Perturbation of the Wigner equation in inner product C-modules

Why Are Symmetries a Universal Language ofPhysics?(on the unreasonable effectiveness of symmetriesin physics)Vladik Kreinovich and Luc Longpr´eAbstractIn this paper,we prove that in some reasonable sense,every possible physical law can be reformulated in terms of symmetries.This result explains the well-known success of group-theoretic approach in physics.1Formulation of the ProblemTraditional physics used differential equations to describe physical laws.Mod-ern physical theories(starting from quarks)are often formulated not in terms of differential equations(as in Newton’s days),but it terms of the corresponding symmetry groups(see,e.g.,[3,9,8]).Moreover,it turned out that theories that have been originally proposed in terms of differential equations,including gen-eral relativity(and scalar-tensor version of gravity theory),quantum mechanics, the electrodynamics,etc.,can be reformulated in terms of symmetries[4,1,2].Symmetry approach has been very successful in physics.Its success raises two natural questions:•How universal is this approach?Can we indeed(as some physicists sug-gest)express any new idea in terms of symmetries.•Is the fact that many(if not all)physical theories can be expressed in terms of symmetries a general mathematical result or a peculiar feature of our physical world?12What We Are Planning to DoIn this paper,we analyze this problem from a mathematical viewpoint.We will show that,in some reasonable sense,every possible physical law can be expressed in terms of symmetries.Comment.This result wasfirst announced in[5].In order to prove the corresponding theorem,we will have to formalize what “law”is and what“symmetry”is.3Motivations of the Following Definitions3.1What do we mean by“law”?In order to describe what we mean by a physical law,let usfirst describe what it means that an object does not satisfy any law at all.For example,let us consider the direction of a linear-shaped molecule.If this molecule is magnetic, then its orientation must follow the magneticfield.If the molecule is an electric dipole,then it must follow the electricfield,etc.In all these cases,there is a physical law that controls the orientation of the molecule.It is also possible that the molecule has neither of these orienting properties (or it has,but the correspondingfields are absent).In this case,there is no physical law to control its orientation;therefore,the orientation is not described by any law,it is random.We still have to formalize what“random”means but so far,we hope,the conclusion sounds reasonable:if something does not satisfy any physical law, then it is random.By inverting this informal statement,we can conclude that an object satisfies a physical law if and only if it is not random.To make this definition precise, we must define what“random”means.3.2What do we mean by“random”?The formal definition of“random”was proposed by Kolmogorov’s student P. Martin-L¨o f[7];for a current state of the art,see,e.g.,[6].(We will be using a version of this definition proposed by P.Benioff.)To describe this definition,let us recall how physicists use the assumption that something is random.For example,what can we conclude if we know that the sequence of heads and tails obtained by tossing the coin is random?One thing we can conclude is that the fraction of heads in this sequence tends to 1/2as the number of tosses tends to infinity.What is the traditional argument behind this conclusion?In mathematical statistics,there is a mathematical theorem saying that w.r.t.the natural probability measure on the set of all infinite sequences,for almost all sequences,the frequency of heads tends to1/2.2In more mathematical terms,this means that the probability measureµ(S)of the set S of all sequences for which the frequency does not tend to1/2is0.Because the property P holds for almost all sequencesω,sequences that do not satisfy this property are(in some sense)exceptional.Because we have assumed that a given sequenceωis random,it is,therefore,not exceptional, and hence,this sequenceωmust satisfy the property P.The informal argument that justifies this conclusion goes something like that: ifωdoes not satisfy the property P,this means thatωpossesses some property (not P)that is very rare(is almost never true),and therefore,ωis not truly random.All other existing applications of statistics to physics follow the same pattern: we know that something is true for almost all elements,and we conclude that it is true for an element that is assumed to be random;in