ENGINEERING QUANTUM MECHANICS

Transcription

1 ENGINEERING QUANTUM MECHANICS Doyeol Ahn Seoung-Hwan Park IEEE PRESS A JOHN WILEY & SONS, INC., PUBLICATION

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3 ENGINEERING QUANTUM MECHANICS

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5 ENGINEERING QUANTUM MECHANICS Doyeol Ahn Seoung-Hwan Park IEEE PRESS A JOHN WILEY & SONS, INC., PUBLICATION

6 Copyrght 2011 by John Wley & Sons, Inc. All rghts reserved. Publshed by John Wley & Sons, Inc., Hoboken, New Jersey. Publshed sultaneously n Canada. No part of ths publcaton ay be reproduced, stored n a retreval syste, or transtted n any for or by any eans, electronc, echancal, photocopyng, recordng, scannng, or otherwse, except as pertted under Secton 107 or 108 of the 1976 Unted States Copyrght Act, wthout ether the pror wrtten persson of the Publsher, or authorzaton through payent of the approprate per-copy fee to the Copyrght Clearance Center, Inc., 222 Rosewood Drve, Danvers, MA 01923, (978) , fax (978) , or on the web at Requests to the Publsher for persson should be addressed to the Perssons Departent, John Wley & Sons, Inc., 111 Rver Street, Hoboken, NJ 07030, (201) , fax (201) , or onlne at Lt of Lablty/Dsclaer of Warranty: Whle the publsher and author have used ther best efforts n preparng ths book, they ake no representatons or warrantes wth respect to the accuracy or copleteness of the contents of ths book and specfcally dscla any pled warrantes of erchantablty or ftness for a partcular purpose. No warranty ay be created or extended by sales representatves or wrtten sales aterals. The advce and strateges contaned heren ay not be sutable for your stuaton. You should consult wth a professonal where approprate. Nether the publsher nor author shall be lable for any loss of proft or any other coercal daages, ncludng but not lted to specal, ncdental, consequental, or other daages. For general nforaton on our other products and servces or for techncal support, please contact our Custoer Care Departent wthn the Unted States at (800) , outsde the Unted States at (317) or fax (317) Wley also publshes ts books n a varety of electronc forats. Soe content that appears n prnt ay not be avalable n electronc forats. For ore nforaton about Wley products, vst our web ste at Lbrary of Congress Catalogng-n-Publcaton Data: Ahn, Doyeol. Engneerng Quantu Mechancs/Doyeol Ahn, Seoung-Hwan Park. p. c. Includes bblographcal references and ndex. ISBN Quantu theory. 2. Stochastc processes. 3. Engneerng atheatcs. 4. Seconductors Electrc propertes Matheatcal odels. I. Park, Seoung-Hwan. II. Ttle. QC A '53012 dc obook ISBN: epdf ISBN: epub ISBN: Prnted n Sngapore

7 CONTENTS Preface v PART I Fundaentals 1 1 Basc Quantu Mechancs Measureents and Probablty Drac Forulaton Bref Detour to Classcal Mechancs A Road to Quantu Mechancs The Uncertanty Prncple The Haronc Oscllator Angular Moentu Egenstates Quantzaton of Electroagnetc Felds Perturbaton Theory 38 Probles 41 References 43 2 Basc Quantu Statstcal Mechancs Eleentary Statstcal Mechancs Second Quantzaton Densty Operators The Coherent State The Squeezed State Coherent Interactons Between Atos and Felds The Jaynes Cungs Model 69 Probles 71 References 72 3 Eleentary Theory of Electronc Band Structure n Seconductors Bloch Theore and Effectve Mass Theory The Luttnger Kohn Haltonan The Znc Blende Haltonan 105 v

