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Unformatted text preview: 2. Evolution of the particle’s wave function Each of the states  tp_ }, with its wave function £0“le describes a station”,
state. which leads to timeindependent physical predictions. Time ﬁlt'ﬂlilllﬂl‘l
appears only when the state vector is a linear comlznination of several kets  [pl y
We shall consider here a very simple case, for which at time t = l} the state vector liltlﬂlDiSI ao>=vle[m.>+ez>] [14}
v2 a. WAVE FUNCTION AT THE INSTANT I Apply formula [II54} of chapter III; we immediately obtain 1 1 —.'.ll.:.Er  illI—Er
lettl}=—;;2[e tw’lo.>+e2"“ Iton] as
V or. emitting a global phase factor of  Ila[t] ): l — :
stsi>cp=ltat>+wwaz>1 as
“v 2
with:
El — s1 hill
is: I w = {it}
I s Ethel
b. EVOLUTION UP THE SHAPE 0F THE WAVE PACKET
The shape of the wave packet is given by the probability density:
1 1
tirtx.t}1 = E {pfix} + E goiter] + to1[_t}q12[xlccs tout {[3} We see that the time variation of the probability density is due to the interference
term in topaz. Only one Bohr frequency appears, a“ = {.332 — Ellg'it. since the
initial state [I4] is composed only ol‘ the two states I cpl } and I up: }_ The curves
corresponding to the variation of the functions of. [pi and mm are traced in
ﬁgures d—a. b and c. 114 ‘FL I: H‘ Hr EIGLIRE 4 Graphical representation of the reactions of idle probability density of tlie particle in the
ground state}. pi {the probability density at the particle In the ﬁrst excited state} and [plth
[the cross terr respcnslhle lor the evolution cl" the shape of the nave packet}. ————r—_——....___.—___— Using these ﬁgures and relation {13}. it is not difﬁcult to represent graphically
a variation in time of the shape or the wave packet {11"}: ﬁg. 5]: we see that the
We packet oscillates between the two walls of the well. Pariah: motion era W3“! packet obtained hy Hperposing the ground state and the ﬁrst excited state
ﬁfe particle in an ltdltil‘te well. The lreqneney oi the motion is the Bohr frequerlcy (a: “2n. Hhh—‘—_—. . I.” MOTION 9F THE genran 0.: THE wauE PACKET The yariation of (1')“) is represented iti ﬁgure e. in dashed lines. the variation of the position ofa classical particle has been traced, for a particle rooting
to and fro in the well with an angular frequency cfo:2 {since it is not subjected to
any three except at the walls, its position varies linearly with r between it
and a during each halfperiod }. We immediately notice a very clear difference between these two types of
motion, classical and quantum mechanical. The center of the quantum waye Let us calculate the mean value <‘_ If )[rl of the position of the particle at
time t. It is convenient to take: it" = .r ~ the [19} sitter. hy symmetry1 the diagonal matrix elements of Jr" are zero: ﬂ . packet, instead of turning back at the walls of the well, executes a increment of
(I i Y'l >0: s — 5 sin1 E dx = 1;] smaller amplitude and rctraces its steps before reaching the regions where the
m‘ ’ in] u ' 2 ‘ a potential is not zero. We see again here a result of §D2 of chapter 1: since the
"a 1 potential varies inﬁnitely quickly at x = fl and x = c. its variation within a domain
( fin" I X: 1 <02 } 1 J (x _ 3') 5m? dx = t) {20.} of the order of the dimension of the wave pacltet is not negligible. and the motion
\ ‘ n 2 a of the center of the wave packet does not obey the laws of classical mechanics
W tlrn haye'  [see also chapter 111, §Dldy'}t The physical explanation of this phenomenon
f L ' .. is the following: before the center of the wave packet has touched the wall, the
{1” )[ﬁ = Re i e"'"’'"{ on] I X’ I it’s} } ill} action of the potential on the “edges” of this packet is sufﬁcient to make it tttrn
with: bank
ﬁl'
{tpl.3f'lqsz}=<rp1 XitP3>—§<tl’1¢'2>
s COMMENT:
 2 " m 'inﬂdx
_ 3 “HM a b n The mean Value of the energy of the particle in the state Iu'riri}
I ﬂ ' calculated in {15} is easy to obtain:
= _ _: m] 1 1 s
9“ (H}=§EI+EEZ=§EJ {24}
Therefore :
a ' .'
(It so; = E — E as (out [23} 515
2 9:12 2 1 2 I I? .
