Physics
Hamiltonian Density
The Hamiltonian density is a concept in physics that represents the energy density of a system. It is used in the field of quantum field theory to describe the dynamics of fields and particles. The Hamiltonian density is a key quantity in understanding the behavior of physical systems at a fundamental level.
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5 Key excerpts on "Hamiltonian Density"
eBook - PDF- Barton Zwiebach(Author)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
The Hamiltonian Density is then con-structed as H = ∂ 0 φ − L . (10.9) 197 10.3 Classical plane-wave solutions Quick calculation 10.1 Show that the Hamiltonian Density takes the form H = 1 2 2 + 1 2 ( ∇ φ) 2 + 1 2 m 2 φ 2 . (10.10) The three terms in H are identified as T , V , and V , respectively. This is what we expected physically for the energy density. The total energy E is given by the Hamiltonian H , which in turn, is the spatial integral of the Hamiltonian Density H : E = H = d d x 1 2 ∂ 0 φ ∂ 0 φ + 1 2 ( ∇ φ) 2 + 1 2 m 2 φ 2 . (10.11) To find the equations of motion from the action ( 10.7 ), we consider a variation δφ of the field and set the variation of the action equal to zero. After discarding a total derivative we find δ S = d D x − η μν ∂ μ (δφ)∂ ν φ − m 2 φδφ = d D x δφ η μν ∂ μ ∂ ν φ − m 2 φ = 0 . (10.12) The equation of motion for φ is therefore η μν ∂ μ ∂ ν φ − m 2 φ = 0 . (10.13) If we define ∂ 2 ≡ η μν ∂ μ ∂ ν , then we have (∂ 2 − m 2 ) φ = 0 . (10.14) Separating out time and space derivatives, this equation is recognized as the Klein–Gordon equation: − ∂ 2 φ ∂ t 2 + ∇ 2 φ − m 2 φ = 0 . (10.15) We will now study some classical solutions of this equation. 10.3 Classical plane-wave solutions We can find plane-wave solutions to the classical scalar field equation ( 10.15 ). Consider, for example, the expression φ( t , x ) = a e − i Et + i p · x , (10.16) where a and E are constants and p is an arbitrary vector. The field equation ( 10.15 ) fixes the possible values of E in terms of p and m : E 2 − p 2 − m 2 = 0 −→ E = ± E p , E p ≡ p 2 + m 2 . (10.17) 198 Light-cone fields and particles The square root is chosen to be positive, so E p > 0. There is a small problem with the solution in ( 10.16 ). While φ is a real field, the solution ( 10.16 ) is not real. To make it real, we just add to it its complex conjugate: φ( t , x ) = a e − i E p t + i p · x + a ∗ e i E p t − i p · x . (10.18) This solution depends on the complex number a .
eBook - PDF- J. Mulak, Z. Gajek(Authors)
- 2000(Publication Date)
- Elsevier Science(Publisher)
The effectiveness of the theory is illustrated by some examples of recent numerical results reported in the literature and, finally, some short- comings and open questions are outlined. Apart from the mentioned above monographs and numerous original papers, we have found very helpful dur- ing preparation of this chapter theses by Richter [238] and Trygg [239]. 16.1 Electron density as a key variable To start with we rewrite the general Hamiltonian of a system of N in- teracting electrons moving in electrostatic potential of nuclei in a solid in the Born-Oppenheimer approximation (atomic units are used throughout 185 186 this chapter). where /;/- j + 5r + I7, (16.1) N N 2 T- E ii - E Vi (162) 2 ~ / i 1 ;~, 1 (16 3) 0 - .... - f;- ~ v(ri) - ~ Iri- Rtl (16.4) i i The potential V in Eq.16.4 may be specified to be the Coulomb poten- tial due to the point nuclei but the general theory admits a wide class of arbitrary multiplicative single-particle operators. Leaving imprecise V fa- cilitates presentation of the theory. Thus we stay with the general Hamiltonian Eqs 16.1-16.4 describing a large variety of the electronic systems. It is noteworthy that only the num- ber of electrons N and v(r) are needed to fix it completely and to deter- mine all the electronic properties related to the ground and excited states. In the traditional quantum mechanics the many-electron wave function is the quantity that describes the state of the system. The density functional theory provides an alternative, and complementary, approach. In this re- formulation of wave mechanics, the density of the electrons is a fundamen- tal quantity. The complicated N-electron wave function ~(xl,x2,...,XN) (where xi comprises space coordinate ri and spin coordinate ~ri) defined in the 3N-dimensional real space and the N-dimensional spin space is replaced with much simpler quantity n(r) defined in merely the three-dimensional real space.