8 v CONTENTS 3.4 The Wurtzte Haltonan Band Structure of Znc Blende and Wurtzte Seconductors Crystal Orentaton Effects on a Znc Blende Haltonan Crystal Orentaton Effects on a Wurtzte Haltonan 152 Probles 168 References 169 PART II Modern Applcatons Quantu Inforaton Scence Quantu Bts and Tensor Products Quantu Entangleent Quantu Teleportaton Evoluton of the Quantu State: Quantu Inforaton Processng A Measure of Inforaton Quantu Black Holes 184 Appendx A: Dervaton of Equaton (4.82) 202 Appendx B: Dervaton of Equatons (4.93) and (4.106) 203 Probles 204 References Modern Seconductor Laser Theory Densty Operator Descrpton of Optcal Interactons The Te-Convolutonless Equaton The Theory of Non-Markovan Optcal Gan n Seconductor Lasers Optcal Gan of a Quantu Well Laser wth Non-Markovan Relaxaton and Many-Body Effects Nuercal Methods for Valence Band Structure n Nanostructures Znc Blende Bulk and Quantu Well Structures Wurtzte Bulk and Quantu Well Structures Quantu Wres and Quantu Dots 265 Appendx: Fortran 77 Code for the Band Structure 274 Probles 286 References 287 Index 289

9 Preface Quantu echancs s becong ore portant n appled scence and engneerng, especally wth the recent developents n quantu coputng, as well as the rapd progress n optoelectronc devces. Ths textbook s ntended for graduate students and advanced undergraduate students n electrcal engneerng, physcs, and aterals scence and engneerng. It also provdes the necessary theoretcal background for researchers n optoelectroncs or seconductor devces. In the task of provdng advanced nstructon for both students and researchers, quantu echancs presents specal dffcultes because of ts herarchcal structures. The ore abstract foralss and technques are qute eanngless untl one has astered the earler stages n classcal physcs, whch ost engneerng students are lackng. Quantu echancs has becoe an essental tool for odern engneerng. Ths book covers topcs such as seconductors and laser physcs, whch are tradtonally quantu echancal, as well as relatvely new topcs n the feld, such as quantu coputaton and quantu nforaton. These felds have seen an explosve growth durng the past 10 years, as quantu coputng or quantu nforaton processng can have a sgnfcant pact on today s electroncs and coputatons. The essence of quantu coputng s the drect usage of the superposton and entangleent of quantu echancs. The ost challengng research topcs nclude the generaton and anpulaton of quantu entangled systes, developng the fundaental theory of entangleent, decoherence control, and the deonstraton of the scalablty of quantu nforaton processng. In laser physcs, there has been a growng nterest n the odel of seconductor lasers wth non - Markovan relaxaton partally because of the dssatsfacton wth the conventonal odel for optcal gan n predctng the correct gan spectru and the therodynac relatons. Ths s anly due to the poor convergence propertes of the lneshape functon, that s, the Lorentzan lneshape, used n the conventonal odel. In ths book, a non - Markovan odel for the optcal gan of seconductors s developed, takng nto account the rgorous electronc band structure, any - body effects, and the non - Markovan v

10 v PREFACE relaxaton usng the quantu statstcal reduced - densty operator forals for an arbtrary drven syste coupled to a stochastc reservor. Exaple progras based on Fortran 77 wll also be provded for band structures of znc blende quantu wells. Many - body effects are taken nto account wthn the te - dependent Hartree Fock. Varous seconductor lasers ncludng straned - layer quantu well lasers and wurtzte GaN blue - green quantu well lasers are dscussed. We thank Professor Shun - Len Chuang, Doyeol Ahn s Ph.D. thess advser, for extensve enlghtenng and encourageent over any years. We are also grateful to any colleagues and frends, especally Frank Stern, B. D. Choe, Han Jo L, H. S. Mn, M. S. K, Robert Mann, T Ralph, K. S. Seo, Y. S. Cho, and Chancellor Sa Bu Lee. The support of our research by the Korean Mnstry of Educaton, Scence and Technology s greatly apprecated. Ths book would not have been copleted wthout the patence and contnued encourageent of our edtors at Wley and above all the encourageent and understandng of Taeyeon Y and Young - Mee An. Thanks for puttng up wth us. Doyeol Ahn Seoung-Hwan Park

