Table Of ContentComplete Reconstruction of the Wavefunction of a Reacting Molecule
by Four-Wave Mixing Spectroscopy
David Avisar and David J. Tannor
Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
(Dated: January 25, 2011)
Probing the real time dynamics of a reacting molecule remains one of the central challenges in
chemistry. Inthisletterweshowhowthetime-dependentwavefunctionofanexcited-statereacting
molecule can becompletely reconstructed from resonant coherent anti-StokesRaman spectroscopy.
Themethodassumesknowledgeoftheground-statepotentialbutnotofanyexcited-statepotential,
althoughweshowthatthelattercanbecomputedoncethetime-dependentexcited-statewavefunc-
tion is known. The formulation applies to polyatomics as well as diatomics and to bound as well
1 as dissociative excited potentials. We demonstrate the method on the Li2 molecule with its bound
1
first excited-state, and on a model Li2-like system with a dissociative excited state potential.
0
2
PACSnumbers: 03.65.Wj,31.50.Df,82.53.-k,78.47.nj
n
a
J For several decades now, femtosecond pump-probe can be of any dimension. For simplicity, we consider
3 spectroscopies have been employed to study transition a δ-pulse excitation as well as a coordinate-independent
2 states ofmolecules reacting onexcitedpotential surfaces electronic transition dipole, µ (Condon approximation).
[1–5]. Although these studies have shed a tremendous Applyingfirst-ordertime-dependentperturbationtheory,
]
h amount of light on excited-state dynamics, none of the the wavepacketthat we want to reconstruct is [13]
p methods in use provides complete information on the
t- excited-state wavefunction. The need for an experimen- |Ψ(t)i=−ie−iHet{−µε1}|ψ0i, (2)
n tal method that will provide this information is com-
a where the initial state, |ψ i, is the vibrational ground-
poundedbythefactthattheoreticalabinitiocalculations 0
u
state of H with the eigenfrequency ω , ε is the ampli-
q for excited states are difficult and of limited accuracy. g 0 1
tude of the pulse and t is the propagation time on the
[ Therehavebeenseveraltheoreticalproposalsforcom-
excitedstatemeasuredfromthetimeofpulseexcitation.
plete reconstruction of an excited-state molecular wave-
1 (Here and henceforth we take ~=1.)
function from spectroscopic signals [6, 7]. These studies,
v
Substituting Eq. (2) into the definition of C (t), we
4 however,generallyassumethatoneormoreexcited-state g
5 potentials (or the corresponding vibrational eigenstates) find that the superposition coefficients are given by
3 is known. A notable exception is a recently developed
4 iterative method for excited-state potential reconstruc- Cg(t)=iµε1hψg|e−iHet|ψ0i≡iµε1cg(t). (3)
.
1 tionfromelectronictransitiondipolematrixelements [8]
Hence, the central quantities required for reconstructing
0
but this method does not appear to be applicable to
1 dissociative potentials. Experimental work has focused |Ψ(t)i are the cross-correlation functions cg(t). It has
1 longbeenrecognizedthatthesecorrelationfunctions ap-
v: onwavepacketinterferometryofvibrationalwavepackets pear(uptoµ)inthetime-dependentformulationofreso-
[9,10]aswellaselectronicRydbergwavepackets[11,12].
i nance Ramanscattering(RRS) [14]; however,the exper-
X The approach we present here assumes knowledge of
imental RRS signal involves the absolute-value-squared
r the ground-state potential but not of any excited poten- of the half-Fourier transform of the correlation function,
a tial. In principle, the approach is completely general for
hence the latter cannot be recoveredfrom that signal.
polyatomics. Our strategy is to express the reacting-
Fully resonant coherent anti-Stokes Raman scattering
molecule wavefunction, |Ψ(t)i, as a superposition of the
(CARS)hasbeenshowntobeapowerfulprobeofground
vibrational eigenstates of the ground-state Hamiltonian,
and excited electronic states properties [15, 16]. In this
{|ψ i}:
g letter we show that the correlation functions {c (t)}
g
may be completely recovered from femtosecond reso-
|Ψ(t)i= |ψ ihψ |Ψ(t)i≡ C (t)|ψ i. (1)
g g g g
nant CARS spectroscopy, allowing complete reconstruc-
Xg Xg
tion of the excited-state wavepacket. The formula for
Since the vibrational eigenstates {|ψ i} are assumed theCARSsignalproducedbyathree-pulsepump-dump-
g
known, the challenge is to find the time-dependent su- pump sequence is P(3)(τ)=hψ(0)(τ)|µˆ|ψ(3)(τ)i+c.c.
