's hlackbod Ch)In Iune 1900 Ravleig pp plied the on the rem of classical statistica wed the radiation intensity on of th Planck's blackh y the R formula.Hint:Look up the Taylor series expansion ofin powers of The classical mechanical Rayleigh-Jeans result is physically absurd.since it predicts the emitted energy to ncrea e without limit asincreases.) 1.6 Calculate the de Broglie wavelength of an electron moving at 1/137th the speed of light.(At this speed, the rela correction to the mass is negligible. 1.7 tecguaiond/h 8 oin the paragraph afte Eq.(1.11)to find nte tion constant and show that) 1.8 where a and b are constants and m is the particle's mass.Find the potential-energy function Vfor this system. Hint:Use the time-dependent Schrodinger equation. 1.9 True or false?(a)For all quantum-mechanical states.)2=()2.(b)For all quan- tum-mechanical states,V(r,t)is the product of a function of x and a function of t 1.10 A certain one-particle,one-dimer where b is a constant,c 2.00 nm and icle's energy
Problems | 19 at time t will find it between x and x + dx. The state function is normalized according to 1 - 0 0 2 dx = 1. If the system’s potential-energy function does not depend on t, then the system can exist in one of a number of stationary states of fixed energy. For a stationary state of a one-particle, one-dimensional system, 1x, t2 = e-iEt>U c1x2, where the timeindependent wave function c1x2 is a solution of the time-independent Schrödinger equation (1.19). Problems Answers to numerical problems are given at the end of the book. Sec. 1.2 1.4 1.5 1.6 1.7 1.8 1.9 general Probs. 1.1–1.6 1.7–1.8 1.9–1.11 1.12–1.19 1.20–1.29 1.30–1.31 1.32 1.33 1.1 True or false? (a) All photons have the same energy. (b) As the frequency of light increases, its wavelength decreases. (c) If violet light with l = 400 nm does not cause the photoelectric effect in a certain metal, then it is certain that red light with l = 700 nm will not cause the photoelectric effect in that metal. 1.2 (a) Calculate the energy of one photon of infrared radiation whose wavelength is 1064 nm. (b) An Nd:YAG laser emits a pulse of 1064-nm radiation of average power 5 * 106 W and duration 2 * 10-8 s. Find the number of photons emitted in this pulse. (Recall that 1 W = 1 J>s.) 1.3 Calculate the energy of one mole of UV photons of wavelength 300 nm and compare it with a typical single-bond energy of 400 kJ/mol. 1.4 The work function of very pure Na is 2.75 eV, where 1 eV = 1.602 * 10-19 J. (a) Calculate the maximum kinetic energy of photoelectrons emitted from Na exposed to 200 nm ultraviolet radiation. (b) Calculate the longest wavelength that will cause the photoelectric effect in pure Na. (c) The work function of sodium that has not been very carefully purified is substantially less than 2.75 eV, because of adsorbed sulfur and other substances derived from atmospheric gases. When impure Na is exposed to 200-nm radiation, will the maximum photoelectron kinetic energy be less than or greater than that for pure Na exposed to 200-nm radiation? 1.5 (a) Verify that at high frequencies Wien’s law is a good approximation to Planck’s blackbody equation. (b) In June 1900 Rayleigh applied the equipartition theorem of classical statistical mechanics to derive an equation for blackbody radiation that showed the radiation intensity to be proportional to n2 T. In 1905, Jeans pointed out an error in Rayleigh’s derivation of the proportionality constant and corrected the Rayleigh formula to I = 2pn2 kT>c2 . Show that at low frequencies, Planck’s blackbody formula can be approximated by the Rayleigh–Jeans formula. Hint: Look up the Taylor series expansion of ex in powers of x. (The classicalmechanical Rayleigh–Jeans result is physically absurd, since it predicts the emitted energy to increase without limit as n increases.) 