this manner,we estimate thefluctuations,apply random processes,etc.So,a random object is an object that satisfies all the properties that are true for almost all objects(almost all with respect to some reasonable probability measure).To give a definition of randomness,we must somewhat reformulate this statement.To every property P that is true almost always,we can put into correspon-dence a set S P of all objects that do not satisfy P;this set has,therefore, measure0.An object satisfies the property P if and only if it does not belong to the set S P.Vice versa,if we have a definable set S of measure0,then the property“not to belong to S”is almost always true.In terms of such sets,we can reformulate the above statement as follows:if an object is random,then it does not belong to any definable set of measure0. So,if an object does not belong to any definable set of measure0,we can thus conclude that it has all the properties that are normally deduced for random objects,and therefore,it can reasonably be called random.Thus,we arrive at the following definition:an object is random if and only if it does not belong to any definable set S of measure0µ(S)=0,with respect to some natural probability measureµ.This,in effect,is the definition proposed by Martin-L¨o f.Example.In the above example,the state of possible orientations can be represented as a unit sphere.The natural probability measure on this sphere is as follows:µ(A)is the area of the set A divided by the total area(4π)of the sphere.3.3What do we mean by“symmetry”?A symmetry is a transformation on the set of all objects.For example,for orientations,typical symmetries are rotations.A symmetry is usually assumed to be invertible and therefore,it must be a one-to-one function.A symmetry must also be non-trivial in the sense that being symmetric must be a very informative property(i.e.,only very few elements must be symmetric).3In other words,almost all elements must not be invariant with respect to a symmetry.It is also natural to assume that a symmetry transformation preserves the a priori(natural)probability measure.This is not absolutely necessary;however, our goal is to prove that every physical law can be expressed in terms of sym-metries.Therefore,if we managed to prove that it can expressed in terms of symmetries that preserve the a priori measure,we have proven what we intended to.Now,we are ready for the formal definitions.4Definitions•Let a language L befixed(e.g.,the language of set theory).We say thata set X is definable if in the language L there is a formula P(Z)with onefree set variable Z that is true for only one object:this set X.•Let U be a set with a probability measureµ.We say that an element u∈U is random w.r.t.µif it does not belong to any definable set of µ−measure0.We say that an element u satisfies some law if it is not random.•By a symmetry S,we mean a1-1,measure preserving,definable mapping S:U→U for whichµ{u|S(u)=u}=0(i.e.,almost always S(u)=u).•A probability space(U,µ)is non-trivial if it has at least one symmetry S0.•We say that an element u is invariant(or symmetric)w.r.t.S if S(u)=u. 5A Simple Physical Example Illustrating the Definitions5.1Example in Physical TermsTo illustrate how the above mathematical definition is related to real physical symmetries,let’s use the following simple physical example:Let us describe the fact that a given spherical molecule has no dipole moment d.5.2Reformulation in Terms of Symmetries:Physicists’ViewpointFrom the physical viewpoint,this fact can be easily reformulated in terms of symmetries:namely,it means that the dipole moment vector d is invariant w.r.t. arbitrary rotations around the center of the molecule.45.3Reformulation in Terms of the Above MathematicalDefinitionIn this case,the set U is clearly the set of all possible3-D vectors.If we choose a coordinate system in such a way that the center of the molecule is at a point 0=(0,0,0),then this set is in1-1correspondence with the3-D space R3.To apply our definition,we must choose a probability measureµon the set U=R3that is invariant w.r.t.all rotations around0.The easiest way to do that is to assume that the components d x,d y,and d z of the vector d are independent Gaussian random variables with one and the same standard deviationσ.The probability densityρ( d)of the resulting measure is equal toconst·exp(−d2x+d2y+d2z2σ2),and is,therefore,invariant w.r.t.rotations.Let us check that our definition of symmetry is satisfied.First,each rotation is measure preserving(this is how we chose a measure). Second,each rotation S around0is a rotation around a line.The set of all points u that are invariant under this rotation S coincides with this line and has,therefore,measure0.Thus,according to our definition,S is a symmetry.Since rotations exist,the probability space(R3,µ)has at least one symmetry and is,therefore,non-trivial in the sense of our definition.The fact that u= d= 0can be now expressed as S(u)=u for all rotations S.6The Main ResultTheorem.An element u of a non-trivial probability space(U,µ)satisfies some law iffu is invariant w.r.t.some symmetry.Comments.