perposition coefficients C (t). [17], where ψ(3)(τ) is the third-order wavefunction and
g
Consider a two-state molecular system within the ψ(0)(τ)=e−iHgτψ0. Within the δ-pulseandCondonap-
Born-Oppenheimer approximation. The nuclear Hamil- proximations, P(3) takes the form
tonians H and H correspond, respectively, to the
g e
e
(known)groundand(unknown)excitedpotentials,which P(3)(τ)=εhψ |e−iHeτ43e−iHgτ32e−iHeτ21|ψ i, (4)
0 0
e
2
where τ =τ −τ is the (positive) time-delay between theoremweobtainasinc-typeofspectrumwithpeaksat
ij i j
the centers of the ith and jth pulses and τ =τ −τ the frequencies ω =ω :
43 3 g
with τ being the time of signal measurement. We have
N
denoted Hg =Hg−ω0, ε=i3µ4ε1ε2ε3eiω0(τ21+τ43) with P(3)(τ ,ω,τ )e= S(ω,g)P(3)(τ ,τ ), (6)
43 21 g 43 21
ε as the first, second and third pulse amplitudes, re-
sp1,e2c,3tivelye,andτ ≡[τ ,τe ,τ ]. InwritingP(3)(τ) asa e Xg=0
complexquantityweh2a1ve3a2ssu4m3 edthesignalismeasured where S(ω,g)=2Tεei(ω−ωeg)(τˇ32+T)sinc[(ω−ωg)T],
2T =τˆ −τˇ , and τˇ (τˆ ) is the minimal (maximal)
in a heterodyne fashion. 32 32 32 32
value of τ . Fixing (τ ,eτ ), Eq. (6) can be writeten as
AsillustratedinFig. 1,Eq. (4)hasthefollowingphys- 32 43 21
a matrix equation:
ical interpretation: A first laser pulse, the pump pulse,
transfers amplitude to the excited potential surface cre- P(ω3) =SωgP(g3). (7)
ating a wavepacket whose time-dependence we are in-
3. Invert the meatrix equation (7). The equation
terested in reconstructing. After evolving on the excited
stateforsometime,asecondlaserpulse,thedumppulse, P(g3) = S−gω1P(ω3) isolates the two-dimensional functions
transferspartofthis amplitude backto the groundstate P(3)(τ ,τ ). In inverting S we choose the number of
g 43 21e
where it evolves for a second interval of time. Finally, a frequency elements (ω) equal to the number of the g el-
third laser pulse excites part of the second-order ampli- ements so that the matrix is square. For numerical ac-
tude to the excited state, generating the third-order po- curacy, the inversion is implemented separately around
larization that produces the CARS signal, measured at each of the peaks at ω .
g
later times. The desired wavefunction |Ψ(t)i (Eq. (2)) 4. Take the square-root of P(3). Assuming the func-
g
may already be recognized in Eq. (4). tions {ψ (x)} are reael, we can rewrite P(3) as
g g
Pg(3)(τ43,τ21)=hψg|e−iHeτ43|ψ0ihψg|e−iHeτ21|ψ0i. (8)
Taking the square-root of the diagonal of P(3)(τ ,τ )
g 43 21
(i.e. τ =τ =t), we recover the c (t) up to a sign:
43 21 g
Pg(3)(t)=aghψg|e−iHet|ψ0i≡hψg|e−iHet|ψ0i, (9)
q
where a = ±1 and the sign of ψ (xe) is as yet undeter-
g g
mined. By demandingcontinuityofthe cross-correlation
e
functions (and their derivatives), the coefficients a can
g
be regarded as time-independent. Substituting Eq. (9)
insteadofc (t)intoEq. (3)andusingtheresultingC (t)
g g
FIG. 1: Color online. The pump-dump-pump CARS scheme. in Eq. (1) yields
Ψ(t)isthedesiredwavefunction.
