Ergodic Properties of Infinite Harmonic Crystals an Analytic Approach

Lecture 10-11Schrödinger(薛定谔)equationsPrior to 1925 quantum physics was a “hodgepodge” of hypotheses, principles, theorems and recipes . It was not a logically consistent theory.Once we know this wavefunction we know “everything” about the system!Part 1Dynamic EquationsIf we know the forces acting upon the particle than,according to classical physics , we know everything about a particle at any moment in the future.22,(),()d r F ma F U r U r m dt ==-∇-∇=r r r r r r r r22221()()0E r E r c t∂∇-=∂r r r r A differential equation by itself does not fully determine theunknown function ()(,)r t or E r t r r r .Part 2Dynamic Equation of Wave function---- Schrödinger equations用21()sin cos 2x kx kx ψ⎡⎤=+⎢⎥⎣⎦描述的粒子，只能有5种动量取值，分别是0,2,2,,k k k k -- ，对应的几率分别是11111,,,,28888，这些几率总和应该为1。

()()1111111()0220,2888811111()128888ki i i k i i p P p p k k k k P p ====⋅+⋅+-⋅+⋅+-⋅==++++=∑∑h h h h 1212(),()1(),()1(),()1k ki i i i i x P x x P x x p x xd x x p x dx x x xd x x d ψψ=======⇒=∑∑⎰⎰⎰⎰¡¡¡¡Do we have the same recipe for calculation of average momentum by using wave function in position representation? Yes, of course, we have!To find the expectation (average) value of p , we first need to represent p in terms of x and t . Consider the derivative of the wave function of a free particle with respect to x:0001(,)exp ()p i x t p x E t ⎡⎤ψ=-⎢⎥⎣⎦h We find that0000000000**000(,)(,)(,)(,)(,)()(,)(,)(,)(0)δ∂ψ=ψ⇒ψ=ψ∂∂==ψ-ψ=ψψ=∂∂-∂⎰⎰p p p p p p p p i x t p x t x t p x t x p p x t i x t dx p x t x t dx i xp x　h h h This suggests we define the momentum operator asThe expectation value of the momentum is20022220221()sin cos 2()()()()()=()()()()()ψϕϕϕϕϕ------⎡⎤=+⎢⎥⎣⎦=--++--++k k k k k k k k k k k k x kx kx C x C x C x C x C x x x x x x h h h h h h h h h h h h1()px i p x e ϕ=h ()()111111()022028888===⋅+⋅+-⋅+⋅+-⋅=∑ki i i p P p p k k k k h h h h So,we can not have definite values for the dynamical variables, such as the momentum, when the state of a particle is determined by the wave function with respect to x. We have to find the other way to describe thedynamical variables in Quantum Mechanics.For every dynamical variable or any observable thereis a corresponding Quantum Mechanical OperatorPhysical Quantities →OperatorsOperators are important in quantum mechanics. All observables have corresponding operators.Operators ↔Symbols for mathematical operation✧ The position x is its own operator ˆxx =. Done. Other operators are simpler and just involve multiplication 22x x x x ∧==⋅. ✧ The potential energy operator is just multiplication by V(x).✧ The momentum operator is defined as ˆp i x ∂=-∂h00000002211ˆˆˆˆ1()()[11ˆ(()](())()222ˆ()()())ˆ()()1ˆ()2pp xipp x p xp xix x piix px p pp xix p pp p ppx eipip x i e p eexp x p xx xpT x p p x p p x xpxx eT x T xϕϕϕϕμμμϕϕϕϕϕϕϕμϕ=====⎛⎫∂-=-==⎪∂⎝⎭===hh hhhhh()px xϕ=h000000001(,)exp()1ˆ(,)exp()(,)ˆ(,)(,)pp pp pix t p x E ti iEE x t i p x E t i x ttE x t E x t⎡⎤ψ=-⎢⎥⎣⎦∂⎧⎫⎡⎤⎛⎫ψ=-=ψ-⎬ ⎪⎢⎥∂⎝⎭⎣⎦⎭ψ=ψhh hh h Eigenvalue equation of an operatoreigenvalueDeriving the Schrödinger Equation using operators: This was a plausibility argument, not a derivation. We believe the Schrödinger equation not because of this argument, but because its predictions agree withexperiments.Schrödinger EquationNotes:The Schrödinger Equation is THE fundamental equation of Quantum Mechanics.There are limits to its validity. In this form it applies only to a single, non-relativistic particle (i.e. one withnon-zero rest mass and speed much less than c.)●On the left hand picture 13 velocity vectors of an individual fly are shown; the chain●On the right hand picture the same 13velocity vectors are assigned to 1 fly each todemonstrate that the ensemble average yields the same result, i.e. <v e> = 0,provided that each and every fly does the same thing on average.●i.e. time average = ensemble average. The new subscripts "e" and "r" denote ensemble and space, respectively. This is a simple version of a very far reaching concept in stochastic physics known under the catch word "ergodic hypothesis".●As long as every fly does - on average - the same thing, the vector average overtime of the ensemble is identical to that of an individual fly - if we sum up a fewthousand vectors for one fly, or a few million for lots of flies does not make anydifference. However, we also may obtain this average in a different way:●We do not average one fly in time obtaining <v i>t , but at any given time all flies inspace.