•In other words,every physical law can be reformulated in terms of some symmetries.•Another corollary is that we can now reformulate the definition of ran-domness in terms of symmetries:Corollary.An element u of a non-trivial probability space U is random w.r.t.µiffu is not invariant w.r.t.any symmetry.7Proof of the TheoremLet usfirst prove that if a given object u is invariant w.r.t.some symmetry S, then u is not random.Indeed,if u is invariant w.r.t.some symmetry S,then u5belongs to the set of invariant elements of the symmetry S;let us denote this set by I(S).Since S is definable,this set I(S)is also definable,and by definition of a symmetry,this set is of measure0.Therefore,the element u is not random.Let us now prove that if u is not random,then u is invariant w.r.t some symmetry.Indeed,by definition of randomness,the fact that u is not random means that u belongs to a definable set E of measure0.Since the probability space(U,µ)is non-trivial,by definition,it has at least one symmetry S0.Let us now define a symmetry S(u)as follows:•S(v)=v for all v from the set E∪S0(E)∪S20(E)∪...∪S−1(E)∪...,and•S(v)=S0(v)for all other v.It is easy to check that S(u)=u.Let us show that S is a symmetry:•By construction,the function S is1-1.•Outside the set E∪S0(E)∪S20(E)∪...∪S−10(E)∪...,the function Scoincides with the measure-preserving transformation S0.The set E has measure0;since S0is a symmetry and hence,measure-preserving and 1-1,the sets S0(E),...,all have measure0.Thus,the union set is a union of countably many sets of measure0and hence,has a measure0.So, S coincides with S0everywhere except a set of measure0.Since S0is measure-preserving,we can conclude that S is measure-preserving.•An element v is an invariant element of S iffeither v∈E,or v∈S0(E), ...,or S0(v)=v.Thus,I(S)⊆E∪S0(E)∪...∪I(S0).We have alreadyshown that the set E∪S0(E)∪S20(E)∪...∪S−10(E)∪...has measure0.The set I(S0)has a measure0since S0is a symmetry.Therefore,I(S)is contained in the union of the sets of measure0,and is,hence,itself a set of measure0.So,for almost all v,we have S(v)=v.So,S is a symmetry,and thus,u is invariant w.r.t.some symmetry.Q.E.D. Acknowledgments.This work was partially supported by NSF Grants No. EEC-9322370and CCR-9211174,and by NASA Research Grant No.9-757.The authors are thankful to Michael Gelfond and to all the participants of the1995 Structures Conference,especially to Detlef Seese,for the encouragement. References[1]A.M.Finkelstein and V.Kreinovich,“Derivation of Einstein’s,Brans-Dickeand other equations from group considerations,”On Relativity Theory.Pro-ceedings of the Sir Arthur Eddington Centenary Symposium,Nagpur India 1984,Vol.2,Y.Choque-Bruhat and T.M.Karade(eds),World Scientific, Singapore,1985,pp.138–146.6[2]A.M.Finkelstein,V.Kreinovich,and R.R.Zapatrin,“Fundamental phys-ical equations uniquely determined by their symmetry groups,”Lecture Notes in Mathematics,Springer-Verlag,Berlin-Heidelberg-N.Y.,Vol.1214, 1986,pp.159–170.[3]Group theory in physics:proceedings of the international symposium held inhonor of Prof.Marcos Moshinsky,Cocoyoc,Morelos,Mexico,1991,Amer-ican Institute of Physics,N.Y.,1992.[4]V.Kreinovich.“Derivation of the Schroedinger equations from scale in-variance,”Theoretical and Mathematical Physics,1976,Vol.8,No.3,pp.282–285.[5]V.Kreinovich and L.Longpr´e,“Why Are Symmetries a Universal Languageof Physics?A Remark”,1995Structures Conference Research Abstracts, June1995,Abstract No.95–30.[6]M.Li and P.Vit´a nyi,An introduction to Kolmogorov complexity and itsapplications,Springer-Verlag,N.Y.,1993.[7]P.Martin-L¨o f,“The definition of random sequences”,Inform.Control,1966,Vol.9,pp.602–619.[8]P.J.Olver,Equivalence,invariants,and symmetry,Cambridge UniversityPress,Cambridge,N.Y.,1995.[9]Symmetries in physics:proceedings of the international symposium heldin honor of Prof.Marcos Moshinsky,Cocoyoc,Morelos,Mexico,1991, Springer-Verlag,Berlin,N.Y.,1992.Department of Computer ScienceUniversity of Texas at El Paso,El Paso,TX79968{vladik,longpre}@7。