N
Thereconstructionof|Ψ(t)ifromP(3) proceedsinfive |Ψ(t)i=iµε1 |ψgihψg|e−iHet|ψ0i. (10)
Xg=0
steps: e e
1. Insert a complete set of ground vibrational states. The different sign combinations of ψg(x) generate 2N+1
Introducing |ψ ihψ |=1ˆ into Eq. (4) we obtain possible superpositions. (In fact, only 2N are physically
g g g e
P meaningful since we are free to set the sign of one of the
N g-components.) Only one out of the 2N |Ψ(t)i coincides
P(3)(τ)=ε e−iωegτ32Pg(3)(τ43,τ21), (5) with |Ψ(t)i: the |Ψ(t)i for which the sign combination
e
Xg=0 satisfies |ψ ihψ |= .
e g g eg 1
where Pg(3)(τ43,τ21)=hψ0|e−iHeτ43|ψgihψg|e−iHeτ21|ψ0i, 5. DisPcriminatieng |Ψ(t)i from the set {|Ψ(t)i}. The
and ωg =ωg−ω0. N is determined by the number set of wavefunctions {|Ψ(t)i} are all consisteent with the
of ground vibrational states required to expand |Ψ(t)i. CARS signal at a speciefic value of τ43 =τ21 [18]. How-
Noteethat the desired correlationfunctions c (t) may al- ever,only one |Ψ(t)i is consistentwith the signal deriva-
g
ready be recognized in P(3). tives. To see this, consider the nth derivative of the ex-
g e
2. Fourier-transform P(3) with respect to τ . The perimental signal, Eq. (4), with respect to τ21:
32
transformationresolvesP(3) intoindividualground-state ∂nP(3)(τ)
components, Pg(3). Since τ32 is defined to be positive, we ∂τn =ε†hΨ∗(τ43)|e−iHgτ32Hen|Ψ(τ21)i
21
multiply Eq. (5), prior to the transformation, by the e
rectangular function that takes the value 1 for the τ32 =ε† e−iωgτ32Cg(τ43)Cg′(τ21)Hen,gg′, (11)
domain and 0 elsewhere. Using the Fourier convolution Xg,g′
e
3
where ε† = (−i)n−1µ2ε−11ε2ε3eiω0τ41, τ41 = τ − τ1, on a spatial grid of 256 points in the range of 2–12a.u.
Hen =(He−ω0)n, and Hen,gg′ =hψg|Hen|ψg′i. Substitut- with time spacing of ∆t=0.1fs. A constant transition-
dipole of 2a.u. was used, and the pulse amplitudes ε
ing |Ψ(t)i instead of |Ψ(t)i, into Eq. (11) gives 1,2,3
e e e were taken to be 10−4a.u. The ranges of time-delay for
∂nP(e3)(τ) the Li2 (d-Li2) system were τ21,43 =0−200fs (0−80fs)
∂τn = ε† e−iωgτ32agag′Cg(τ43)Cg′(τ21)Hen,gg′. with spacing of 0.2fs. For both systems, we took τ32 =
g21 Xg,g′ 3−6000fs with 1fs spacing.
e
(12) For Li , we inverted Eq. (7) for each of the first 25
2
Accordingly, the |Ψ(t)i for which ∂nP∂gτ(3n)(τ) = ∂nP∂τ(3n)(τ) pgreiadkspooifntPs3c(eωn)teurseidngartohuenmdatthreicpeseaSkswaitthω2g5. fTrehqisuepnrcoy-
21 21 e
for all n, is the waevefunction that coincides with |Ψ(t)i duced 25 two-dimensional functions Pg(3), g = 0,...,24.
of Eq. (2), and hence, is the reconstruction solution. For d-Li the procedure was performedefor the first
2
Inpractice,weproceedasfollows. Weinvertthetime- 40 peaks, producing 40 two-dimensional functions P(3),
g
dependent Schr¨odingerequation to calculate a set of po-
g =0,...,39.
tentials from each |Ψ(t)i:
1e ∂ 1 ∂2
V(x) = i + Ψ(x,t), (13)
Ψ(x,t)(cid:20) ∂t 2m∂x2(cid:21)
e
where m is theesystem’s reduced mass. One can show
that the potentials calculated by the |Ψ(t)i that do not
coincide with |Ψ(t)i, are time-dependent [19]. Only
e
the potential calculated with |Ψ(t)i = |Ψ(t)i is time-
independent and hence corresponds to the excited-state
e
Hamiltonian H of the measured system. Thus, in order
e
to find the correct wavefunction we use the set of cal-
culated potentials, as if they were time-independent, to
propagate the corresponding {|Ψ(t)i} back to time zero.