1.6 Calculate the de Broglie wavelength of an electron moving at 1>137th the speed of light. (At this speed, the relativistic correction to the mass is negligible.) 1.7 Integrate the equation dx>dt = -gt + gt0 + v0 in the paragraph after Eq. (1.11) to find x as a function of time. Use the condition that the particle was at x0 at time t0 to evaluate the integration constant and show that x = x0 - 1 2 g1t - t022 + v01t - t02. 1.8 A certain one-particle, one-dimensional system has = ae-ibt e-bmx2 >U , where a and b are constants and m is the particle’s mass. Find the potential-energy function V for this system. Hint: Use the time-dependent Schrödinger equation. 1.9 True or false? (a) For all quantum-mechanical states, 0 1x, t2 0 2 = 0 c1x2 0 2 . (b) For all quantum-mechanical states, 1x, t2 is the product of a function of x and a function of t. 1.10 A certain one-particle, one-dimensional system has the potential energy V = 2c2 U2x2>m and is in a stationary state with c1x2 = bxe-cx2 , where b is a constant, c = 2.00 nm-2 , and m = 1.00 * 10-27 g. Find the particle’s energy
20 Chapter 1 The Schrodinger Equation 1.11 Which of the Schrodinger equations is applicable to all nonrelativistic quantum-mechanical nt off e.n-piclenme( ability that the result (a)lies between 0000 nm and(treat this interval as infini- tesimal):(b)lies between0and2nm(use the table of integrals in the Appendix.if necessary). erfy thati ormaihre is no nccd to use ca o answer th 1.13 A one-particle,one-dimensional system has the state function =(sin at)(2/m)el+(cosat)(32/m)Axel wherea isaconstant and=2.000A.If the particle's position is measured at=0.estimate the probability that the result will lie between 2.000 A and 2.001 A. 1.14 Use Eq.(1.23)to find the answer to part (a)of the example at the end of Section 1.6 and 1.15 Which of the foloin fumctns mcct a eofay-demity fuction nts)?(a)e :(b c)e 1.16 ()Fram two chil t least one female child.Wha the odds of giving birth to a boy or girl are equal.) 1.17 If the peak in the mass spectrum of CFat mass number 138 is 100 units high,calculate the C.98.896:C each of the four players A,B,( ,D) C have g two spades are ted so that b and d have one sn ade 119 What im orobability-de sity function occurs in(a)the kinetic theory of gases?(b)the analysis of random errors of measurement? 1.20 Classify each of the follo as a real number or an imaginary number:(a)-17:(b)2 i: (c):(d):(e)V-:(()2/3:(g):(h)(i)(a+bi)(a-bi).where a and b are real 1.21 Plot these points in the complex plane:(a)3:(b)-i;(c)-2+3i. 1.22 Show that 1/i =-i. 123 Simplify(a):b).(c):(dr(e)(1+5i)(2-30:(0(1-3i)/(4+2i).Hint:In (f).multiply numerator and denominator by the complex conjugate of the denominator. 1.24 Find the complex conjugate of (a)-4:(b)-2i:(c)6+3i;(d)2e. 1.25 Find the absolute value and the phase of (a)i:(b)(c)-2e3:(d)1-2i. 1.26 Where in the complex plane are all points whose absolute value is 5 located?Where are all points with phase /4 located? 1.27 Write each of the following in the form re:(a)i;(b)-1:(c)1-2i;(d)-1-i. 1.28 (a)Find the cube roots of 1.(b)Explain why the n nth roots of I when plotted in the complex plane lie on a circle of radius 1 and are separated by an angle 2n from one another. 1.29 Verify that sin em-e c0s9=”+e 2i 1.30 Expresseach of the following units in terms of fundamental SI units (m.kg.s):(a)newton:(b)joule. 1.31 Calculate the force on an alpha particle passing a gold atomic nucleus at a distance of 0.00300 A. 1.32 Find (a)dl2rsin(3x)+51/dr (b)(3x2+1)dxt. 133 True or false?(a)A probability density can pever be never be negative.(c)The state functionmust be a real function.(d)If=,thenz must be