f¨u r Mathematikin den NaturwissenschaftenLeipzigRandom perturbations of spiking activity in apair of coupled neuronsbyBoris Gutkin,J¨u rgen Jost,and Henry TuckwellPreprint no.:492007Random perturbations of spiking activity in apair of coupled neuronsBoris Gutkin∗,J¨u rgen Jost and Henry C.Tuckwell†May14,2007AbstractWe examine the effects of stochastic input currents on thefiring be-haviour of two coupled Type1or Type2neurons.In Hodgkin-Huxleymodel neurons with standard parameters,which are Type2,in the bistableregime,synaptic transmission can initiate oscillatory joint spiking,butwhite noise can terminate it.In Type1cells(models),typified by aquadratic integrate andfire model,synaptic coupling can cause oscilla-tory behaviour in excitatory cells,but Gaussian white noise can againterminate it.We locally determine an approximate basin of attraction,A,of the periodic orbit and explain thefiring behaviour in terms of theeffects of noise on the probability of escape of trajectories from A.1IntroductionHodgkin(1948)found that various squid axon preparations responded in quali-tatively different ways to applied currents.Some preparations gave a frequency offiring which rose smoothly from zero as the current increased whereas oth-ers manifested the sudden appearance of a train of spikes at a particular input current.Cells that responded in thefirst manner were called Class1(which we refer to as Type1)whereas cells with a discontinuous frequency-current curve were called Class2(Type2).Mathematical explanations for the two types are found in the bifurcation which accompanies the transition from rest state to a periodicfiring mode.For Type1behaviour,a resting potential vanishes via a saddle-node bifurcation whereas for Type2behaviour the instability of the rest point is due to an Andronov-Hopf bifurcation,see Rinzel and Ermentrout (1989).Stochastic effects in thefiring behaviour of neurons have been widely reported, discussed and analyzed since their discovery in the1940’s.One of thefirst reports for the central nervous system was by Frank and Fuortes(1955)for catX1X3X2X4X1X2TIMEFigure1:On the left are shown the solutions of(1)-(4)for two coupled QIF model neurons with the standard parameters.X1and X2are the potential variables of neurons1and2and X3and X4are the inputs to neurons1and2, respectively.On the right is shown the periodic orbit in the(x1,x2)-plane.The square marked P was explored in detail in reference to the extent of the basin of attraction of the periodic orbit.spinal neurons.Although there have been many single neuron studies,the effect of noise on systems of coupled neurons have not been extensively investigated. Some preliminary studies are those of Gutkin,Hely and Jost(2004)and Casado and Baltan´a s(2003).2The quadratic integrate andfire modelA relatively simple neural model which exhibits Type1firing behaviour is the quadratic integrate andfire(QIF)model.We couple two model neurons in the following manner(Gutkin,Hely and Jost,2004).Let{X1(t),X2(t),t≥0}be the depolarizations of neurons1and2,where t is the time index.Then the model equations are,for subthreshold states of two identical neurons,dX1=[(X1−x R)2+β+g s X3]dt+σdW1(1)dX2=[(X2−x R)2+β+g s X4]dt+σdW2(2)dX3=−X3τ+F(X1)(4)2where X3is the synaptic input to neuron1from neuron2and X4is the synaptic input to neuron2from neuron1.The quantity x R is a resting value.g s is the coupling strength.βis the mean background input.W1and W2are independent standard Wiener processes which enter with strengthσ.This term may model variations in nonspecific inputs to the circuit as well as possibly intrinsic membrane and channel noise.By construction,we take this term to be much weaker than the mutual coupling between the cells in our circuit.The function F is given byF(x)=1+tanh(α(x−θ))whereθcharacterizes the threshold effect of synaptic activation.Since when a QIF neuron is excited and it receives no inhibition,its potential reaches an infinite value in afinite time,for numerical simulations a cutoffvalue x max is introduced so that the above model equations for the potential apply only if X1 or X2are below x max.To complete a“spike”in any neuron,taken as occurring when its potential reaches x max,its potential is instantaneously reset to some value x reset which may be taken as−x max.At the bifurcation point g s=g∗s, two heteroclinic orbits between unstable rest points turn into a periodic orbit of antiphase oscillations.3Results and theoryIn the numerical work,the following constants are employed throughout.x R= 0,x max=20,θ=10,α=1,β=−1,g s=100andτ=0.25.The initial values of the neural potentials are X1(0)=1.1,X2(0)=0and the initial values of the synaptic variables are X3(0)=X4(0)=0.When there is no noise,σ=0,the results of Figure1are obtained.The spike trains of the two coupled neurons and their synaptic inputs are shown on the left.Thefiring settles down to be quite regular and the periodic orbit,S,is shown on the right.The patch marked P is the location of the region explored in detail below.The effects of a small amount of noise are shown in Figure2.The neural excitation variables are shown on the left and the corresponding trajectories in the(x1,x2)-plane are shown on the right.In the top portion an example of the trajectory forσ=0.1is shown.Here three spikes arise in neuron1and two in neuron2,but the time between spikes increases and eventually the orbit collapses away from the periodic orbit.In the example(lower part)forσ=0.2 there are no spikes in either neuron.In10trials,the average numbers of spikes obtained for the pair of neurons were(2.5,2.2)forσ=0.1,(1.4,1.1)forσ=0.2 and(1.3,0.9)forσ=0.3;these may be compared with(5,5)for zero noise. 