Ofallthepotentials,onlythetrulytime-independentone
e
willpropagatethecorresponding|Ψ(t)icorrectlybackto
|ψ i, and therefore this |Ψ(t)i is the correct wavefunc- FIG. 2: Color online. Snapshots of the real part of the recon-
0 e structed (circles, red) vs. the exact (dots, blue) wavefunction, at
tion. Note that the above procedure requires knowing
e varioustimesontheexcited (A)potential (solidline)ofLi2.
the signal as a function only of τ and τ =τ .
32 21 43
TABLE I: The parameters, in atomic units, for the X, A and Ae In Figs. 2 and 3 we present snapshots of the real part
potentials usedinsimulatingtheCARSsignals. of the reconstructed first-order wavefunction for the Li2
X A Ae andthe d-Li2 molecules,respectively. For Li2 (d-Li2) we
superposethefirst25(40)eigenfunctionsψ (x)usingthe
D 0.0378492 0.0426108 9.11267×10−5 g
cross-correlationfunctions obtainedby the CARSanaly-
b 0.4730844 0.3175063 1.5875317
sis and maintaining |ψ ihψ |= . The reconstructed
x0 5.0493478 5.8713786 7.3699313 g g g 1
T 0 0.0640074 0.0640074 P e
To test the above reconstruction methodology,
we simulated the CARS signal by calculating
hψ(0)(τ)|µˆ|ψ(3)(τ)i as a function of the three time-
delays, for two one-dimensional systems. The first
is the Li molecule, with its ground (X) and first-
2
excited (A) electronic states as Morse-type potentials,
V(x)=D(1−e−b(x−x0))2+T. The second system,
FIG. 3: Color online. Snapshots of the real part of the recon-
henceforth denoted d-Li , has the Li ground state
2 2 structed (circles, red) vs. the exact (dots, blue) wavefunction, at
(X) but a dissociative excited potential of the form varioustimesontheexcited (Ae)potential (solidline)ofd-Li2.
V(x)=De−b(x−x0)+T (denoted A). Table I gives the
potential parameters in atomic units used for the simu- wavefunctionsareseento be inexcellentagreementwith
e
lations. The parameters for the Morse-type potentials theexactones,obtainedbydirectcalculationofthefirst-
are based on data published in [20]. order population, for all propagation times. For the Li
2
The wavepacket propagations employed in simulating system,ahighqualityreconstructionisalreadyobtained
P(3) wereperformedusingthesplit-operatormethod[21] by superposing just 20 basis functions.
4
since the frequency shift between the pump and dump
pulses will be more effective in discriminating unwanted
processes that may contribute to the measured signal at
k=k −k +k . We simplified matters by considering
1 2 3
δ-function pulse excitations, a coordinate-independent
transition dipole moment and only one excited-state po-
tential. In future work we will test the removal of all
these assumptions.
We have shown that once the time-dependent wave-
function is found, the excited potential can be recon-
structed with quite high accuracy. It will be of great
FIG. 4: Color online. The reconstructed (circles, red) vs. the interest to test the method on polyatomics, where ob-
exact(dots, blue)Apotential ofLi2. tainingmultidimensionalpotentialsurfacesfromspectro-
scopic data has been one of the longstanding challenges
of molecular spectroscopy. An important application of
excited-state potential reconstruction will be the ab ini-
tiosimulationsoflasercontrolofchemicalbondbreaking.
Experimental laser control has been greatly hindered by
thelackofdetailedtheoreticalguidance,whichinturnis
due to the lack of accurate excited-state potentials. The
present methodology could have a significant impact in
this field by providing the necessary information about
excited-state potentials.
This research was supported by the Minerva Founda-
tionandmadepossible,inpart,bythehistoricgenerosity
of the Harold Perlman family.
FIG. 5: Color online. The reconstructed (circles, red) vs. the
exact(dots, blue)Aepotential ofd-Li2.