●This means, we just add up the velocity vectors of all flies at some moment in timeand obtain <v e>r , the ensemble average. It is evident (but not easy to prove for general cases) thata) Schrödinger equation is a linear homogeneous partialdifferential equation.b) The Schrödinger equation contains the complex number i.Therefore its solutions are essentially complex (unlikeclassical waves, where the use of complex numbers isjust a mathematical convenience.)c) The wave equation has infinite number of solutions,someof which do not correspond to any physical or chemical reality.1. For an electron bound to an atom/molecule, the wavefunction must be everywhere finite, and it must vanish in the boundaries2. Single valued3. Continuous4. Gradient (dψ/dr) must be continuous5. ψψ*dτ is finite, so that ψ can be normalizedd) Solutions that do not satisfy these properties(above)DONOT generally correspond to physicallyrealizable circumstances.e) Conditions on the wave function(波函数的三个基本条件——有限、单值、连续)1. In order to avoid infinite probabilities, the wave functionmust be finite everywhere.2. The wave function must be single valued.3. The wave function must be twice differentiable. Thismeans that it and its derivative must be continuous. (An exception to this rule occurs when V is infinite.)4. In order to normalize a wave function, it must approachzero as x approaches infinity.f) Only the physically measurable quantities must be real.These include the probability, momentum and energy.Can think of the LHS of the Schrödinger equation as a differentialoperator that represents the energy of the particle ?This operator is called the Hamiltonian of the particle , and usually given the symbolˆH.Hamiltonian is a linear differential operator .222ˆ(,)2d V x t H m dx ⎡⎤-+ψ≡ψ⎢⎥⎣⎦h Kineticenergy Potential energyHence there is an alternative (shorthand) form for thetime-dependent Schrödinger equation:Part 3Time-independent Schrödinger equation (TISE), i.e.stationary state(定态)Schrödinger equationSuppose potential is independent of time(),()U x t U x =Look for a separated solution, substitute (,)()()x t x T t ψψ=into• This only tells us that T(t) depends on the energy E . It doesn’t tell us what the energy actually is. For that we have to solve the space part.• T(t) does not depend explicitly on the potential U(x). But there is an implicit dependencebecause the potential affectsthe possible values for the energy E .This is the time-independent Schrödinger equation (TISE) or so-called stationary state Schrödinger equation.Solution to full TDSE isEven though the potential is independent of time the wavefunction still oscillates in time . But probability distribution is static()()2*//2*,,()()()()()iEt iEt P x t x t x e x e x x x ψψψψψ+-=ψ===h hFor this reason a solution of the TISE is known as a StationaryState(定态)Stationary state Schrödinger Equation Notes:• In one-dimension space, the TISE is an ordinary differential equation (not a partial differential equation)• The TISE is an eigenvalue equation for the Hamiltonianoperator:ˆ()()Hx E x ψψ=Part 4 Probability current density and continuity equation Definition of probability current densityIn non-relativistic quantum mechanics, the probability current of the wave function Ψ is defined asin the position basis and satisfies the quantum mechanical continuity equationwith the probability density defined as.If one were to integrate both sides of the continuity equation with respect to volume, so thatthen the divergence theorem implies the continuity equation is equivalent to the integral equationwhere the V is any volume and S is the boundary of V. This is the conservation law for probability in quantum mechanics.In particular, if is a wavefunction describing a single particle, theintegral in the first term of the preceding equation (without the time derivative) is the probability of obtaining a value within V when the position of the particle is measured. The second term is then the rate at which probability is flowing out of the volume V. Altogether the equation states that the time derivative of the change of the probability of the particle being measured in V is equal to the rate at which probability flows into V. Derivation of continuity equationThe continuity equation is derived from the definition of probability current and the basic principles of quantum mechanics. Suppose is the wavefunction for a single particle in the positionbasis (i.e. is a function of x, y, and z). Thenis the probability that a measurement of the particle's position will yield a value within V. The time derivative of this iswhere the last equality follows from the product rule and the fact that the shape of V is presumed to be independent of time (i.e. the time derivative can be moved through the integral). In order to simplify this further, consider the time dependent Schrödinger equationand use it to solve for the time derivative of :When substituted back into the preceding equation for this gives.Now from the product rule for the divergence operatorand since the first and third terms cancel:If we now recall the expression for P and note that the argumentof the divergence operator is justthis becomeswhich is the integral form of the continuity equation .The differential form follows from the fact that the preceding equation holds for all V, and as the integrand is a continuousfunction of space, it must vanish everywhere:For all whole space we have()()2lim lim 0lim lim 0V V V V V S V S dV j dV t j dV j ds →∞→∞→→∞→∞⎛⎫∂ψ ⎪=-∇⋅= ⎪∂⎝⎭∇⋅=⋅=⎰⎰⎰⎰⎰⎰⎰⎰⎰⎰⎰r r r r rwhich meansthat must be continuousat any positionin the whole space.So the wavefunction and its derivative must be continuous.