11 PART I Fundaentals

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13 1 Basc Quantu Mechancs 1.1 MEASUREMENTS AND PROBABILITY In the begnnng of 20th century, t was dscovered that the behavor of very sall partcles, such as electrons, the nucle of atos, and olecules, cannot be descrbed by classcal echancs, whch had been qute successful n explanng the acroscopc world untl then. Nonetheless, t was soon dscovered that the descrpton of these phenoena on the atoc scale s possble by the set of laws descrbed by quantu echancs. Both classcal echancs and quantu echancs are based on the descrpton of easureents of observable quanttes called dynacal varables, such as poston, oentu, and energy. Consder an experent n whch we can ake three easureents successve n te. Let s denote the frst of observable quanttes A, the second B, and the thrd C. We also denote a, b, and c as one of a nuber of possble results that could coe fro the easureent of A, B, and C, respectvely. Let P( ba ) be the condtonal probablty that f the easureent of A results n a, then the easureent of B wll result n b. Fro the eleentary probablty theory, the condtonal probablty P( ba ) can be wrtten as follows: Pab Pba ( ) (, ) Pa (), (1.1) where P(, ab) s the jont probablty that easureents of both A and B wll gve a and b, sultaneously, and P() a s the probablty that the easureent of A wll gve the outcoe a. For three successve easureents A, B, and C, the condtonal probablty P( cba ) that f the easureent of A results n a, then the easureent of B wll result n b, then the easureent of C wll result n c s gven by: Pcba ( ) PcbPba ( ) ( ). (1.2) Engneerng Quantu Mechancs, Frst Edton. Doyeol Ahn, Seoung-Hwan Park John Wley & Sons, Inc. Publshed 2011 by John Wley & Sons, Inc. 3

14 4 BASIC QUANTUM MECHANICS Moreover, f we su Equaton (1.2) over all the utually exclusve alternatves for b, we obtan the condtonal probablty P ca ( ): Pca ( ) PcbPba ( ) ( ). (1.3) b In classcal echancs, the above relaton descrbed by Equaton (1.3) s always true. However, t was found that the above relaton soetes fals on the atoc scale, and one needs to odfy Equatons (1.1) to (1.3) by ntroducng new coplex quanttes ϕ ba, ϕ cb, and ϕ ca, called probablty apltudes, whch are related to probabltes by [1,2] Pba ( ) ϕ 2, (1.4) ba and Pca ϕcbϕba ( ). b 2 (1.5) Equatons (1.4) and (1.5) descrbe the probablty of easureent outcoe n quantu echancs. Fro the atheatcal pont of vew, the probablty apltude s found to be the nner product of vectors n a specal knd of vector space called the Hlbert space: ϕ ba b a, (1.6) where a s the colun vector, called the ket vector, correspondng to the observable a, and b s the row vector, called the bra vector, correspondng to the observable b. 1.2 DIRAC FORMULATION In quantu echancs, a physcal state correspondng to the observable quantty a s represented by the ket vector, a, n a coplex vector space H wth denson N. For exaple, when N 2, the ket vector a s a colun vector gven by [1,3] a a 1 a, (1.7) 2

15 DIRAC FORMULATION 5 where a 1 and a 2 are coplex nubers. We can also consder the case where the denson of the vector space s nfnte. In ths case, the ket vector a s represented by a colun vector gven by a a1 a 2., (1.8).. a wth coplex quanttes a1, a2, a3,, a. Fro now on, we wll denote ket vectors sply as vectors. If a and b are vectors n H, and α and β are coplex nubers, then the lnear superposton of these two vectors α a + β b s also a vector n H. For each vector a n H, we can relate a row vector a called bra vector, whch s gven by and * a ( a a * ), for N, (1.9) * * * * a ( a a a... a ), for N, (1.10) where * s the coplex conjugate. By coparng Equatons (1.7) wth (1.10), one can see that the ket vector a and bra vector b are related by a ( a ), (1.11) where s the adjont operaton, whch s the transposton, followed by the coplex conjugaton. The nner product between two vectors a and b s a coplex nuber ba, whch s gven by * * * ba ba + ba + + ba ab*. (1.12) The vector space where one can defne the nner product relaton of Equaton (1.12) s called the Hlbert space. The length of the vector a s defned by

