Operator of Time and Properties of Solutions of Schroedinger Equation for Time Dependent Hamiltonian

0.08 MB·

by  Slobodan Prvanovic

Operator of Time and Properties of Solutions of Schr¨odinger Equation for Time
Dependent Hamiltonian
Slobodan Prvanovi´c and Duˇsan Arsenovi´c
Scientific Computing Laboratory,
Center for the Study of Complex Systems,
Institute of Physics Belgrade,
University of Belgrade, Pregrevica 118, 11080 Belgrade,
Serbia
(Dated: January 26, 2017)
Withintheframeworkofself-adjointoperatoroftimeinnon-relativisticquantummechanicssome
7 propertiesofsolutions ofSchr¨odingerequation,related toHilbertspaceformalism, areinvestigated
1 for two typesof time dependentHamiltonian.
0 PACS number: 03.65.Ca, 03.65.Ta, 02.30.Sa
2

n 1
Ja dreEsvsienrgsitnimceeSdcehpr¨oenddinagnetrS[1c]h,rt¨ohdeirnegearreeqmuaantyioanrtaincdlestiamde- i~[Iˆ⊗tˆ,Iˆ⊗sˆ]=Iˆ⊗Iˆ
4 dependent Hamiltonians [2–8]. Within these investiga- The other commutators vanish. The operator of time tˆ
2 tions,oneisusuallyconcernedwiththeinfluenceofenvi- has continuous spectrum ,+ , just like the oper-
ronmentona systemunder considerationandproperties {−∞ ∞}
ators of coordinate and momentum qˆ and pˆ. So is the
] of evolving states. Treatment of these problems entirely
h case for the operator sˆ, that is conjugate to time, and
p belongs to the standard formalism of quantum mechan- which is the operator of energy. After noticing complete
- ics where time is treated as parameter. This means,
similarity among coordinate and momentum on one side
t
n in particular, that time dependence of the solutions of
andtimeandenergyontheotherside,onecanintroduce
a Schr¨odinger equation has not been systematically ana- eigenvectors of tˆ:
u
lyzedfrom the pointof view of the Hilbert space formal-
q
[ ism. Very often in standard quantum mechanics, time tˆt =tt , t R
dependence of solutions of Schr¨odinger equation is seen | i | i ∀ ∈