(An exception to this rule occurs when V is infinite.)One more, if the(,)()Et i x t x e ϕ-ψ=hand ()x ϕis real, the probability current*Im ()()0j x x m ϕϕ⎡⎤=∇≡⎣⎦r r h over the whole 1D space which means j r is always continuous whatever the wavefunction ()x ϕand its derivative ()x ϕ'are continuous or not. However, ()x ϕhas to be continuous for an acceptable physical solution for that the probability density is uniquely defined(唯一确定). As to ()x ϕ', it may not be continuous especially at the point where the potential energy is infinite.It is easy to prove that ()x ϕ' has to be continuous at the point 0x where the potential energy just has a limited high step.Have a fun!。

万物与虚无名句英文英文回答：All and Nothing: A Philosophical Exploration of Existence and Void.In the realm of philosophy, the concepts of "all" and "nothing" have captivated thinkers for centuries. These two seemingly contradictory notions challenge our understanding of reality and existence itself.The notion of "all" implies a totality, a comprehensive embrace of everything that exists. It suggests a boundless realm where every entity, from the tiniest particle to the vastest galaxy, finds its place. However, this all-encompassing perspective raises profound questions about the nature of limits and the boundaries of the universe.On the other side of the philosophical spectrum lies the concept of "nothing." It evokes a void, an absence ofexistence, a realm devoid of any substance or property. The idea of nothingness challenges our preconceptions about the nature of space and time, as it implies an unfathomable void that defies our understanding of reality.The interplay between "all" and "nothing" has ignited countless philosophical debates throughout history. Some thinkers, such as the ancient Greek philosopher Parmenides, argued for the primacy of "all," asserting that existenceis fundamentally indivisible and unchanging. Others, like the Buddhist philosopher Nagarjuna, embraced the concept of "nothingness," positing that the true nature of reality is an empty void beyond all conceptualizations.In contemporary philosophy, the discussion of "all" and "nothing" continues to engage thinkers in diverse fields. For instance, in metaphysics, the question of the "all" relates to the nature of the universe and its possible finitude or infinitude. In epistemology, the concept of "nothing" challenges our understanding of knowledge and the limits of human perception.Exploring the interplay between "all" and "nothing" invites us to confront the fundamental questions of existence. It encourages us to ponder the nature of reality, the limits of our knowledge, and the enigmatic void thatlies beyond our comprehension. By delving into this philosophical realm, we embark on a journey of intellectual exploration and self-discovery, seeking a deeper understanding of the cosmos and our place within it.中文回答：万物与虚无，关于存在与空性的哲学探讨。

Harmony is the essence of a thriving ecosystem,and it is a concept that has been valued across various cultures and societies for centuries.The idea of a harmonious coexistence with nature is not just a romantic notion but a practical necessity for the sustainability of our planet.Here is an essay that delves into the importance of a harmonious coecology and how it can be achieved.Title:The Symphony of a Harmonious CoEcologyIn the grand orchestra of life,every creature,every plant,and every ecosystem plays a unique melody that contributes to the overall harmony of our planet.A harmonious coecology is one where all components of the environment work in unison,respecting each others roles and spaces,and ensuring the balance of life is maintained.The Importance of BiodiversityBiodiversity is the foundation of a harmonious coecology.It encompasses the variety of life forms,genes,and ecosystems that make up our planet.Each species plays a specific role in the ecosystem,contributing to its health and resilience.For instance,bees pollinate plants,ensuring the growth of crops and the continuation of plant life.The loss of any species can disrupt this delicate balance,leading to a cascade of negative effects.Sustainable PracticesTo foster a harmonious coecology,sustainable practices must be adopted.This includes responsible farming techniques that preserve soil health,reduce pesticide use,and promote biodiversity.Urban planning that incorporates green spaces,such as parks and community gardens,also plays a crucial role in maintaining ecological balance.These spaces not only provide habitats for wildlife but also improve air quality and offer recreational areas for humans,enhancing the overall wellbeing of the community.Conservation EffortsConservation is a vital component of a harmonious coecology.Protecting endangered species and their habitats,as well as restoring damaged ecosystems,are essential steps in preserving the natural world.This can be achieved through the establishment of protected