Havingdeterminedthe wavefunctionswecalculatethe
corresponding excited potential surfaces from Eq. (13)
[1] A.H. Zewail, Science 242, 1645 (1988).
using eight-point (three-point) central finite-differencing
[2] J.C. Polanyi and A.H. Zewail, Acc. Chem. Res. 28, 119
for the time (spatial) derivatives. The time-step used
(1995).
was 0.2fs but very good results were also obtained us-
[3] P.Kukura,D.W.McCamant,S.Yoon,D.B.Wandschnei-
ing 0.5fs. Figures 4 and 5 compare the reconstructed vs. der and R.A. Mathies, Science310, 1006 (2005).
the exact potentials. The wavefunction (absolute value) [4] S. Takeuchi, S. Ruhman, T. Tsuneda, M. Chiba, T.
usedincalculatingthepotentialisshownbyablacksolid Taketsugu and T. Tahara, Science322, 1073 (2008).
line. Note from Figs. 4 and 5 that combining the re- [5] U. Banin and S. Ruhman, J. Chem. Phys. 99, 9318
(1993).
constructed potential from two points in time (e.g. 5
[6] M. Shapiro, J. Chem. Phys. 103, 1748 (1995); C. Leich-
and 70fs for Li and 5 and 79fs for d-Li ) is sufficient to
2 2 tle,W.P.Schleich,I.Sh.AverbukhandM.Shapiro,Phys.
reconstruct the potential over the full range of interest
Rev. Lett.80, 1418 (1998).
(2–5˚A). Once the potential is known one can calculate [7] T.S.HumbleandJ.A.Cina,Phys.Rev.Lett.93,060402-
the excited-state wavefunction as a function of time for 1 (2004); J.A. Cina, Annu. Rev. Phys. Chem. 59, 319
any excitation pulse sequence without the need for any (2008).
additional laboratory experiments. [8] X.Li,C.Menzel-Jones,D.AvisarandM.Shapiro,Phys.
Chem. Chem. Phys.12, 15760 (2010).
To conclude, we have presented a methodology for
[9] N.F. Scherer et al., J. Chem. Phys.95, 1487 (1991).
the complete reconstruction of the excited-state wave-
[10] K.Ohmorietal.,Phys.Rev.Lett.96,093002(2006);K.
function of a reacting molecule by analyzing a multi-
Ohmori, Annu.Rev.Phys. Chem. 60, 487 (2009).
dimensional resonant CARS signal. The methodol- [11] T.C.Weinacht,J.AhnandP.H.Bucksbaum,Phys.Rev.
ogy applies to polyatomics as well as diatomics. We Lett. 80, 5508 (1998).
have assumed that only the ground-state potential is [12] A. Monmayrant, B. Chatel and B. Girard, Phys. Rev.
known. The approach is very compelling since the de- Lett. 96, 103002 (2006).
[13] D.J. Tannor, Introduction to Quantum Mechanics: A
sired excited-state wavefunction is explicitly contained
Time-Dependent Perspective (University Science Books
in the formula for the CARS signal. Highly accurate
Sausalito, 2007), Eq. (13.8).
reconstruction is obtained even far from the Franck-
[14] Soo-Y. Lee and E.J. Heller, J. Chem. Phys. 71, 4777
Condon region. In fact, in practice the method may (1979); E.J. Heller, R.L. Sundberg and D. Tannor, J.
be more accurate far from the Franck-Condon region, Phys.Chem.86,1822(1982);A.B.Myers,R.A.Mathies,
5
D.J. Tannor and E.J. Heller, J. Chem. Phys. 77, 3857
(1982); D.Imre, J.L. Kinsey,A. Sinhaand J. Krenos, J.
Phys.Chem. 88, 3956 (1983).
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Chem. Phys.73, 4695 (1980).
[16] N.A.Mathew et al., J. Phys.Chem. A 114, 817 (2010).
[17] J.Faeder,I.Pinkas,G.Knopp,Y.PriorandD.J.Tannor,
J. Chem. Phys. 115, 8440 (2001).
[18] In fact, the set of wavefunctions given by (14) are con-
sistent with theCARS signal for any pair (τ21,τ43).
[19] SeeSupplementary OnlineMaterial.
[20] G.Herzberg, in Molecular Spectraand Molecular Struc-
ture; I. Spectra of Diatomic Molecules, Krieger Publish-
ing Company,Malabar, Florida, (1950).
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Phys.47, 412 (1982).