16 6 BASIC QUANTUM MECHANICS a a a, (1.13) and when the vector has the unt length, we call t a noral vector. The vectors n the Hlbert space H are apped to dfferent vectors n the Hlbert space H by the lnear transforaton Â, whch s often called the lnear operator n quantu echancs. Matheatcally, the act of the lnear operator  on the vectors of the Hlbert space H s expressed as A ˆ : H H. In ost cases, the ntal Hlbert space H and the fnal Hlbert space H are the sae, and unless otherwse specfed explctly, we wll assue that s the case. Any two vectors a and b n H are transfored by  as and Aˆ ( α a + β b ) αaˆ a + βaˆ b, (1.14) ( Aˆ α a ) α* a A ˆ, (1.15) where α and β are coplex constants. If we set c A ˆ a fro Equaton (1.11), we obtan c ( c ) a Aˆ, and then, usng Equaton (1.12), we get baa ˆ * * bc cb aaˆ b. (1.16) For a gven operator Â, there exsts partcular set of vectors { a 1, a 2,, a n, } called egenstates, whch satsfy Aa ˆ a a, Aa ˆ a a,, Aa ˆ a a, (1.17) n n n Here, the nubers a 1, a 2,, a n, are called egenvalues. Of the any knds of operators, one partcular type of operator, called the Hertan operator, plays an portant role n quantu echancs. If  s a Hertan operator, then t satsfes the followng propertes: (I) ˆ A Aˆ. (II) Egenvalues a 1, a 2,, a n, are real. (III) Egenvectors correspondng to dfferent egenvalues are orthogonal. (IV) a a I, where I s an dentty atrx. n n n The proof s as follows: Fro Âa we have a a and Âa a a, j j j

17 DIRAC FORMULATION 7 a Aˆ a Aˆ Aˆ * a a a. ( ) (1.18) j j j j j Then, we get a A ˆ a a a a, (1.19) j j and a Aˆ * a a a a. (1.20) j j j Subtractng Equaton (1.20) fro Equaton (1.19), we obtan * ( a a ) a a 0. (1.21) j j When j, we get aj a 0, whch proves the property (III). On the other hand, f j, then we have a a *, thus gvng the property (II). The property (III) dctates that the set of egenvectors { a } of a Hertan operator  for an orthogonal bass f the state a s properly noralzed that s, a a 1 and a aj 0 for j. Let the Hlbert space spanned by these bass vectors be denoted as H. Then any vector ψ that belongs to H can be expressed as ψ C a, (1.22) where C s an expanson coeffcent that can be calculated by takng an nner product between the state vectors ψ and a : C a ψ. (1.23) By substtutng Equaton (1.23) nto Equaton (1.22), we obtan ψ C a a a ψ a a I ψ, ψ (1.24) whch proves the property (IV). It would be handy to eorze that gven a chan of vectors or operators, we can nsert the dentty operator

18 8 BASIC QUANTUM MECHANICS defned by (IV) n any place at our convenence. For exaple, the Hertan operator  can be wrtten as Aˆ I AI ˆ a a Aˆ n n a a n a a Aˆ a a n, n, n n n a ( a a a ) a n n a a a n n n, (1.25) whch s also called the spectral representaton of an operator Â. Let  and ˆB be lnear operators, and we defne ˆD as D ˆ AB ˆ ˆ. Moreover, we defne b B ˆ a and d Dˆ a Aˆ b. Then, fro Equatons (1.11) and (1.15), we obtan d a Dˆ baˆ (1.26) aba ˆ ˆ, or ˆˆ ˆ ( AB) B Aˆ. (1.27) 1.3 BRIEF DETOUR TO CLASSICAL MECHANICS Alost seven decades ago, Drac ade a connecton between classcal echancs and quantu echancs by assung that the lnear operators correspond to the dynacal varables at that te. By dynacal varables, we ean quanttes such as the coordnates and the coponents of velocty, oentu, and angular oentu of partcles, and functons of these quanttes, the varables n ters of whch classcal echancs s bult. Even now, 70 years later, hs postulates of quantu echancs are stll vald and perhaps only plausble approaches. Drac s postulates requre that those dynacal varables shall also occur n quantu echancs, but wth the dfference that they are now subject to an algebra n whch the coutatve axo of ultplcaton does

