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1 A La Rsa Lecture Ntes PSU-Physics PH 4/5 ECE 598 I N T R O D U C T I O N Q U A N T U M T O M E C H A N I C S CHAPTER- OVERVIEW: CONTRASTING CLASSICAL AND QUANTUM MECHANICS FORMULATIONS 3 Prbabilistic interpretatin f the wavefunctin 3A Max Brn s prbabilistic interpretatin f the wavefunctin 3B The QM wavefunctin represents an ensemble f systems 3C Deterministic evlutin f the wave functin 3D Nrmalizatin cnditin fr the wave functin The Hilbert space 3E The Philsphy f Quantum Thery The Neils Bhr and Heisenberg s schl De Brglie and Einstein schl Feyman s alternative frmulatin f quantum mechanics: Path integrals References L D Landau and E M Lifshitz, Quantum Mechanics (Nn-relativistic Thery), Butterwrth Heinemann (003) Chapter Richard Feynman, The Feynman Lectures n Physics, Vlume III, Chapter Vlume I, Chapters 30 and 38

2 3 PROBABILISTIC INTERPRETATION OF THE WAVEFUNCTION 3A Max Brn s prbabilistic interpretatin f the wavefunctin In Chapter 5 we will see that harmnic functins are used t describe the mtin f a free particle The idea may have been triggered in analgy t electrmagnetic waves In effect, since a prper cmbinatin f the Maxwell s equatins lead t the wave equatin (which admits travelling harmnic waves as slutins), a typical electrmagnetic wave is described b π ε(x, ε Cs [ x πνt] electrmagnetic wave λ whse intensity I (energy per unit time crssing a unit area perpendicular t the directin f radiatin prpagatin) is prprtinal t ε ( x, Similarl in chapter 5 we will see that t QM describe the mtin f a particle f classical linear mmentum mmentum p p mv ne can use a wave-packet k x ωt x t A k e i [( )] (, ) dk π ( ) QM wavefunctin f a free-particle k f width k (that depends n the uncertainty f v ), and with a dminant harmnic cmpnent at k= k, where k / ( / h)mv Einstein (in the cntext f explaining the results frm the phtelectric effec intrduced the granularity interpretatin f the electrmagnetic waves (later called phtns), abandning the mre classical cntinuum interpretatin In Einstein s view, the intensity f radiatin is interpreted as a statistical variable I cε N h ν Here N cnstitutes the average number f phtns per secnd crssing a unit area perpendicular t the directin f radiatin prpagatin Ntice ε ~ N Average values are used in this interpretatin because the emissin prcess f phtns by a given surce is statistical in nature The exact number f phtns crssing a unit area per unit time fluctuates arund an average value N Max Brn prpsed a similar view t interpret a particle s wave-functins In his view, ( x, plays a rle similar t ε ( x,, ( x, is a measure f the prbability f (6) finding the particle arund a given place x and at a given time t That is, ( x, plays the rle f density prbability (Pictriall the particle must be at sme lcatin where the wavefunctin has an appreciable value) This interpretatin

3 was intrduced years after Schrdinger (96) had develped a frmal quantum mechanics descriptin 3B The QM wavefunctin represents an ensemble f systems Let s expand the descriptin f the statistical interpretatin descriptin a bit further This time we emphasize that a wave functin des nt represent a single system but a bunch f systems; hence the intrductin f the cncept f ensemble Ensemble Imagine a very large number f identical, independent systems, each f them cnsisting f a single particle mving under the influence f a given external frce In the language used in Sectin, this is a system f 3 degrees f freedm The quantum state f these systems are identically prepared (Nte: We will have t explain what we mean when we say identically prepared quantum state This will be addressed in Chapter 0, after intrducing the cncept f QM peratrs and their cmmutatin prperties that allws defining states cmmn t a set f peratrs, hence narrwing the selectin f states) At a given time t, the whle ensemble is assumed t be described by a single cmplex-variable wave functin ( x, It is a pstulate f quantum mechanics that cntains all the infrmatin that can be btained abut the ensemble Every prperty f the ensemble can be derived frm its wave functin (We emphasize, hwever, that even fully knwing the wave functin, nly prbabilistic predictins abut physical measurement can be made) N Ensemble describes the whle ensemble is used t make a prbabilistic predictin n what may happen in a particular member f the ensemble It is pstulated that: If measurement f the particle s psitin are made n each f the N member f the ensemble, the fractin f times the particle will be 3