6
Supplementary Online Material – Determining where we emphasize that V is time-independent. Let us
e
the Correct Wavefunction out of the Set {Ψ (t)} now define the related quantity
i
e
In this supplement we explain how we determiene the 1 ∂
V = i −T Ψ(t), (19)
correct wavefunction out of the set of wavefunctions Ψ(t)(cid:20) ∂t (cid:21)
{Ψi(t)}, i = 1,2,...,2N, where N is the number of ba- where T is the usual kinetic energy opeerator. Obviously,
sis functions {ψg} needed to span Ψ(t) (ref [22] in the for Ψ(t)≡Ψ(t) Eq. (1e9) is equivalent to the usual time-
e
paper). dependent Schr¨odinger equation for Ψ(t) and therefore
e
Wehavedefinedasetofwavefunctionsthatcanbecon- V ≡V istime-independent. Weclaimthatforanyother,
e
structed using the information obtained from the CARS incorrect, wavefunction Ψ(t), Eq. (19) results in a time-
signal: dependent potential V.
e
In order to show this we substitute T = T +∆T in
|Ψ(t)i ≡ |ψgihψg|e−iHet|ψ0i Eq. (18), where ∆T =T −T, and obtain:
Xg e
e e e
1 ∂
= |ψ ihψ |Ψ(t)i≡ |Ψ(t)i. (14)
g g V = i −(T +∆T) Ψ(t)
Xg 1 e Ψ(t)(cid:20) ∂t (cid:21)
e e e 1 e
(InwritingEq.(14)weomittedthe proportionalitycoef- = Ve− [∆T]Ψ(t). (20)
Ψ(t)
ficientiµε relativetoEq.(10)inthepaper.) Recallthat
1 e
1≡ g|ψgihψg|≡ g|ψgiaghψg|whereag maytakeone The term 1 [∆T]Ψ(te) is time-dependent (unless Ψ(t)
out oPf two possiblePvalues: ±1. A useful property of the Ψe(t)
eoperator is tehat its square equals the identity operator is an eigenfunction oef ∆T, which has no general reaeson
: 1 to hold. In addition, in the Appendix we show that ∆T
1 e is generally different from zero). Therefore, in order to
= agag′|ψgihψg|ψg′ihψg′| preserve the time-independence of Ve ≡ Ve , V must
11 1 1
Xgg′ also be time-dependent.
ee e e e
= a2|ψ ihψ |= |ψ ihψ |= . (15) Tosummarize: inordertodeterminethecorrectwave-
g g g g g
1 function out of the set of wavefunctions {Ψ (t)},i =
Xg Xg i
1,2,...,2N, we use the fictitious Schr¨odinger equation,
e
We can derive an equation of motion for Ψ(t): (19), to calculate a potential, V, from each wavefunc-
tion Ψ(t) of the set. At different times, t, the wavefunc-
∂ ∂ ∂ e
|Ψ(t)i = |Ψ(t)i= |Ψ(t)i=−i He|Ψ(t)i tionseΨ(t) will give different potentials V except for the
∂t ∂t1 1∂t 1 one correct wavefunction, Ψ(t), that corresponds to the
e = −ieHe |Ψ(te)i=−i He |Ψe(t)i correcet Schr¨odinger equation and therefore will give the
1 11 1 1
same potential, V ≡ V , at all times. Thus, the correct
≡ −iHee|Ψe(et)i, e e e (16) e
wavefunction Ψ(t) can be selected from the set {Ψ (t)}
i
where, we have useed tehe fact that is time-independent astheonethatprovidesatime-independent potentialvia
and therefore commutes with ∂ . 1Equation (16) shows Eq. (19). Alternatively, as described in the papeer, the
∂t e
that Ψ(t) obeys a time-dependent Schr¨odinger equation wavefunction Ψ(t) that propagates back to the known
Ψ(0)≡ ψ , using the corresponding potential calculated
with tehe effective Hamiltonian He =1He1. byEq.(190),iseguaranteedtobethecorrectreconstructed
The Hamiltonian H has the conventional form of
e
e e e wavefunction, Ψ(t).
H = V +T, where T is the kinetic-energy operator.
e e
The Hamiltonian H therefore takes the form:
e
H ≡ H e = V + T ≡V +T, (17) Appendix
e e e e
1 1 1 1 1 1
Note thatetheeopereatore deoesenoet comemutee with Ve, T We show that ∆T 6=0:
1
or H since it does not share a common basis of eigen-
e
e
vectors with the last three operators. Note also that the ∆T = (T −T)= ( T −T)
operators V , T and H are all time-independent. 11 1 1
e e = (T − T)= [T, ]. (21)
e ee e e
Rearranging Eq. (16), we obtain: 1 1 1 1 1
e e e
e e e e e
Thecommutator[T, ]isnotidenticallyzero. Therefore,
1 ∂ 1
V = i −T Ψ(t). (18) [T, ]≡∆T is not identically zero as well.
e
Ψ(t)(cid:20) ∂t (cid:21) 1 1 e
e e e e e
e