19 BRIEF DETOUR TO CLASSICAL MECHANICS 9 not hold. Nonetheless, the dynacal varables of quantu echancs stll have any propertes n coon wth ther classcal counterparts, and t wll stll be possble to buld up a theory of the closely analogous to the classcal theory and for a generalzaton of t. In ths sprt, the transton fro classcal echancs to quantu echancs can be ade ost convenently and easly usng the Haltonan forulaton of classcal echancs [4,5]. Classcal echancs s based on the assupton that any physcally nterestng varable, that s, dynacal varable, can be easured wth arbtrary precson and wthout utual nterference fro any other such easureent. On the other hand, quantu echancs s based on the realzaton that the easurng process ay affect the physcal syste. The easureent of one varable affects other varables n such a way that t prevents us fro knowng what ther values ght have been. The atheatcal forulaton of the law of physcs that takes ths basc dea nto account s very dfferent fro the atheatcal forulaton of classcal echancs. Halton s least acton prncple, whch s equvalent to Newton s law, s forulated as follows. The laws of physcs are such that the te ntegral over a certan functon L( q, q, t), called Lagrangan of the physcal syste, assues a nu. For echancal systes, the varables q on whch the Lagrangan depends on are the coordnates of all ndependent parts of the syste. A syste wth f degrees of freedo has f coordnates q 1, q 2,, q f and the te ntegral J, whch s defned by t2 J L( q, q, t) dt, s nu. In other words, when δq 0 at t δj t2 t1 t1 δl( q, q ; t) dt t2 L δ δ q q L + q q dt t 1 t2 L d L q dt q δqdt t1 0. t 1, t 2, we get (1.28) (1.29) Equaton (1.29) dctates that the Lagrangan satsfes the followng Euler equaton:

20 10 BASIC QUANTUM MECHANICS L d L q dt 0, q 1, 2, 3,, f. (1.30) The behavor of the physcal syste s thus copletely specfed by Euler s equaton once the Lagrangan s known. The proper Lagrangan s the one that leads to a descrpton of the physcal syste that s n agreeent wth experental observatons. 1 2 For exaple, f we choose L T V x 2 V( x) for a one - densonal partcle wth ass n the potental feld V(x), we obtan Then the Euler equaton yelds L V( x) L and x x x x. (1.31) V x x ( ) F( x), (1.32) x whch s the faous Newton s frst law of echancs. For later purpose, we ake the followng canoncal transforaton: p L, H pq L q, (1.33) where H s called the Haltonan of the syste. Then we obtan the followng, Halton s equaton of oton: H q, p d L L p dt q q q j pq H H j j, q (1.34) or, q H H, p. (1.35) p q Here, the Haltonan H represents the total energy of the syste. For any dynacal varable F that depends on canoncal varables ( q, p ), the te dervatve of F s gven by

21 BRIEF DETOUR TO CLASSICAL MECHANICS 11 df dt F q q F F + p t + F H F H q p p q + F t { FH}+ F,, t (1.36) where the Posson bracket { AB, } for dynacal varables A,B s defned by A B A B { AB, } BA,. q p p q { } (1.37) Equatons (1.36) and (1.37) ply that dh dt { H H HH, }+ t t, (1.38) whch says f H does not explctly dependents on the te t, the Haltonan, the total energy of the syste, s conserved. Aong any nterestng propertes of the Posson bracket, the followng relaton s especally useful for the latter purpose: { q, pj}δ j. (1.39) For a partcle ovng n a one - densonal world specfed by the 2 coordnate x, the Lagrangan s gven by L ( 12 / ) x V( x), as before. Then the canoncal transforaton yelds, L p x x, (1.40) p H px L x + V( x) + V ( x ). 2 2 Equaton (1.40) allows us to nterpret the Haltonan H as the total energy of the syste and p as the oentu. Halton s equaton of oton, Equatons (1.34) and (1.35), gves H p ẋ, p