4 fund within a vlume element d 3 r =dx dy dz arund the psitin r ( x, is given b * ( x, ( x, d 3 r (7) where * stands fr the cmplex cnjugate number Ntice that this is nthing but the language f prbability; in this case, psitin prbability density P P( x, ( x, Cautin: Fr cnvenience, we shall ften speak f the wave functin f a particular system, BUT it must always be understd that this is shrthand fr the wave functin assciated with an ensemble f identical and identically prepared systems, as required by the statistical nature f the thery Mre general, An ensemble f systems (each therwise classically described as a n degrees f freedm system) is cmpletely specified quantum mechanically by a wave functin ( q, q,, q n ) which cntains an arbitrary factr f mdulus and satisfies,, q,, qn ) dq dq dqn ( q where the integratin extends ver all the accessible values f the crdinates q, q,, q n All pssible infrmatin abut the system can be derived frm this wave functin (8) The evlutin with time f the wave-functin is gverned by the Schrdinger equatin, i t V ( x, x m Schrdinger Equatin (9) 4

5 3C Deterministic evlutin f the wave functin The predictins f quantum mechanics are prbabilistic It makes prbabilistic predictins n the utcme f measurements t be perfrmed n a system These measurements necessarily disturbs the system in a way that, in general, cannt be cmpletely determined On the ther hand, being the slutin f a differential equatin, the Schrdinger equatin, varies with time in a way that is cmpletely deterministic That is, if were knwn at t=0, the Schrdinger equatin determines precisely its frm at any future time Ntice, hwever, that the wave functin at t=0 cannt be uniquely determined This is because, a set f measurements at t=0 at mst may lead t the determinatin f but d nt uniquely defines Further, it turns ut that the assumptin that we culd knw a system at t=0 actually represents a theretical abstractin One cannt measure a state directly in any way What ne des d is t measure certain physical quantities such as energies, mmenta, etc, which Dirac referred t as bservables Frm these bservatins ne then has t infer the state f the system We will address this issue in Chapter 0 3D Nrmalizatin cnditin fr the wave functin The Hilbert space The interpretatin f the wave functin as a density prbability implied the requirement f being square integrable functins, all space, q,, qn ) dq dq dqn ( q (0) because the likelihd t find a given particle anywhere shuld be ne Ntice that if is a slutin f the Schrdinger equatin, the functin cnstan will be als a slutin The multiplicative factr c therefre has t be chsen such that the functin c satisfies the cnditin (6) This prcess is called nrmalizing the wavefunctin c (c being a In general, there will be slutins f the Schrdinger equatin (9) but that d nt satisfy the cnditin (0) This means they are nn-nrmalizable and therefre cannt represent a prbability density Such functins must be rejected n the grunds f Bhr s prbability interpretatin Quantum mechanics states are represented by square-integrable functins that satisfy the Schrdinger equatin () Such particular subset f functins that are square integrable frm a vectr space call the Hilbert space G 5

