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Unformatted text preview: (om; . Speciﬁcally. forlight. the enetgy E and the frequency u are Hill E =hv : (g'klj'+[kyjt+fkriyﬂefiit.t+E.v+tycMt and for matter wares, the momentum p and the wavelength Jr. are related by = “"4" Since we want p = Mt to multiply tit on the righthand side. we multiply both a = Ml sides by not; We will assume that berth of these properties apply to matter waves. The equation _yyl1yqy = HILL. _ 4,.
for energy. E = uh LI1 Et'tt'l'ctﬁpt'tndﬁ. [n — p
In summary. we now have two operators for which I]! is an eigenfunctiun; an: E = (hizﬂlhul = “'5” gives the WW E 33 I“: Eigenvalue. and the outer gives the momentum F as the eigenvalue;
and the momentum equation. p 2 ML becomes
_ Elli"
a =tﬁtlrrtt1=rtlt= at: [33} tHﬁr— =. go
We can generalize this equation from one oirrtention to three dimensions. In three —t'ﬁVHl" = pill dimensions. It points in the direction of— mrrtimt ol' the pﬂﬂiciﬁ, which is also the
direction of the momentum vector. p. Hence. Equation (3.7) becomes We now derive an equation which conesponds to Equation (3 El. Applying the
momentum operator —:'.w twice to o produces two factors of p: {—ih?) " {ﬁih‘ﬁ'tllt = P 1 pt? ——. 3lII'
For a non—relativistic particle with mass till end with no potential energy. the rela— '0
. . . . . Hence,
ttonshtp between p and E; is given by
at??? = ?
2m mica" “it” ‘5' 1“ themed for ‘i'  ‘i' = ﬁltaxi + stray? +32I3zZ—J Then
_It 1'5 now possible to generate expressions for E and p from Equation til.5} hr 1 p:
by taking the appropriate derivatives. Taki ng the derivative of Equation {3.5‘] Wiﬂt _ E v' ll’ = m 4' [3.10;
 'l to t' . ct. '   
reaper 1rne We at wﬂi I} = Heitmpmj le'lﬁ righthand srtles ofIEquatLOns {3.9J and [3.10) are manifest” equal. Since
E — —ituBr”"""°“" mictitan Euiuriililitmlirtmmm Sides “"1”: eEllmime aim us a dif
ﬁr _ n so s y :
3.5 s
Multiplying both sides by in gites energy on the righthand side'., I i _ Fl' v2!” _ In Hill
g ‘ 3 [3.11;
ii'lII' . . . . .
“l.— = hmﬂf'tt'rnm So far. all we have shown is that the wave funenon given In Equation {3.5)
d: _' satisfies Equation {3.1 l). and further. that il is an eigenfunction oflhe operams on
= e o {1'50 “1.: Momma mi .1 gh'thsmd sides of Equation {3. I t l with eigenvalllﬂﬁ Fitz»: and E, respectively. It is at this point that we malu: an. unjustiﬁed leap: what happens
if we now add a potential energy V to the system? In a classical system, the total
energy is just the sum ofthc kinetic and potential energies: Note that ihtiiifiiltl is a linear operator. and '4' is an. eigenfunction of This operator
with cigsnt'm'ue E.
New weneedto ﬁnd asimilar operator which gives the momenmm p. Expanding the clot product in Equation {3.5} gives * dwl ~ m 5 M~ 2
II'L “it t . “‘5‘ p— + V = E (fr*BJE 551345 Eur EM “ﬁﬁjpcq will; win” : Bantt.t'+t.y+t_:—oﬂ Th' . st that mod'f t' 3.” totearl
50th“! I if? E? {Pry‘ tssugge s we IyEquatoni ]I ' it? 53L
ﬂu! is +t +s t ..'—"L
—' I 1‘ VJ' ri'4“
_..rk_,b'e (inf? 1—
.” :52. shin
ﬂ =ikyﬂEIHlllJty_'t'+it;1IIT: # E's aw film; f rig? _ = ik‘Beiu.xlk.JI~k_.: mr
ii: " Hy [Notethatunlikethe ltjnetit: energy ahdtotal energy. which etuTespondto operators
containing yarims derivatives. the operator mrresptntding to the potential energy is simply multiplication ol' '4' by VJ
Equation (3.1 1} is theSchrﬁdingcr equatitut, Tutsst'hly the most important equa rP lye. 3 1572;; 2.1 STA'I'IDI‘HHRY STATES [def/'72s) In Chapter I we tallted a lot about the wave function. and how you use ll to
calculate various quantities of interest. The time has come to stop procrastinating. and confront what is. logically. the prior question: How do you get tilts. t'] in the
ﬁrst place? We need to solve the Schrodinger equation. ,hao n1 taintI W a} 1
t —— = ————.
lit 2m 312 + ' [" J for a speciﬁed potentialt 'v'fx. .r}. In this chapter {and most of this book} I shall
assume that V is independent of r. to that case the Schrodinger equation can be
solved by the method of separation of variables tthe physicist’s ﬁrst line of attack
on any partial differential equation}: We look for solutions that are simple products. tl'f.r.r} = artstoot. t2 1t where tit {hovercase} is a function of it alone. and to is a function of l' alone. [In
its face. this is an absurd restriction. and we came: hope to get more than a tiny subset of all solutions in this way. But hang on. because the solutions we do obtain
turn out to be of great interest. Moreover {as is typically the case with separation
of variablesl we will be able at the end to patch together the separable solutions
tn such a way as to construct the most general solution. For separable solquns we have so_ as shit die
at at' as2 as! [ordinary derivatives, now}. and the Schrodinger equation reads as slain
'it —=—_=— v .