{H )=—E.+—s£=_s§ (as;
2 2  2
which gives:
_ " 3
AH 2 E E I {26] EDIE in particular that { H ). { H2 } and AH are not time—dependent; since H
15 El constant of the motion. this could have been foreseen. In addition. we see from
'31? preceding discussion that the wave packet evolves appreciably over a time ol‘ order of :
i __ __ _._.'__.i "L. ‘1' 3” _‘ {Ell
0i isn’an I “’21
FIGURE 6  Using {26] and [2?], We ﬁnd:
Time variation of the mean value ( Jr” > corresponding to the nave packet or ﬁgure 5. The dashd AH A! m _3_ E X i _ E 23
line represents the position of a classical particle aimﬁg with the same period. Quaomnt neebadn ‘g ' — 1 1 SEI — 2 i } predicts that the courier of the oate packet will turn back before reaching the wall. as explained hit
the action of the potential on the “edges” of the wave packet. We again ﬁnd the timeenergy uncertainty relation. 't
""—'—————....—___—u _ 1. Description of the model In the ammonia molecule NH}, the three hydrogen atoms form the base of
a pyramid whose apex is the nitrogen atom is}: ﬁg. 1]. We shall study this
molecule by using a simpliﬁed model with the following features: the nitrogen
atom, much heavier than its partners, is motionless; the hydrogen atoms form
a rigid equilateral triangle whose axis always passes through the nitrogen
atom. The potential energy of the system is thus a function of only ricotta I
H H
Schematic drawing cfthe ammonia nitIncite; x is the alge
hraic distance between the plane oi the hydrogen atoms and
H the nitrogen atom, which is assumed to be motionless. 0M parameter. the [algebraic] distance it between the nitrogen atom and
the plane deﬁned by the three hydrogen atoms“. The shape of this potential
“nasty Fix} is given by the solidline curve in ﬁgure 2. The symmetry of the
Problem with respect to the x = 1} plane requires Vix} to be an even function of x.
The two minima of Fix] correspond to two symmetrical conﬁgurations of the
molecule in which, classically, it is stable; we shall choose the energy origin such that its energy is then zero. The potential barrier at .r = t}, of height F1,
expresses the fact that, if the nitrogen atom is in the plane of the hydrogen atoms,
they repel it. Finally. the increase in Vin} when x is greater than it corresponds
to the chemical bonding force which insures the cohesion of the molecule. IIfitIHE 2 1Variation with respect to .r of the potential energy Fix] ofﬂle Mannie. Vial has Inc nirliraa [chis
sleal equilibrium positions}. EL'FIJIIEIEI by a potential barrier due to the repulsion for small IxI between the nitrccgrn atom and the three hydrogen atoms. The “ square potential "' tnnid in approximate VIN]
E shown in dashed lines. This model therefore reduces the problem to a onedimensional one in which a ﬁctitious particle of mass m (it can be shown that the “ reduced mass “ m of the Shh—m”) is under the inﬂuence of the potential Fix}. Under
lint” + in" system is equal to
these conditions, what are the energy levels predicted by quantum mechanics”
With respect to classical predictions, two major differences appear : {ii The Heisenberg uncertainty relation forbids the molecule to have an
energy equal to the minimum of Vin} [Vm = [I in our case]. We have already seen,
in complements C, and Mm why this energy must be greater than me. {if} Classically, the potential barrier at. :r = 0 cannot be cleared by a particle
whose energy is less than V, : the nitrogen atom thus always remains on the same
side of the plane of the hydrogen atoms, and the molecule cannot invert itself.
Quantum mechanically, such a particle can cross this barrier by the tunnel
elfect it}. chap. 1, {tillc}: the inversion of the molecule is therefore always
possible. We are going to discuss the consequences of this effect. We shall be concerned here only with a qualitative dimussion of the physical
phenomena and not with an exact quantitative calculation which would not have
mitt—h :ioniﬁrtance in this approximate model. For example, We shall try to
demonstrate the existence of an inversion frequency of the ammonia molecule. without giving an exact or even an approximate value of this frequency. We shall therefore simplify the problem still more by replacing the function Vix] by the squa re
potential drawn in dashed lines in ﬁgure 2 [two inﬁnite potential steps
at x = i to + slit and a potential barrier of height PE, centered at x 2 ti and of
width [24!)  cl]. 2. Eigenfunetiona and eigenvalues of the Hamiltonian 3. INFINITE POTENTIAL BARRIER Before calculating the eigenfunctions and eigenvalues of the Hamiltonian
corresponding to the “square” potential or" ﬁgure 2, we are going to assume, in
this ﬁrst stage, that the potential barrier 11;, is inﬁnite [in which case, no tunnel
sheet is possible]. This will lead us to a better understanding of the consequences
of the tunnel effect across the ﬁnite potential barrier of figure 2. We shall therefore
consider, ﬁrst of all, a particle in a potential pix} composed of two inﬁnite wells
of width a centered at x = i b (ﬁg. 3]. If the particle is in one of these two wells.