22 12 BASIC QUANTUM MECHANICS H ṗ x V x ), ( x Newton s law. If we have a charged partcle n an electroagnetc feld, the stuaton s a bt ore coplex. The electrc feld E and the agnetc flux densty B can be expressed by a vector potental A and a scalar potental φ as A E φ, t B A. (1.41) The dvergence of a agnetc flux densty s zero, so t can be wrtten as the curl of the vector, the well - known vector potental A. In statc, the curl of the electrc feld s zero, and t can be wrtten as the gradent of a scalar functon. In the te - varyng case, the electrc feld and the agnetc feld are related by the followng Maxwell s equatons: B E, t B 0, D ρ, H J + D, t (1.42) wth D ε E and Bμ H. Here, D s the electrc fulx densty, H s the agnetc feld, ρ s the charge densty, J s the current densty, ε s the perttvty, and μ s the pereablty. We have used the nternatonal syste of unts (SI unts) to wrte Maxwell s equaton. Fro substtutng Equaton (1.41) nto (1.42), we fnd that + E A t 0. Then t s obvous that we have A E + t φ. (1.43)

23 BRIEF DETOUR TO CLASSICAL MECHANICS 13 Equaton (1.30) ples that the generalzed force Q s defned by Q V q d V + dt q. (1.44) The force actng on the charged partcle, called the Lorentz force, s F e{ E+ r B}. (1.45) Substtutng Equaton (1.41) nto Equaton (1.45), we obtan F e ( r A) da φ, (1.46) dt where we have used the followng relaton: r ( A) ( r A) r A. ( ) Equatons (1.44) and (1.46) ndcate that the generalzed potental n the case of a charged partcle ovng n an electroagnetc feld s gven by V e φ r A, (1.47) ( ) and the correspondng Lagrangan s 1 L r 2 + er A eφ. (1.48) 2 The generalzed canoncal oentu s then p L x + ea, q x, (1.49) q and the Haltonan s gven by H pq L 1 ( x + ea) x r 2 er A + eφ 2 1 r 2 + eφ 2 2 ( p ea) + eφ. 2 (1.50)

24 14 BASIC QUANTUM MECHANICS 1.4 A ROAD TO QUANTUM MECHANICS In classcal echancs, we dealt wth functons of the coordnates q and oentu p such as energy; we descrbe these quanttes collectvely as observable. The ter observable descrbes any quantty accessble to the easureent processes. We assue that every physcal observable s atheatcally represented by a Hertan operator, and every easureent of the physcal observable wll result n one of the egenvalues of the correspondng Hertan operator. The egenvector s used to characterze the state of the physcal syste. Quantu echancs assues that any arbtrary state of the physcal syste s characterzed by a state vector that s not necessarly an egenvector of any partcular Hertan operator. After a easureent has been perfored, the state vector collapses to one of the egenvectors wth an egenvalue E n. If we descrbe the state of the syste by a ket vector Ψ, then we can represent ths state vector by Ψ n E n E n Ψ, where E n s an egenvector of a partcular Hertan operator correspondng to the easureent done on the syste. How does Ψ relate to possble easureents when the result of a easureent ust be one of the egenvalues { E n }? For ths, we need the followng postulate Postulate The quantty Ψ H Ψ represents the average value of a seres of easureents on an enseble of systes that are all descrbed by the state vector Ψ and s gven by Ψ H Ψ 2 En Ψ En, (1.51) n where Pn En Ψ 2 s the probablty of obtanng the value of E n as a result of a easureent. Prevously, we descrbed that n quantu echancs the dynacal varables of classcal echancs are replaced by correspondng Hertan operators. In classcal echancs, the dynacs of an observable are descrbed by the Posson bracket. We would lke to extend the Posson bracket to descrbe the dynacs of a quantu echancal operator. The Posson bracket for the classcal observables A, Bs defned n Equaton (1.37) as A B A B { AB, } BA,. q p p q { } (1.37)