3.1Exit-time and orbit stabilityIf a basin of attraction for a periodic orbit can be found,then the probabil-ity that the process with noise escapes from the region of attraction gives the probability,in the present context,that spiking will cease.Since the system3TIMEX1X21 X2Figure2:On the left are shown examples of the neuronal potentials for neurons 1and2(QIF model)for two values of the noise,σ=0.1andσ=0.2.On the right are shown the trajectories corresponding to the results on the left,showing how noise pushes or keeps the trajectories out of the basin of attraction of the periodic orbit.(1)-(4)is Markovian,we may apply standardfirst-exit time theory(Tuckwell, 1989).Letting A be a set in R4and letting x=(x1,x2,x3,x4)∈A be a values of X1,X2,X3,X4)at some given time,the probability p(x1,x2,x3,x4)that the process ever escapes from A is given byL p≡σ2∂x21+σ2∂x22(5)+[(x1−x R)2+β+g s x3]∂p∂x2+ F(x2)−x3∂x3+ F(x1)−x4∂x4=0,x∈Awith boundary condition that p=1on the boundary of A(since the process is continuous).If one also adds an arbitrarily small amount of noise for X3and X4(or considers those solutions of(5)that arise from the limit of vanishing noise for X3,X4),the solution of the linear elliptic partial differential equation (5)is unique and≡1,that is,the process will eventually excape from A with probability1.Hence,the expected time f(x)of exit of the process from A satisfies L f=−1,x∈A with boundary condition f=0on the boundary of A.In fact,for small noise,the logarithm of the expected exit time from A,that4is,the time at whichfiring stops,behaves like the inverse of the square of the noise amplitude(Freidlin and Wentzell,1998).These linear partial differential equations can be solved numerically,for example by Monte-Carlo techniques.The basin of attraction A must be found in order to identify the domain of(5).We have done this approximately for the square P in Figure1.The effects of perturbations of the periodic orbit S within P on the spiking activity were found by solving(1)-(4)with various initial conditions in the absence of noise.The values of x1were from−0.43to1.57in steps of0.2and the values of x2were from-4to2also in steps of0.2.For this particular region, as expected from geometrical considerations,the system responded sensitively to to variations in x1but not x2.For example,to the left of S there tended to be no spiking activity whereas just to the right there was a full complement of spikes and further to the right(but still inside P)one spike.4Coupled Hodgkin-Huxley neuronsAs an example of a Type2neuron,we use the standard Hodgkin-Huxley(HH) model augmented with synaptic input variables as in the model for coupled QIF neurons given by equations(3)and(4),but with different parameter values. It has been long known that additive noise has a facilitative effect on single HH neurons(Yu and Lewis,1989).Coupled pairs of HH neurons have been employed with a different approach using conductance noise in order to analyze synchronization properties(e.g.Casado and Balt´a nas,2003).For the present approach,with X1and X2as the depolarizations of the two cells,we putdX1=1g K n4(V K−X1)+it was found that transient synchronization can terminate sustained activity. For Type2neurons,we have investigated coupled Hodgkin-Huxley neurons and found that in the bistable regime,noise can again terminate sustained spiking activity initiated by synaptic connections.We have investigated a minimal cir-cuit model of sustained neural activity.Such sustained activity in the prefrontal cortex has been proposed as a neural correlate of working memory(Fuster and Alexander,1973).ReferencesCasado,J.M.,Balt´a nas,J.P.(2003).Phase switching in a system of two noisy Hodgkin-Huxley neurons coupled by a diffusive interaction.Phys.Rev.E68,061917,Frank,K.,Fuortes,M.G.(1955).Potentials recorded from the spinal cord with microelectrodes,J.Physiol.130,625-654.Freidlin,M.I.,Wentzell,A.D.(1998),Random Perturbations of Dynamical Sys-tems,2nd ed.,Springer,New York Fuster,J.M.and Alexander,G.E.(1971),Neuron activity related to short-term memory.Science652-654 Gutkin,B.,Ermentrout,G.B.(1998).Dynamics of membrane excitability de-termine interval variability:a link between spike generation mechanismsand cortical spike train statistics.Neural Comp.10,1047-1065. Gutkin,B.S.et al.(2001)Turning on and offwith p.Neurosc.11:2,121-134Gutkin,B.,Hely,T.,Jost,J.(2004).Noise delays onset of sustainedfiring in a minimal model of persistent activity.Neurocomputing58-60,753-760. Hodgkin,A.L.(1948).The local changes associated with repetitive action in a non-medullated axon.J.Physiol.107,165-181.Rinzel,J.,Ermentrout,G.B.(1989).Analysis of neural excitability and oscilla-tions;in:Koch C.&Segev I.,eds.MIT Press.Tateno,T.,Harsch,A.,Robinson,H.P.C.(2004).Thresholdfiring frequency-current relationships of neurons in rat somatosensory cortex:Type1and Type2dynamics.J.Neurophysiol.92,2283-2294.Tuckwell,H.C.(1989).Stochastic Processes in the Neurosciences.SIAM,Philadel-phia.Yu,X.,Lewis,E.R.(1989).Studies with spike initiators:linearization by noise allows continuous signal modulation in neural networks.IEEE Trans.Biomed.Eng.36,36-43.6。

