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What is variational approximation method in quantum mechanics?
Wiki User
∙ 11y ago
When dealing with certain quantum systems, an absolutely quantitative and accurate description of the system is impossible and requires physicists and chemists to make approximations. For example, one may calculate the Hamiltonian of a single hydrogen atom or a molecule of diatomic helium with a single electron (after invoking the Born-Oppenheimer Approximation of course), but cannot solve a multi-electron problem such as benzene.
Although we cannot calculate the Hamiltonian for benzene, we can approximate it and receive an answer which is very close (and according to the Variation Principal, higher than) the actual energy (Hamiltonian).
One way that computational chemists do this is by using variational approximations. One of these which is most popular is the Hartree-Fock method. Here, chemists say that there exists a ground state wavefunction which describes the benzene system that may be approximated by a single Slater Determinant. We chose a candidate wavefunction which we think suits the system (think e^ikx for SHOs) and which depends on a set of parameters. We then calculate the Hamiltonian for sets of parameters and find the lowest energy. This is a gross oversimplification, but the idea holds.
A simpler way to think about this would be: "What is the shape of a rope tied to a bucket of water?"
We could answer this question by starting with an equation for the rope in 2 dimensions, calculate the potential energy of the bucket as the rope changes coordinates, and eventually find that it's potential energy is minimized when the rope extends completely along the y axis.
Variational approximations work quite the same way for quantum systems where, due to the entangled nature of quantized particles (such as fermions or bosons) we cannot derive an exact answer.
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∙ 11y ago
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Write the calculation of second excited state of Simple harmonic oscillator by variational method?
For help with solving quantum mechanics homework problems Google "physics forums". Providing an answer to this question will yield no value to the community and the answer so long that I would have spend a too long writing it. To help you get started; use the corresponding normalized |psi> (Dirac notation), build the Hamiltonian for the SHO then find the expectation value of the Hamiltonian.
In what way is quantum mechanics and NLP similar?
There is a certain level of similarity between NLP & QMNLP = Neuro-linguistic programming: This is a method invented in the 1970's by a linguist, physiologist, and anthropologist. It is a method used to change behavior, i.e. develop good habits and eliminate negative ones. It relies on the assumption that the intuition of the surrounding universe is a system of beliefs, inside which we act and that we can call universe. But every one has it own system of beliefs. Communication problems can be generated by the differences between those system of beliefsQuantum Mechanics is a mathematically rigorous, and extremely successful theory used to explain and predict behavior of extremely small (subatomic) particles.It relies on the assumption that observations of the surrounding universe is organized in a set of operators called the observables. The fact that those operators do not commute creates fuzziness in classical simultaneous observations. Following the leading interpretation, (the Copenhagen interpretation) a measure is an interaction between a macroscopic entity (the observer) and the microscopic system that decides the result of the observation. Therefore, the observer plays a fundamental role in the construction of the universe, therefore the universe is linked to the existence of observersWe see here a possible similarity. The systems of beliefs of NLP can be considered as set of operators of the quantum mechanics. They create an illusion of surrounding universe and they are linked to designated actors (observers). The fuzziness following the non commutation of operators in quantum mechanics is analogous to discrepancies in the meaning (interpretation) of facts insides different system of beliefs of NLP
How long does it take from Trinidad to haiti?
a long time more than 10 hrs ... That depends on Method of Transportation really... it can take less then a sec. If quantum leaped there.what??
How would you know the hypothesis for growing plants?