25 A ROAD TO QUANTUM MECHANICS 15 It s straghtforward to show the followng propertes: { A+ C, B} { A, B}+ { C, B}, (1.52a) { AB, + C} { AB, }+ { AC, }, (1.52b) { AB, C} { A, C} B + A{ B, C}, (1.53a) { ABC, } { AB, } C+ B{ AC, }. (1.53b) Let us assue that a quantu echancal Posson bracket s analogous to a classcal one. So we assue that a quantu echancal Posson bracket satsfes all the condtons of Equatons (1.37), (1.52), and (1.53). We shall denote the quantu echancal Posson bracket for operators  and ˆB as Aˆ, Bˆ. We use these condtons to deterne the functonal for of a quantu Posson bracket by evaluatng AB ˆˆ, CD ˆˆ n two dfferent ways fro Equaton (1.53a,b): and ˆˆ, ˆˆ ˆ, ˆˆ ˆ ˆ ˆ, ˆˆ AB CD A CD B + A B CD { AC ˆ, ˆ Dˆ Cˆ Aˆ +, ˆ ˆ ˆ ˆ, ˆ ˆ ˆ ˆ, ˆ D } B+ A{ B C D+ C B D } (1.54) AC ˆ, ˆ DB ˆ ˆ Cˆ ˆ + AD, ˆ B ˆ A ˆ BC ˆ, ˆ D ˆ AC ˆ ˆ BD ˆ, ˆ + +, ˆˆ, ˆˆ ˆˆ, ˆ ˆ ˆ ˆ ˆ, ˆ AB CD AB C D + C AB D { AC ˆ, ˆ Bˆ Aˆ Bˆ +, ˆ ˆ ˆ ˆ, ˆ ˆ ˆ ˆ, ˆ C } D+ C{ A D B+ A B D } (1.55) AC ˆ, ˆ BD ˆ ˆ Aˆ ˆ + BC, ˆ D ˆ C ˆ AD ˆ, ˆ B ˆ CA ˆ ˆ BD ˆ, ˆ + +. Equatng Equatons (1.54) and (1.55), we obtan A ˆ, C ˆ ( BD ˆ ˆ DB ˆ ˆ ) + ( AC ˆ ˆ CA ˆ ˆ ) ˆ, ˆ B D 0. (1.56) Snce the above condton holds for arbtrary Hertan operators ˆ, ˆ, ˆ ABC, and ˆD, we ust have ( ) ( ) Aˆ, ˆ ˆ ˆ ˆ ˆ ˆ, ˆ ˆ ˆ ˆ ˆ C AC CA and B D BD DB. (1.57) Fro now on, we shall call a quantu Posson bracket a coutator. We further assue that a coutator or a quantu Posson bracket s proportonal to the correspondng classcal Posson bracket:

26 16 BASIC QUANTUM MECHANICS Aˆ, ˆ B ħ { AB, }, (1.58) where ħ s Planck s constant dvded by 2π. Ths suggests that quantu echancs s based on the assupton that the Posson bracket assues the sae physcal eanng and the sae nuercal values as n classcal echancs. In partcular, [ ] { } qˆ, pˆ ħ q, p ħδ, (1.59) j j j whch s the fundaental quantu condton. The te evoluton of a quantu echancal operator can be obtaned fro the equaton of oton for the classcal observable: df dt dfˆ dt F FH t + {, }, (1.36) Fˆ FH ˆ, ˆ, t + 1 ħ (1.60) where Ĥ s the energy operator correspondng to the classcal Haltonan. Equaton (1.60) s called the equaton of oton n the Hesenberg pcture. Ths partcular quantu echancal representaton assues that the operators vary wth te, whle the state vector Ψ s te ndependent. Ths pcture s forally analogous to classcal echancs, snce the equatons of oton for the operators closely reseble the correspondng classcal equatons. Now, let us study a dfferent pcture, called the Schrodnger pcture, where the operators are constant n te, whle the te varaton s expressed by the state vectors. In ths pcture, the explct te dependence of an operator stll reans. Dfferent quantu echancal pctures follow fro each other by a untary transforaton. Fro now on, we denote a quantu echancal operator  sply as A when we don t need to dstngush t fro the correspondng classcal observable. Let us denote the state vector and operator n the new pcture by Ψ and A, respectvely. Then, they are related to Ψ and A of an old pcture by and Ψ U Ψ, A UAU (1.61) Ψ A Ψ Ψ U UAU U Ψ Ψ A Ψ, (1.62)