薛定谔方程英语English: The Schrödinger equation, also known as the Schrödinger wave equation, is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is named after the Austrian physicist Erwin Schrödinger, who first formulated it in 1926. The equation is a partial differential equation that describes how the wave function of a physical system evolves over time, and it is central to the understanding of quantum mechanics. The Schrödinger eq uation is used to calculate the probability distribution of a particle in a given region of space, and it has been immensely successful in explaining the behavior of particles at the atomic and subatomic levels.中文翻译: 薛定谔方程，也被称为薛定谔波动方程，是量子力学中的基本方程，描述了一个物理系统的量子状态随时间的变化。

它以奥地利物理学家埃尔温·薛定谔的名字命名，他在1926年首次提出了这个方程。

DOI: 10.11858/gywlxb.20230722固体氢在极端压强下的超导性能杜 昱1,2，孙 莹1,2，王彦超1,2，钟 鑫1,2（1. 吉林大学物质模拟方法与软件教育部重点实验室, 吉林 长春　130012；2. 吉林大学物理学院, 吉林 长春　130012）摘要：氢元素在常压下具有最简单的晶体结构和物理性质。

随着压强增加，氢单质发生相变，由绝缘体转变为金属，被称为金属氢。

数值模拟表明，金属氢具有高温超导电性，因此，金属氢研究也被称为高压物理领域的“圣杯”课题。

利用基于密度泛函理论的第一性原理计算方法，对固体氢在极端高压（0.5～5.0 TPa ）下的结构和超导电性开展了系统研究。

研究结果表明：固体氢的高压相变序列为I 41/amd →oC12→cI16；对于同一种结构，随着压强增加，电声耦合系数减小，费米面处电子态密度减小，特征振动频率增加，超导转变温度发生小幅变化；在2.0 TPa 压强下，固体氢的超导转变温度高达418 K （库伦赝势经验值μ*=0.10）。