1 just as exponential phase factor, so, as such, it has not In t representation, operator of energy becomes i~∂ ,
6v attracted much attention. Regarding the formalism of whi|leiitseigenvectors E becomeei1~Et,foreveryE ∂Rt.
| i ∈
7 Hilbert spaces, one is usually more concerned with the Accordingto Pauli,there is no self-adjointoperatorof
0 other part of solutions. That is, in the coordinate rep- time that is conjugate to the Hamiltonian H(qˆ,pˆ) which
7 resentation, one is interested in properties of ψ(q) and hasboundedfrombelowspectrum. Withinourproposal,
0 not much in eif(t). But, after introducing the operator self-adjointoperatorof time is conjugate to the operator
.
1 oftime,propertiesofthepartofsolutionthatdependon ofenergy,whichhasunboundedspectrum. TheHamilto-
0 time become equally well interesting andthis is what we nianandoperatorofenergyareactingindifferentHilbert
7
are going to discuss in some detail. spaces, but there is subspace of the total Hilbert space
1
: There is a whole variety of topics and approaches re- where:
v
lated to the operator of time, e.g. [9–11] and references
Xi therein, but let us shortly review our approach that we sˆψ =H(qˆ,pˆ)ψ
| i | i
r haveproposedin[12–15]. Ourapproachissimilarto[16],
a The states that satisfy this equation are physical since
and references therein, and [17].
theyhavenon-negativeenergy. Thelastequationisnoth-
In order to fulfill demand coming from general rela-
ingelsebuttheSchr¨odingerequation. Bytaking q t
tivity, that space and time should be treated on equal | i⊗| i
representationofpreviousequation,onegetsthefamiliar
footing, just like for every spatial degree of freedom a
form of Schr¨odinger equation:
separate Hilbert space is introduced, one should intro-
duce Hilbert space where operator of time tˆacts. More ∂
i~ ψ(q,t)=Hˆψ(q,t)
concretely,forthecaseofonedegreeoffreedom,besideqˆ ∂t
and conjugate momentum pˆ, acting non-trivially in ,
q
tjuhgeareteshtooutˆld, abcetHntonw-htreirveiatˆl,lyt.ogeStoh,erinwith sˆthatfiosrcHothne- oWthitehr wthoerdssh,oortphearnatdornootfaetnioenrgHyˆh=as hnqe|gHa(tqiˆv,epˆ)e|qig′ie.nvaIln-
q t
self-adjoint operators qˆ Iˆ, pˆ Iˆ, Iˆ Htˆa⊗ndHIˆ sˆ the ues as well as non-negative, but the Schr¨odinger equa-
⊗ ⊗ ⊗ ⊗ tionappearsasaconstraintthatselectsphysicallymean-
following commutation relations hold:
ingful states. That is, states with non-negative en-
1 ergy, due to the non-negative spectra of H(qˆ,pˆ), are se-
i~[qˆ⊗Iˆ,pˆ⊗Iˆ]=Iˆ⊗Iˆ lected by Schr¨odinger equation. For time independent
2
Hamiltonian, the typical solution of Schr¨odinger equa- of the solution can be called generalized Dirac delta
tion ψE(q)ei1~Et is q t representationof ψE E , function with explanation similar to the above given
| i⊗| i | i⊗| i
where H(qˆ,pˆ)ψ = E ψ and sˆE = E E . The en- one. The properties of generalized Dirac delta func-
E E
| i | i | i | i
ergy eigenvectors E have the same formal characteris- tions tE,h = 1 e iEh(t) are as follows. Their bi-
| i h | i √2π −
tics as, say, the momentum eigenvectors (they are nor- orthogonalset is:
malized to δ(0) and, for different values of energy, they
are mutually orthogonal). tE ,h, = 1 h(t)e iE′h(t)
′ ′ −
Now, let us discuss properties of solutions of h | ⊥i √2π
Schr¨odinger equation with time dependent Hamiltonian. since:
In order to be more systematic, we shall firstly ana-
lyze situation when time dependence appears through E ,h, E,h = 1 +∞dth(t)e ih(t)(E E′) =
′ ′ − −
h ⊥| i 2π Z
the term that is added to the time independent Hamil-
−∞
ttoimnieadne,pi.ened.,enHt(tqˆe,rpˆm)+mugl(ttˆi)p,ltiehsentimtheeinsidteupaetniodnenwthpeanrtthoef = 21π Z +∞dye−iy(E−E′) =δ(E−E′).
the Hamiltonian, i.e., H(qˆ,pˆ) g(tˆ), and finally we shall −∞
· Resolution of unity is:
briefly comment combination of these two. (We are not
going to discuss the case when g(tˆ) depends on qˆand pˆ + +
since it is beyond the scope of this article.) Iˆ= ∞dE E,h, E,h = ∞ E,h E,h, .
Z | ⊥ih | Z | ih ⊥|