27 A ROAD TO QUANTUM MECHANICS 17 snce UU UU I, (1.63) when U s a untary operator. Equaton (1.62) ndcates that the untary transforaton does not change the physcal content of the theory. Also, we have da du AU U da + dt dt dt U + UAdU dt du AU U A dt t U 1 + U A, H U ħ + [ ] + UA du dt. (1.64) If A s not explctly dependent on te, A/ t 0, and the operator An the Schrodnger pcture s te ndependent except for an explct te dependence. Ths ples that da U A dt t U 0 (1.65). Fro Equatons (1.64) and (1.65), we get du AU U A H U UA du 1 + [, ] + dt ħ dt du AU UA du U( AH HA) U dt dt ħ du dt U A A U du ( AH HA ) dt ħ du dt U H A A U du H ħ dt ħ 0. (1.66) Equaton (1.66) should hold for any quantu echancal operators A n the Schrodnger pcture, and, as a consequence, we have du ħ HU. (1.67) dt

28 18 BASIC QUANTUM MECHANICS Snce Ψ U Ψ the state vector, we obtan the followng Schrodnger equaton for d ħ Ψ H Ψ. (1.68) dt The Schrodnger pcture s perhaps the ost wdely used because t leads drectly to the forulaton of wave echancs. In ths pcture, egenvectors of the poston operator q are used to represent the state vector of the syste. We consder the contnuous set of egenvalues q of the poston operator q gven by qq q q. (1.69) We assue that correspondng egenvectors { q } for a coplete set. For splcty, we consder the one - densonal case only, but the extenson to the hgher densonal case s straghtforward. The poston representaton of the state vector s defned by Ψ( q ) q Ψ. (1.70) Fro Equaton (1.59), the canoncal oentu operator p and the poston operator q satsfy the followng quantu Posson s bracket or the coutator relaton and [ q, p] ( qp pq) ħ, (1.71) ( qp pq) Ψ ħ Ψ. (1.72) If we take an nner product of Equaton (1.72) by the bra vector q, we obtan Also, q ( qp pq) Ψ q q p Ψ q pq Ψ. (1.73) q ( qp pq) Ψ ħ q Ψ. (1.74) Equatons (1.73) and (1.74) are consstent only f the followng condton s satsfed

29 A ROAD TO QUANTUM MECHANICS 19 q p Ψ ħ q q Ψ ħ Ψ( q ), q (1.75) snce Equaton (1.75) ples q pq ( ) q q q (1.76) Moreover, we have the followng relatons: q p q ħ q q q ħ δ( q q ), q (1.77) and Then, we get I dq q q. (1.78) Φ Ψ dq Φ q q Ψ dq Φ*( q) Ψ( q), Φ p Ψ dq dq Φ q q p q q Ψ dq dq Φ*( q ) ħ δ( ) ( ) q q q Ψ q dq dq ( ) ( ħ Φ* q δ q q ) Ψ( q ) q (1.79) (1.80) dq Φ ( q ) Ψ( q ) ħ * q dq Φ* ( q ) ħ ( q ), q Ψ and q F( q, p) Ψ F q, ħ Ψ( q ). (1.81) q

30 20 BASIC QUANTUM MECHANICS As a specal case, we consder F H, where H s the Haltonan of the syste. Then the Schrodnger equaton (1.68) becoes ħ Ψ( q, t) H( q, ħ / q ) Ψ( q, t), (1.82) t of the wave echancs. As an exaple, we consder a partcle ovng n a one - densonal potental V( x). If we set the poston operator as q x, the classcal Haltonan s gven by Equaton (1.40): H p 2 + V ( 2 x ). Then, the correspondng Schrodnger equaton s gven by 2 2 ħ ħ Ψ( xt, ) + V x x t t ( ) x Ψ(, ). 2 2 (1.83) For three densons, we sply replace q wth r and / q wth. For exaple, the Schrodnger equaton for a charged partcle ovng n an electroagnetc feld s ħ 1 rt t ħ ea r t 2 r t e r r Ψ(,) ( (,)) Ψ(,) + φ () Ψ(,). 2 t (1.84) When the Haltonan H s te ndependent, the te - dependent Schrodnger equaton can be rewrtten n te - ndependent for by tryng the soluton of the for Ψ( rt, ) exp( Et/ ħ) ψ ( r) (1.85) nto Equaton (1.83). Then, we obtan the followng te - ndependent Schrodnger equaton 2 ħ 2 ψe() r + V() r ψe() r EψE(), r (1.86) 2 where E s the energy of the partcle. One can also consder the egenvectors of the oentu operator p, whch are gven by E