’ “but anax2w+ W Dr. dividing through by itrw: _ [dip s ldzitr 1
__=__.__._. _ s
mth thllretxl +5; [ 1 Now. the left side is a function of t alone. and the right side is a function of
x alone.2 The only way this can possibly be true is if both sides are in fact
cemtont—dd'terwise. by varying t. [ could change the left side wintout touching
the right side. and the two would no longer be equal. {That’s a subtle but crucial
argument. so it‘ it's new to you. be sure to pause and think it through.) For reasons
that will appear in a moment. we shall call the separation constant E. Then 1 do: 'h——=E.
I melt or .
it” = _Eg~ [2.41
dt it and CIT [15] Separation of variables has turned a partial differential equation into two units
nary differential equations {Equations 2.4 and 2.5}. The ﬁrst of these [Equation 2.4: it? HS)" to solve [jue'l multiply through by tit and integrate}: the general solittittti is
C exptt' Erin}. but we might as well absorb the constant E into it [since the quantity
of interest is the product drip}. Then —r.ElI."h left] = e [lo] The HEt‘Ond [Equation 2.5.1 ts called the timevindEpendent Schrodinger equation;
We can go no further With it until the potential Hit} is speciﬁed. The real or lhiﬁ Chapter will he devoted to solving the tithe—independent
Schrodinger actuation. for a variety of simple potentials. But before i get to
that you have everyr right to ask: What's so great about segutt'ttbic striations ."
After all. most solutions to the {time dependent} Schrodinger equation {to our
take the form tttid'lyttt‘l. I offer three answers—two of them physical. and one
mathematical: 1. "llicy are stationary states. Although the wave function itself. 4r{_tlr}=1ﬁ['x}E—fﬁfifﬂl "'1 does {obviously} depend on t. the pmbttbt'it'ry density.
salami = ufo = ats+i5‘i”as"'5'i“ = Iann3. i2.:.~t does t‘ltl!1I'IE lime—dependence cancels out.‘ The same thing happens in calculat
iog ll'lt: expcewliﬂn value of any dynatttiea] ‘t'ﬂIiliblc: Equation tJo reduces to ill of
too. to} = f we (r. 2'") it or. 12.91 refit Every caperrattan: value it murmur in tinre; we might as wei] drop the factor tutti
altogether. and simply use tit in plane of Iit". tindecd. it is common to relet' to tit an
:‘the wave function." but this is sloppy language that can be dangerous. and it is
Il'l'tpDI‘L'IJll to remember that the one wave function always carries that exponential
tttne~dependent I‘actont In particular. tr} is constant. and hence iiiquatton 1.33::
{p} 2 ti. Nothing ever happens in a stationary state. 2. They are states of definite nitnt energy. In classical mechanics. the total
enters")r {kinetic plus potential] is called the Hamiltonian: 1
Ht.t.n‘t= g—m+vta‘]. _ [lint THE INFINITE SQUARE WELL _.______'_'—.________u_'__ Suppose
if t] 5 .r 5 a. . _ LL
H” ' I oo. otherwise [1W] {Figure 2.1}. A particle in this potential is completely free. except at the two ends
is = I] and .t = a}. where an inﬁnite I'orce prevents it from escaping A classical
model would be a can on a frictionless horizontal air track, with pertEttly elastic
butnpers— it just keeps bouncing back and forth forever. [This potential in urtiliv
cial. of course. but I urge you to treat it with respect. Despite its simplicity—or
rather. precisely because of its simplicity—it serves as a wonderfully aCCd‘s‘ni ble test case for all the Fancy machinery that comes later. We'll refer back to it
frequently.) utxi EilGURE 2.1: The inﬁnite square well poten
” x nal {Equation 2.13}. (Janine the well. arts] = t! (the probability of ﬁnding the particle mere is
zero}. inside the well. where l’ = D. the timesindiependent Schrodinger equation
{Equation 1.5i reads a? iiia
_EE = so, [2.2a]
131' 1
d—
E‘i—j = 421a. where t a ﬂaw. [2.21] the general solution is
tltfri = A sin it: + B cos let. [2.31 where A and B are arbitrary constants. Typically. these constants are ﬁxed by the
boundary conditlons of the problem. What are the appropriate boundary oorn
ditiona for shirt)? Ordinarily. he!!! hit and dihde are continuous. hill Where the
potential goes to inﬁnity only the ﬁrst of these applies. tl'll prove these boundary
conditions. and amount [or the exception when V = no. in Seclion 2.5; for new I hope you will trust me.)