20 Chapter 1 | The Schrödinger Equation 1.11 Which of the Schrödinger equations is applicable to all nonrelativistic quantum-mechanical systems? (a) Only the time-dependent equation. (b) Only the time-independent equation. (c) Both the time-dependent and the time-independent equations. 1.12 At a certain instant of time, a one-particle, one-dimensional system has = 12>b3 21>2 xe- 0 x 0>b , where b = 3.000 nm. If a measurement of x is made at this time in the system, find the probability that the result (a) lies between 0.9000 nm and 0.9001 nm (treat this interval as infinitesimal); (b) lies between 0 and 2 nm (use the table of integrals in the Appendix, if necessary). (c) For what value of x is the probability density a minimum? (There is no need to use calculus to answer this.) (d) Verify that is normalized. 1.13 A one-particle, one-dimensional system has the state function = 1sin at212>pc2 21>4 e-x2 >c2 + 1cos at2132>pc6 21>4 xe-x2 >c2 where a is a constant and c = 2.000 Å. If the particle’s position is measured at t = 0, estimate the probability that the result will lie between 2.000 Å and 2.001 Å. 1.14 Use Eq. (1.23) to find the answer to part (a) of the example at the end of Section 1.6 and compare it with the approximate answer found in the example. 1.15 Which of the following functions meet all the requirements of a probability-density function (a and b are positive constants)? (a) eiax ; (b) xe-bx2 ; (c) e-bx2 . 1.16 (a) Frank and Phyllis Eisenberg have two children; they have at least one female child. What is the probability that both their children are girls? (b) Bob and Barbara Shrodinger have two children. The older child is a girl. What is the probability the younger child is a girl? (Assume the odds of giving birth to a boy or girl are equal.) 1.17 If the peak in the mass spectrum of C2F6 at mass number 138 is 100 units high, calculate the heights of the peaks at mass numbers 139 and 140. Isotopic abundances: 12 C, 98.89%; 13 C, 1.11%; 19 F, 100%. 1.18 In bridge, each of the four players (A, B, C, D) receives 13 cards. Suppose A and C have 11 of the 13 spades between them. What is the probability that the remaining two spades are distributed so that B and D have one spade apiece? 1.19 What important probability-density function occurs in (a) the kinetic theory of gases? (b) the analysis of random errors of measurement? 1.20 Classify each of the following as a real number or an imaginary number: (a) -17; (b) 2 + i; (c) 27; (d) 2-1; (e) 2-6; (f) 2>3; (g) p; (h) i 2 ; (i) 1a + bi21a - bi2, where a and b are real numbers. 1.21 Plot these points in the complex plane: (a) 3; (b) -i; (c) -2 + 3i. 1.22 Show that 1>i = -i. 1.23 Simplify (a) i 2 ; (b) i 3 ; (c) i 4 ; (d) i*i; (e) 11 + 5i212 - 3i); (f) 11 - 3i2> 14 + 2i2. Hint: In (f), multiply numerator and denominator by the complex conjugate of the denominator. 1.24 Find the complex conjugate of (a) -4; (b) -2i; (c) 6 + 3i; (d) 2e-ip>5 . 1.25 Find the absolute value and the phase of (a) i; (b) 2eip>3 ; (c) -2eip>3 ; (d) 1 - 2i. 1.26 Where in the complex plane are all points whose absolute value is 5 located? Where are all points with phase p>4 located? 1.27 Write each of the following in the form reiu : (a) i; (b) -1; (c) 1 - 2i; (d) -1 - i. 1.28 (a) Find the cube roots of 1. (b) Explain why the n nth roots of 1 when plotted in the complex plane lie on a circle of radius 1 and are separated by an angle 2p>n from one another. 1.29 Verify that sin u = eiu - e-iu 2i , cos u = eiu + e-iu 2 . 1.30 Express each of the following units in terms of fundamental SI units (m, kg, s): (a) newton; (b) joule. 1.31 Calculate the force on an alpha particle passing a gold atomic nucleus at a distance of 0.00300 Å. 1.32 Find (a) d32x2 sin13x4 2 + 54>dx; (b) 1 2 1 13x2 + 12 dx. 1.33 True or false? (a) A probability density can never be negative. (b) The state function can never be negative. (c) The state function must be a real function. (d) If z = z*, then z must be a real number. (e) 1 - dx = 1 for a one-particle, one-dimensional system. (f) The product of a number and its complex conjugate is always a real number