it obviously cannot go into the other one. House 3 when the height Vt, ml the potential
banleu of figure 2 Is large. tee haw: tau — b
b + x practically inﬁnite potential wells of IIItitlth
'3 a «hose centers are separated by : tlistanee
+—— —&——— ——.’. ‘nhIh.II* DI. lb _ Each of the two wells ofﬁgure 3 is similar to the one studied itt complement H J.
m illbit. We can therefore use the results obtained in this eornplernent. The
PUSSIbIe energies of the particle are; E = a; l
1. 2m { }
with:
r“ en [a
u it is ooutrenient to change bases, in each of
of the particle. Since the function Vtx‘t parity operator Htef. complement F...
or odd; In the rest of the calculations, the eigensuhspaoes of the Hamiltonian
is even. this Hamiltonian H commutes with the I
hit]. In this ease. a basis of eigenveetors of H can he found. which are even the wase functions of these sectors are the symmetrical and antisymmetrieal linear
combinations '_ as.ch = ta".th + estr1]
“i 3 {4} rest = + [ «use — taste]
as 2 in the states ] (p: h and  (p: In the particle can he found in one or the other of the the two potential wells.
In what follows~ we shall oonﬁne ourselves to the study of the ground state, for which the wave functions with], :pﬂx}, to: {I} and (19,1th are shown in ﬁgure S. I.._—.....,—._._...———' FJﬁURF. S The States ﬁtsl and miLrthhtmn in ligtl'e a. are tartanietnlrg.I states ntth the same energy. resneeﬁt'eljr
brilliant 'm the righthand well. and the lefthand well or ﬁgure 3. To use the symmetry of the pro
hlent. It ts more oohrmieat to chum: as stationary statues the symmetrical state toﬂh'] amt the anti
ﬁll'll‘IInEtrteal state oﬂtt. llrtelr combinations of to: [It and tpHA'l [tinn: ti}. __. __ _
E: _._¥_
i soa
53 _i_
‘ t‘rtjttate I}
When one takes the ﬁite height I}, of the barrier into amount,
one ﬁnds that the energt.I spectrum at ﬁgtI'e 4 is modiﬁed: each
5: in] lewd Splits htu tern distinct tales. Th Bah frequencies [EL2::
El I and ﬁlth ourrﬁpcmlilg to tmnellia; from me 1well to the other
i are the inletsine frequencies of the ammonia moleeuie tor the
I] ﬁrst 11w vibration levels. The tunnel I.th it some important in the higher vibration level. so I}: 1: 12,. Finally. in ﬁgure 1', we have shown the shape of the eigenfunctions I:{.t}
and xﬂx} which are given by equations [5}, [i]: and till}. once it; and t; have
been determined from {I4} and [15]. We see that theyr greatly resemble the func— Fictute ? Wave I'unoﬂnns associated with the Ierels E;
and E: in ﬁgure in. Nate the analogy with the
functions ' l'egtre 5b: however, these leiiI
functions do not ranisl. on the interval
—b+ot2£x$b—n.l'2. tions tonic] and tails} of ﬁgure 5, the essential difference being that the wave
t'unction is no longer zero in the interval a {it — itle a .t e; in — ci'El. The reason
for introducing the is; and or: basis in the preceding paragraph can now be
understood : the eigenfunctions x; and Li, in the presence of the tunnel eli‘oet,
resemble or; and to: much more than ad and all. e. E'tI'ﬂLUTIDhI or THE MotchLe. Invenslon Faeouencr
Assume that at time i = [I1 the molecule is in the state : i.lll'=Ul}=+[jti}+lid” on
v 2
The state vector  Mt} > at time t can be obtained by using the general formula {D44}
of chapter Ill ; we obtain: E are};
Intel} = L. Eda?" [thaws V”? 13 > + 3"“: I if. >] {17] From this we deduce the probability densilir'T
. .1 I I i
no: or e s [Iil‘ﬁllz + 5 [sitar + commits} tilt} “31 The variation with respect to time of this probability density is simple to
obtain graphically from the curves oi" ﬁgure T. They are shown in ﬁgure 3.
For t = i} {ﬁg iiia}, we see that the initial state chosen in [16] corresponds to
a probability density which is concentrated in the righthand 1arel] tin the left—hand
tied the Functions 1: and y; are of opposite sign and very close in absolute value,
so their sum is practically zero}. It can therefore be said that the particle. initially,
is practically in the right—hand well. t—‘tt time i = allﬂt [ﬁg 3b], it has moved [litE. t}: then It rclurns In the righthand hell (ﬁg. ill all Ille inciilal stale (“2. e}. and 511 m ——._.___ Florrte S _...———l Evolution :11 a war: packet ohlained in. niper
posing the two stationery ware l‘metions ol'ﬁgure '1'.
The particle. initially, is it the righthail net]
mg. I}, ramels isle the lefthand well (ﬁg. hi and.
after a eemln time. becomes Iiicalmed there ...
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 Spring '10
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