areas,reforestation initiatives,and the implementation of policies that discourage harmful human activities.Education and AwarenessRaising awareness about the importance of a harmonious coecology is cation plays a significant role in this,as it equips individuals with the knowledge and understanding needed to make informed decisions that benefit the environment.By teaching the next generation about the interconnectedness of all life and the consequences of disrupting this balance,we can inspire a sense of stewardship and responsibility towards the planet.Technological AdvancementsTechnology can also contribute to a harmonious coecology.Innovations in renewable energy,waste management,and environmental monitoring can help reduce human impact on the environment.For example,solar panels and wind turbines generate clean energy, reducing reliance on fossil fuels and lowering greenhouse gas emissions.Similarly,smart waste management systems can minimize pollution and promote recycling,contributing to a cleaner environment.ConclusionA harmonious coecology is not just a dream but a necessity for the survival of our species and the countless others that share this planet with us.It requires a collective effort, combining sustainable practices,conservation,education,and technology.By working together,we can ensure that the symphony of life continues to play,with each note contributing to the rich tapestry of our world.In conclusion,embracing a harmonious coecology is a step towards a future where humans and nature coexist in a balanced and respectful manner.It is a vision that demands our attention,our action,and our commitment to preserving the beauty and diversity of our planet for generations to come.。

True HarmonyIn the intricate tapestry of human existence, true harmony is the vibrant thread that weaves through the fabric of our lives. It is the harmonious blend of diverse perspectives, the collective power of cooperation, and the unwavering commitment to a common purpose. As the ancient philosopher Plato once observed, "The beginning of wisdom is the definition of terms." True harmony is the wisdom that arises from the clear understanding of our interconnectedness and the shared goal of creating a better world.The concept of true harmony has been a recurring theme in literature and philosophy throughout history. It is the driving force behind the enduring love stories, the inspiration that compels individuals to sacrifice for the well-being of others, and the motivation that fuels the quest for a more harmonious society. In the annals of literature, J.R.R. Tolkien's "The Lord of the Rings" delves into the transformative power of true harmony, illustrating the profound impact it can have on the lives of individuals and communities.True harmony is not merely a passive sentiment; it is an active force that compels us to engage with the world, to challenge the status quo, and to strive for a better future. It is the fire that ignites our passions, the spark that ignites our souls. As the poet Robert Frost wrote, "The only way out is through." True harmony is the way through, the path that leads us from darkness to light, from despair to hope. One of the most compelling examples of true harmony is the story of the United Nations. Founded in the aftermath of World War II, the UN was established with the goal of promoting peace, security, and cooperation among nations. Through its various agencies and initiatives, the UN has worked tirelessly to address global challenges such as poverty, disease, and climate change. The true harmony of its member states has enabled the UN to make significant strides in fostering a more peaceful and prosperous world.In the realm of personal growth, true harmony plays a pivotal role. It is the motivational force that drives us to become better versions of ourselves, to push beyond our comfort zones and embrace change. The true harmony for a common goal is at the heart of many social movements, as seen in the fight for civil rights, gender equality, and environmental conservation.The structure of this essay mirrors the journey of true harmony. It begins with an exploration of its philosophical underpinnings, moves on to its manifestations in real-life achievements and personal growth, and concludes with a reflection on its significance in shaping our lives and the world around us.In conclusion, true harmony is the indomitable force that animates our existence, the unyielding commitment that drives us to achieve greatness.As we navigate the complexities of life, let us harness the power of true harmony, for it is the essence of our humanity, the touchstone of our emotions, and the compass that guides us toward love and connection. As the poet Rilke implores, "Let everything happen to you: beauty and terror. Just keep going. No feeling is too weak, and none too strong, to be worth your attention." May we all embrace the true harmony that makes us vulnerable and, in doing so, find the strength to love deeply and live fully.。