For the Hamiltonian H(qˆ,pˆ) + g(tˆ), the Schr¨odinger −∞ −∞
equation(in q t representation,andwith~=1taken Functions 1 e iEh(t) generate transform:
for simplicity|)ii⊗s|saitisfied for ψ (q) 1 eih(t)eiEt, where √2π −
E √2π
E is the eigenvalue of Hamiltonian and h(t) = g(t)dt. [ f(t)](E) 1 +∞f(t)e iEh(t)dt
h −
The time dependent part of the solution can beRseen as F ≡ √2π Z
−∞
modulated Dirac delta function since it is slightly differ-
which is related to Fourier transform by:
ent from the time representation of energy eigenvector,
which is Dirac delta function in energy representation. dh 1(t)
ThepropertiesofmodulatedDiracdeltafunctionsareas [Fhf(t)](E)=(cid:20)F(cid:26) −dt f(h−1(t))(cid:27)(cid:21)(E).
follows.
Withthe setoffunctions tE,h = 1 e ih(t)e iEt it It is easy to verify that inverse transform is:
h | i √2π − −
iseasytoverifyorthogonalityandcom+pletenessrelations: [Fh−1f(E)](t)= √12πh′(t)Z +∞f(E)eiEh(t)dE.
E,hE ,h =δ(E E ), Iˆ= ∞ E,h E,hdE. −∞
′ ′
h | i − Z | ih | The operator 1 i∂ has 1 e iEh(t) as generalized
−∞ h′(t) ∂t √2π −
The h generated transform: eigenfunctions since(cid:0): (cid:1)
[ hf(t)](E) 1 +∞f(t)e−ih(t)e−iEtdt h1(t)(cid:18)i∂∂t(cid:19)(cid:20)√12πe−iEh(t)(cid:21)=E(cid:20)√12πeiEh(t)(cid:21).
F ≡ √2π Z ′
−∞
This operator is not self-adjoint, which is in accordance
is easily related to the Fourier transform:
with the fact that 1 e iEh(t) are not orthogonal.
√2π −
[ f(t)](E)=[ (f(t)e ih(t)](E). While:
h −
F F
It’s inverse is S(E)= 1 +∞dteiEh(t)
2π Z
[Fh−1f(E)](t)≡ √12π Z +∞f(E)eih(t)eiEtdE. may not be integrable, it m−a∞y exist as a distribution.
−∞ Acting on a Schwartz function ϕ(E), it holds:
Thissetoffunctionsaregeneralisedeigenfunctionsofthe
+
self-adjoint operator i∂ g(t) : S, ϕ = ∞S(E)ϕ(E)dE =
∂t − h i Z
(cid:0) (cid:1)
−∞
(cid:18)i∂∂t −g(t)(cid:19)(cid:20)√12πe−ih(t)e−iEt(cid:21)=E(cid:20)√12πe−ih(t)e−iEt(cid:21). = 21π Z +∞dtZ +∞ϕ(E)e−iEh(t)dE =
−∞ −∞
For the Hamiltonian H(qˆ,pˆ) g(tˆ), the Schr¨odinger = 1 +∞dt[ (ϕ(E))](h(t)) =
equation (in q t represent·ation) is satisfied for √2π Z F
ψE(q)√12πe−iE|h(it),⊗w|hiere E is the eigenvalue of Hamil- = 1 +∞−d∞udh−1(u)[ (ϕ(E))](u).
tonian and h(t) = g(t)dt. The time dependent part √2π Z du F
R −∞
3
Using Parseval’s theorem, the result is:
S, ϕ = 1 +∞ϕ(E) −1 dh−1(t) ∗ (E) ∗ [1] E. Schr¨odinger, Ann.Phys., 81, 109 (1926)
h i √2π Z (cid:26)(cid:20)F (cid:18)(cid:18) dt (cid:19) (cid:19)(cid:21) (cid:27) [2] D. Poulin et al., Phys.Rev.Lett., 106, 170501 (2011)
−∞
[3] D. Chruscinski et al., Phys.Rev.A, 91, 042123 (2015)
so:
[4] A.T. Schmitz and W.A. Schwalm, Phys. Lett. A, 380,
S(E)= 1 −1 dh−1(t) ∗ (E) ∗. [5] C11.2M5.(2S0a1r6ri)s et al.,Phys. Lett.A, 324, 1 (2004)
√2π (cid:26)(cid:20)F (cid:18)(cid:18) dt (cid:19) (cid:19)(cid:21) (cid:27) [6] R.Uzdinetal.,J.Phys.A,Math.Theor,Vol45,415304
(2012)
Finally, let us just mention that in case of H(qˆ,pˆ)· [7] J. S. Briggs et al, J. Phys. A: Math. Theor., 40 1289
g1(tˆ)+g2(tˆ) one can easily combine above results. On (2007)
theotherside,forthecaseofH(qˆ,pˆ) g(tˆ),ifthewholeso- [8] V. Enss and K. Veselic, Ann. Inst. H. Poincare Sect. A,
lutionofSchr¨odingerequationψ (q)· 1 e iEh(t)istaken 39, 159 (1983)
E √2π − [9] G. C. Hegerfeldt et al.,Phys.Rev.A,82, 012113 (2010)
into account,mutualorthogonalityofψ (q) fordifferent
E [10] G. Gour et al.,Phys. Rev.A 69, 014101 (2004)
E implies the orthogonality of the whole ψE(q,t). [11] J. J. Haliwell et al.,Phys. Lett.A, 379, 2445 (2015)
[12] S. Prvanovi´c, Prog. Theor. Phys.,126, 567 (2011)
[13] D. Arsenovi´c et al.,EPL, 97, 20013 (2012)
ACKNOWLEDGEMENT [14] D. Arsenovi´c et al.,Chin. Phys. B, 21, 070302 (2012)
[15] S. Prvanovi´c, arXiv:1701.05244 [quant-ph]
We acknowledge support of the the Serbian Ministry [16] V. Giovannetti et al.,Phys. Rev.D,92, 045033 (2015)
[17] P. M. Morse and H. Feshbach, Methods of Theoretical
ofeducation,scienceandtechnologicaldevelopment,con-
Physics (McGraw-Hill, NewYork,1953) p.248.
tract ON171017.