- Abhijit Chatterjee(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
Another interesting aspect of this wave of representing matter waves is that although massive matter waves can propagate with any speed between 0 and c , noninclusive, nevertheless, the five‑dimensional wave that represents it must always propagate with the same characteristic velocity, like a massless particle. Further work on the physical details of five‑optics was also carried out by Rumer [10], as well. Of course, Klein, along with Kaluza, Jordan, and Thirry, examined the possibility that one could unify the theories of gravitation and electromagnetism by means of the geometry of five‑dimensional Lorentzian manifolds [11, 12]. 3.2 DENSITY MATRIX In quantum mechanics, a density matrix is a self‑adjoin (or Hermitian) positive, semidefinite matrix (possibly infinite dimensional) of trace one that describes the statistical state of a quantum system. The formalism was introduced by John von Neumann (and according to other sources, independently by Lev Landau and Felix Bloch) in 1927. The details have been described elsewhere [13]. It is the quantum mechanical analogue to a phase‑space probability measure (probability distribution of position and momentum) in classical statistical mechanics. The 60 Structure Property Correlations for Nanoporous Materials need for a statistical description via density matrices arises when one consid‑ ers either an ensemble of systems or one system when its preparation history is uncertain and one does not know with 100% certainty which pure quantum state the system is in.
eBook - PDFApproaches to Quantum Gravity
Toward a New Understanding of Space, Time and Matter
- Daniele Oriti(Author)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
The entire Hamiltonian is conserved, but the Hamilton densities, or the partial Hamiltonians, are not, and interacting parts could easily mix positive energy states with negative energy states. Deter- ministic quantum mechanics will only be useful if systems can be found where all states in which parts occur with negative energy, can also be projected out. The subset of Hilbert space where all bits and pieces only carry positive energy is only a very tiny section of the entire Hilbert space, and we will have to demonstrate 20 G. ’t Hooft that a theory exists where this sector evolves all by itself, even in the presence of non-trivial interactions. What kind of mechanism can it be that greatly reduces the set of physical states? It is here that our self-imposed restriction to have strictly deterministic Hamilton equations may now bear fruit. In a deterministic system, we may have information loss. In a quantum world, reducing the dimensionality of Hilbert space would lead to loss of unitarity, but in a deterministic world there is no logical impediment that forbids the possibility that two different initial states may both evolve into the same final state. This gives us a new view on what was once introduced as the ‘holographic prin- ciple’. According to this principle, the number of independent physical variables in a given volume actually scales with the surface area rather than the volume. This may mean that, in every volume element, information concerning the interior dis- sipates away due to information loss, while only the information located on the surface survives, possibly because it stays in contact with the outside world. Information loss forces us to assemble physical states in ‘equivalence classes’. Two states are in the same equivalence classes if, in due time, they eventually evolve into the same final state.
- Sow-Hsin Chen, Michael Kotlarchyk;;;(Authors)
- 2007(Publication Date)
- WSPC(Publisher)
Chapter 7 THE DENSITY OPERATOR AND ITS ROLE IN QUANTUM STATISTICS Up to this point, we have considered quantum systems where it is assumed that complete knowledge of the state vector exists at some instant of time. Quite often this is not the case; instead one may only have information related to certain macroscopic or thermodynamic properties such as temperature or average energy. Thesesituations can be handled by introducing a so-called density operator into the framework of quantum mechanics. The action of this operator is to assign appropriate probabilities to the occupancy of available quantum states. We illustrate how to determine the density operator of a system by means of the principle of maximum entropy . Aspects of the technique are applied to a few examples such as radiation in a laser cavity both below and above threshold. We conclude this chapter by performing a perturbation expansion of the density operator. Transitions caused by a random perturbation are investigated. 7.1 Mixed States and the Density Operator If a system is in a pure quantum state | ψ angbracketright and one measures a physical observable represented by a Hermitian operator ˆ A , then the probability of obtaining eigenvalue a for the result is P ( a )= |angbracketleft a | ψ angbracketright| 2 . A large number of measurements of the variable A performed on an ensemble of systems, each identically prepared in the state | ψ angbracketright , produces the average result angbracketleft A angbracketright = angbracketleft ψ | ˆ A | ψ angbracketright . However, now consider a situation where one’s a priori knowledge of the system’s quantum state is less than perfect. Suppose, instead, that only the likelihood of being in certain pure states is known. In this case, one says that the system is in a mixed state –a member of the ensemble has a certain probability P ( ψ ) of being in pure state | ψ angbracketright .
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