研究工作将为金属氢及其超导电性的后续理论和实验研究提供参考。

关键词：金属氢；超高压强；高温超导；相变中图分类号：O521.2 文献标志码：A氢作为宇宙中含量最丰富的元素，位于元素周期表中第一位，其原子核外只有一个电子，因此，氢应该具有最简单的物理性质。

在标准大气压下，氢与其他第一主族元素不同，以宽带隙绝缘分子相存在，这是由于2个氢原子提供2个电子，形成强共价键。

在极端高压条件下，理论预言氢单质发生高压相变，可能具有金属性和高温超导电性。

因此，氢一直是凝聚态物理领域研究的焦点。

金属氢这一概念最早始于1935年，基于J. D. Bernal 关于“任何元素都应该在足够高的压力下变成金属”的观点，美国普林斯顿大学 Wigner 等[1]预言固体氢分子在25 GPa 下会发生解离并形成全氢原子化的具有立方结构的金属固体氢。

随着密度泛函理论等第一性原理计算方法的大规模应用，理论计算精度和计算速度大大提高。

a r X i v :q u a n t -p h /0211040v 1 8 N o v 2002Exact Solutions of the Time Dependent Schr¨o dinger Equation in One SpaceDimension.B.HamprechtInstitut fr Theoretische Physik,Freie Universitt Berlin,Arnimallee 14,D-14195Berlin,Germanye-mails:bodo.hamprecht@physik.fu-berlin.de(May 16th,1997)A closed expression for the harmonic oscillator wave function after the passage of a linear signal with arbitrary time dependence is derived.Transition probabilities are simple to express in terms of Laguerre polynomials.Spontaneous transitions are neglected.The exact result is of some interest for the physics of short laser pulses,since it may serve as an accuracy test for numerical methods.I.CONSTRUCTION OF THE PROPAGATORWe consider the Schr¨o dinger Equation:− 2∂x 2+mω2∂t(1.1)The driving force j (t )is supposed to be real and of finite duration.The propagator of the above equation is well known [1].Ψ(x,t,y )=α2πi sin ωt exp i 2cos ωt −xy+xx 0+yy 02α2 2 (1.2)where:x 0=−G,y 0=G cos ωt −F sin ωt,chi =G 2cos ωt −(F G +2H )sin ωt,α=(1.3)G (t )=tdt ′j (t ′)sin ωt ′,F (t )=tdt ′j (t ′)cos ωt ′H (t )=12+iB (t )αx −C (t )dtA =iω(1−A 2)dαd√τA,B and C have to be solutions of (1.5)with a singularity at t =0:1A(t)=−i cotωtB(t)=−G(t)αsinωtC(t)=log(2πi sinωt)−iα2y2cotωt−i 2y+G +2i√2α2+i ωt m+1m!2−iH√√√2 (−ir∗)n|a n|2=R nα22 dy(3.1) The integral evaluates to:Ψ(x,t)=4 πexp −α2x 0=−i(F +iG )e −iωt2α2 2+1α2p =−(F cos ωt +G sin ωt )(3.3)IV.APPENDIXWe evaluate equation (2.1),using the generating function for the Hermite polynomials.If γm,n is the coefficient of w m z n in the Taylor expansion of:J =α∞−∞Ψ(x,t,y )e 2(wx +zy )α−w2−z 2−x 2+y2n !m !2γm,n (4.1)Now:J =12πi sin t∞ −∞exp −X T AX +2P T X −Q α2dX = 2i sin tdetAexp P T A −1P −Q(4.2)where:X =αxy P =w −iG2α(F −G cot ωt ) A =12cot ωti2sin ωt12cot ωtQ =w 2+z 2+iα2 2HWe find:det A =−ie iωtα−iwF +iG4α2 2(4.4)Therefore:a m,n =2n +mexp−F 2+G 2+4iHαm −k F −iGk !(m −k )!(n −k )!Extracting common factors from the sum and replacing k by m −k in the summation,we obtain:a n.m =√4α2 2 −i F −iG2α n −m m k =0−F 2+G 2k !(m −k )!(n −m +k )!The finite sum in this expression defines a Laguerre polynomial.Therefore:a n.m = n !exp −|r |2α22(−ir ∗)n −m L (n −m )m (|r |2)(4.5)3where:r=F+iG2α=12αtdt′j(t′)e iωt′which proves equation(2.2).。