Continuity of iglrtix} requires that inﬁll = with} = t]. 11.23] so as to join onto the solution outside the well. Whal does this tell us about A and
B'.‘ Well. with = Asinﬂ+ Email: 3. w B = t}. and hence
1M1”) = A stnlta'. [2.24] Then tarts} = A smite. so either at = I] [in which ease we‘re Jet‘t with the triv—
iaJ—nonnormalizabIe—solutiort ttrtr} = [I], or else sin in = t}. which means that
he =t'l. in". i'En'. iiiIr. [2.15] But I: = [II is no good (again, that would imply ttrtx} = lit}, and the negative
solutions give nothing new. since sini—h‘i = —airttft5"] and we can aheor'h the
minus sign into A. So the distinct solutions are r323, wima=t.2.3. 12.26]
rt i
E z " = 1 _ 2.1?
L 1 2m Euro j I I In radical contrast to the classical case. a quantum particle in the infinite snuaie
well cannot have just any old energy—it has to be one of these special allowed
values.“ To ﬁnd A. we normooze tir: t_I “ 1 . 1: sir 1t
1 IAI' stn“{k.t‘irt‘.t‘ = A‘— = 1. so A‘ = .
u 2 “.1 E_,__ _. 1L"tiJ'J' ValTl titan} = ﬁnd
{I 1.1 THE STATISTICEL INTERPRETATION list slat crane it the "one henna " and can that it in or rate one 1»:
get it'.’ After at]. a particle. by its nature. is localized at a point. whereas the wave
function {as its name suggests} is spread out in space tit's a function of .t. for any
given time ti. How can such an object represent the state of a pru'ra'ie T The answer
is provided by Bern's stat'ﬁﬁcal interpretation of the wave function. which says
that tutti. r}j gives the probability ofﬁntling the particle at point .t. at time .r—or.
more precisely" F‘h—_  t ” probability ol‘ ﬁnding the particle ] I'llI'[.'t.F]II:t.l.t = {in i (I between it and h. at time t. Probability is the iiien under the graph of .l'Ll l. For the wave function in Figure Ll.
you would be quite likely to ﬁnd the particle in the vicinity of point at. where ttt
is large and relatively unlikely to ﬁnd it near point 3. 1M“ FIGURE 1.2: A typical wave function. The shaded area I'CPIEII“! the probability elf ﬁnding the particle between a and it. Tltc particle would be relatively lilter to be found
near A. and unlikely to be found near B. The statistical interpretation introduces a kind of indeterminacy into quan
tum mechanics. for even if you know everything the theory has to tell you about
the particle {to wit: its wave functionl. still you. cannot predict with certainty the
outcome of a staple experiment to meattun: its position—alt quantum mechan
ics has to offer is .rl'ctt'itt'it'af information about the priﬂ'ib‘fr’ results. This inde
terminacy has been profoundly d'tsmrhirtg to physicists and philosophers alike.
and it is natural to wonder whether it is a fact of nature. or a defect in the
theory. Suppose i do measure the positimi of the particle. and I rind it to be at point
C4 Question: Where was the paniele just deﬁne I made the measurement? There
are three plausible answers to this question. and they serve to characterize the ntain
schools of thought regarding quantum indeterminacy: I. The realist position: Titrput'rt'rle was E" C‘. This certainly seems like a sen
sible response. and it is the one Einstein advocated. Note. however. that it‘ this is
true then quantum mechanics is an intimplied theory. since the particle rerriij.‘ tent
at f‘. and yet quantum mechanics was unable to tell us to. To the realist. indeter
minacy is note feet of nature. but a reﬂection of our ignomnce. its tl‘Eepegnet put
it. "the position of the particle was never indeterminate. but was trierer unknown
to the eitpet'irntmter."5 Evidently '41 is not the whole story—some additional infor
mation [known as a hidden variable} is needed to provide a complete description
of the particle. 2. The orthodo't position: The pnnicfe wasn't really nnytt'hrte. [I was the act
of measurement that forced the particle to "take a stand" {though how and why it
decided on the point C we dare not ask]. Jordan said it most starkly: ‘ﬂhservations
not only efiti'ttl'b what is in be measured. they pmduce it We {'tNTJpPl {the 401 Wane. no measuring instrument on pert'eelly precise: what i mom l: thtlt the particle was
found in the intuit}. oi C. to within. the tolerance of the equipment. We return now to the statisticat interpretation of the wave function [Equation 1.3}.
which says that I'I"l.'l.'. l'll2 is the probability density for finding the particle at point
x. at time L It follows {Equation 1.16} dial the integral of i'tl'i‘E must he I (the
particle‘s got to he .t‘emtewherej:  W .
If tet.r.:itt.t=t. [too] it: ...
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