this: where the amplitude of the wave function is large. After the measurement is performed, having obtained some result x, the wave function collapses into a position eigenstate centered at x. The time evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian, the operator corresponding to the total energy of the system, generates time evolution. The time evolution of wave functions is deterministic in the sense that, given a wavefunction at an initial time, it makes a definite prediction of what the wavefunction will be at any later time. During a measurement, on the other hand, the change of the wavefunction into another one is not deterministic, but rather unpredictable, i.e., random. A time-evolution simulation can be seen here. Wave functions can change as time progresses. An equation known as the Schrödinger equation describes how wave functions change in time, a role similar to Newton's second law in classical mechanics. The Schrödinger equation, applied to the aforementioned example of the free particle, predicts that the center of a wave packet will move through space at a constant velocity, like a classical particle with no forces acting on it. However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain. This also has the effect of turning position eigenstates (which can be thought of as infinitely sharp wave packets) into broadened wave packets that are no longer position eigenstates. Some wave functions produce probability distributions that are constant, or independent of time, such as when in a stationary state of constant energy, time drops out of the absolute square of the wave function. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucleus. The Schrödinger equation acts on the entire probability amplitude, not merely its absolute value. Whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states. This gives rise to the wave-like behavior of quantum states. It turns out that analytic solutions of Schrödinger's equation are only available for a small number of model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the hydrogen molecular ion and the hydrogen atom are the most important representatives. Even the helium atom, which contains just one more electron than hydrogen, defies all attempts at a fully analytic treatment. There exist several techniques for generating approximate solutions. For instance, in the method known as perturbation theory one uses the analytic results for a simple quantum mechanical model to generate results for a more complicated model related to the simple model by, for example, the addition of a weak potential energy. Another method is the "semi-classical equation of motion" approach, which applies to systems for which quantum mechanics produces weak deviations from classical behavior. The deviations can be calculated based on the classical motion. This approach is important for the field of quantum chaos. There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly used formulations is the transformation theory proposed by Cambridge theoretical physicist Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics, matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger).In this formulation, the instantaneous state of a quantum system encodes the probabilities of its measurable properties, or "observables". Examples of observables include energy, position, momentum, and angular momentum. Observables can be either continuous (e.g., the position of a particle) or discrete (e.g., the energy of an electron bound to a hydrogen atom). An alternative formulation of quantum mechanics is Feynman's path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over histories between initial and final states; this is the quantum-mechanical counterpart of action principles in classical mechanics. cheers!
What are the factors determining quantum of communication?
Communication medium is often decided by what is available. If all communication methods are available, determining factors would include price, quality, and relationship to the media method presentation required.
5 famous scientist in mechanics?
Osborne Reynolds - Fluid Mechanics Isaac Newton - Vector Mechanics/ Gravitational Physics Gallileo Gallilei - Gravitational Physics Erwin Schrodinger - Quantum Mechanics WIlliam Hamilton - LaGrangian Method Mechanics
What has the author H R Coish written?
H. R. Coish has written: 'Application of the factorization method to problems of quantum mechanics'
Who developed Vogel's approximation method?
Vogel's approximation method was developed by William R. Vogel.
Write the calculation of second excited state of Simple harmonic oscillator by variational method?
For help with solving quantum mechanics homework problems Google "physics forums". Providing an answer to this question will yield no value to the community and the answer so long that I would have spend a too long writing it. To help you get started; use the corresponding normalized |psi> (Dirac notation), build the Hamiltonian for the SHO then find the expectation value of the Hamiltonian.
What is ment by MODI method?
Vogel's Approximation method(verification
Why Vogel Approximation Method gives only feasible solution?
cuz its only an approximation afterall
What has the author C J Umrigar written?
C. J. Umrigar has written: 'Accelerated variational Monte Carlo method for electronic structure simulation'
What has the author J Tinsley Oden written?
J. Tinsley Oden has written: 'Final report on adaptive computational methods for SSME internal flow analysis' 'A Posterori Error Estimation in Finite Element Analysis' 'Vectorizable algorithms for adaptive schemes for rapid analysis of SSME flows' -- subject- s -: Algorithms, Vector analysis 'Variational methods in theoretical mechanics' -- subject- s -: Mechanics, Calculus of variations, Continuum mechanics '[Analysis and development of finite element methods for the study of nonlinear thermomechanical behavior of structural components]' -- subject- s -: Computational fluid dynamics, Nonlinear systems, Boundary value problems, Structural members, Approximation, Thermodynamics, Finite element method, Matrix methods, Numerical stability, Numercial integration 'An introduction to mathematical modeling' -- subject- s -: Analytic Mechanics, MATHEMATICS / General 'Applied functional analysis' -- subject- s -: Functional analysis, Engineering mathematics 'Finite Elements'
Does Islam support meditation using the quantum method taught by the Quantum Foundation in Bangladesh?
Please read following article about this Quantum method and I believe the things will be clear to you. Quantum method is meditation is trying to put us in different direction beside shariah. We Ask Allah to help you InshaAllah and Guide you to the Straight Path Ameen We Make Dua and no we dont use Meditation
Did Einstein use the scientific method when creating the quantum theory?
yes he did
What is the differene between analytical and numerical differentiation?
numerical method 1:numerical method uses finite difference or finite element method approximation to solve differential equation 2:give just approximation of the perfect solution analytical method 1:does not uses finite difference 2:give theoreticaly perfect solution.
What role does common sense play in the scientific method?
Observation is the process of gathering objective data, and inference is the process of making some decisions about what the data...