6 In additin t depending n the crdinates q, q,, q n the wave functin depends als n the time t, but the dependence n the qs and n t are essentially different The Hilbert space G is frmed with respect t the spatial crdinates q, q,, q n nly States f the system at different instants f time t, t, t, are given by different wave functins t ( q, q,, qn ), ( ) t' q, q,, qn, ( ) t" q, q,, qn,, f the Hilbert space This descriptin that treats the time variable different than the psitin crdinates results frm the nn-relativistic way we are describing quantum mechanics 3E The Philsphy f Quantum Thery There has been a cntrversy ver the Quantum Thery s philsphic fundatins Neils Bhr has been the principal architect f what is knwn as the Cpenhagen interpretatin (statistical interpretatin) Einstein was the principal critic f Bhr s interpretatin His statement Gd des nt play dice with the universe, refers t the abandnment f strict causality and individual events by quantum thery Heisenberg cunteracted arguing: We have nt assumed that the quantum thery (as ppsed t the classical thery) is a statistical ther in the sense that nly statistical cnclusins can be drawn frm exact data In the frmulatin f the causal law, namel if we knw the present exactl we can predict the future it is nt the cnclusin, but rather the premise which is false We cannt knw, as a matter f principle, the present in all its details Luis de Brglie, n the ther hand, argues that that limited knwledge f the present may be rather a limitatin f the current measurement methds being used He recgnizes that, a) it is certain that the methds f measurement d nt allw us t determine simultaneusly all the magnitude which wuld be necessary t btain a picture f the classical type, and that b) perturbatins intrduced by the measurement, which are impssible t eliminate, prevent us in general frm predicting precisely the results which it will prduce and allw nly statistical predictins The cnstructin f purely prbabilistic frmulae was thus cmpletely justified But, the assertin that, The uncertainty and incmplete character f the knwledge that experiment at its present stage gives us abut what really happens in micrphysics, is the result f a real indeterminacy f the physical states and f their evlutin, cnstitutes an extraplatin that des nt appear in any way t be justified, accrding t de Brglie De Brglie cnsiders as pssible that lking int the future we will be able t interpret the laws f prbability and quantum physics as being the statistical 6

7 results f the develpment f cmpletely determined values f variables which are at present hidden frm us The views presented abve highlight the bjectin t quantum mechanics philsphic indeterminism On ne hand, accrding t Einstein: The belief f an external wrld independent f the perceiving subject is the basis f all natural science On the ther hand, Quantum Mechanics, states that when prbing the external wrld, the latter gets mdified in the prcess, s that ne can nly expect a prbabilistic measurement frm the prbe It is, therefre, impssible t knw the the external wrk with abslute certainty S, what an electrn des when inside a hydrgen atm (the hydrgen atm is the external wrld)? Fr sure we knw the electrn des nt fllw trajectries (that will make it unstable fr the cntinuum emissin f radiatin that that wuld imply) Quantum mechanics tell us that we can nly prvide prbabilities t find it here and there In that sense, physics has given up n the prblem f trying predicting exactly what will happen in a given circumstance In the cntext f these cntrversial arguments, Feynman s path integral frmulatin f quantum mechanics cmes handy Since the use f the least actin principle is s rted in the descriptin f classical mechanics, a frmulatin f quantum mechanics based n similar (generalized) principles ffers a gd perspective platfrm frm which we culd gain a better understanding f quantum mechanics and its aximatic principles The Feynman apprach is nt standard in part because it came in 948 when all the tls, based n differential equatins, had already been develped since 96 The Feynman apprach instead is based n the calculatin f multiple integrals, which, in cmparisn, less tractable mathematically Hwever, the Feynman s frmulatin ffers a view f the quantum mechanics as a living piece f nature rather than as a fld f arcane algrithms that, while lvely and mysterius and satisfying, ultimately defy understanding and intuitin 3 Feynman s apprach naturally and implicitly satisfies the Landau s demand that a frmulatin f quantum mechanics shuld have classical mechanics as a limiting case We start a brief descriptin f such new frmulatin in chapter 6 3 It is interesting t bserve new strategies beynd the ensemble apprach: B L Altshuler, JETP Lett 4, 648 (985) P A Lee and A D Stne, Phys Rev Lett 55, 6 (985) See als the intrductin article by Igr V Lerner, S Small Yet Still Giant, Science 36, 63 (007) B H Brasden and C J Jachain, Quantum Mechanics, nd Editin, page 56 D F Styer, in the Preface Sectin f his emended versin f Ref 3 7