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The first-order change in the n-th energy eigenket has a contribution from each of the energy eigenstates k ≠ n. Each term is proportional to the matrix element, which is a measure of how much the perturbation mixes eigenstate n with eigenstate k ; it is also inversely proportional to the energy difference between eigenstates k and n, which means that the perturbation deforms the eigenstate to a greater extent if there are more eigenstates at nearby energies.
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first-order and change
The various solid / liquid / gas transitions are classified as first-order transitions because they involve a discontinuous change in density, which is the first derivative of the free energy with respect to chemical potential.
Other all-pole second-order filters may roll off at different rates initially depending on their Q factor, but approach the same final rate of 12 dB per octave ; as with the first-order filters, zeroes in the transfer function can change the high-frequency asymptote.
Such mechanisms, where first-order change successfully produces desired results, are called first-order mechanisms.
Similarly, in social theory, a single direct action producing a social change is a first-order control system.
( 1974 ) differentiated between first-order and second-order change and how second-order change is often the focus of community psychology.
* first-order change: changing the individuals in a setting to attempt to fix a problem
A first-order change to " fix " homelessness would be to offer shelter to one or many homeless people.
The resulting first-order change in the Lagrangian is
Then, to a first-order approximation, the change in caused by a small change in must satisfy:
This pure motion perception is referred to as " first-order " motion perception and is mediated by relatively simple " motion sensors " in the visual system, that have evolved to detect a change in luminance at one point on the retina and correlate it with a change in luminance at a neighbouring point on the retina after a short delay.
Linear predictive analysis is a simple form of first-order extrapolation: if it has been changing at this rate then it will probably continue to change at approximately the same rate, at least in the short term.
The volume change at the transition point is either discrete ( as in a first-order Ehrenfest transition ) or continuous ( second order Ehrenfest analogy ), depending on the degree of ionization of the gel and on the solvent composition.
A change of variables is made to transform into a linear first-order differential equation.
first-order and n-th
Similarly, the set of true formulas of the standard model of second order arithmetic ( or n-th order arithmetic for any n ) can be defined by a formula in first-order ZFC.
first-order and energy
The first-order energy is the Hartree – Fock energy and electron correlation is included at second-order or higher.
This leads to the first-order energy shift:
To obtain the first-order correction to the energy eigenstate, we insert our expression for the first-order energy correction back into the result shown above of equating the first-order coefficients of λ.
The first-order energy shift is not well defined, since there is no unique way to choose a basis of eigenstates for the unperturbed system.
simultaneously for all the degenerate eigenstates, where are first-order corrections to the degenerate energy levels.
1 ( as is often the case for electronic states of molecules ) the first-order energy becomes proportional to the expectation ( average ) value of the dipole operator,
Transition state structures can be determined by searching for first-order saddle points on the potential energy surface ( PES ).
The factor comes about during the calculation of the first-order perturbation in the energy of an atom when a weak uniform magnetic field ( that is, weak in comparison to the system's internal magnetic field ) is applied to the system.
Since the first-order MP energy
This result is the Møller – Plesset theorem: the correlation potential does not contribute in first-order to the exact electronic energy.
Equivalent expressions are obtained by a slightly different partitioning of the Hamiltonian, which results in a different division of energy terms over zeroth-and first-order contributions, while for second-and higher-order energy corrections the two partitionings give identical results.
The zeroth plus first-order correction yields the Hartree – Fock energy.
first-order and has
** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.
Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic.
No first-order theory, however, has the strength to describe fully and categorically structures with an infinite domain, such as the natural numbers or the real line.
In Gödel's paper, he uses a version of first-order predicate calculus which has no function or constant symbols to begin with.
Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is declarative: the program logic is expressed in terms of relations, represented as facts and rules.
Each real set, function, and relation has its natural hyperreal extension, satisfying the same first-order properties.
It has also been shown by L. Berman in 1980 that the problem of verifying / falsifying any first-order statement about real numbers that involves only addition and comparison ( but no multiplication ) is in EXPSPACE.
If the transfer function of a first-order low-pass filter has a zero as well as a pole, the Bode plot will flatten out again, at some maximum attenuation of high frequencies ; such an effect is caused for example by a little bit of the input leaking around the one-pole filter ; this one-pole – one-zero filter is still a first-order low-pass.
A first-order lens has a focal length of 920 mm ( 36 in ) and an optical area 2590 mm ( 8. 5 ft ) high.
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model.
Proof: Fix a first-order language L, and let Σ be a collection of L-sentences such that every finite subcollection of L-sentences, i ⊆ Σ of it has a model.
# A real closed field has all the first-order properties of the real number system, regardless of whether they are usually taken as axiomatic, for statements involving the basic ordered-field relations +, *, and ≤.
For an instrument whose payment stream is described by f ( t ), the value V ( t ) satisfies the inhomogeneous first-order ODE (" inhomogeneous " is because one has f rather than 0, and " first-order " is because one has first derivatives but no higher derivatives ) – this encodes the fact that when any cash flow occurs, the value of the instrument changes by the value of the cash flow ( if you receive a $ 10 coupon, the remaining value decreases by exactly $ 10 ).
It is more expressive than propositional logic but has more efficient decision problems than first-order predicate logic.
For example, while it was once assumed that catagenetic processes were first-order reactions, some research has shown that this may not be the case.
There are no first-order administrative divisions as defined by the United States Government, but Puerto Rico has 78 municipalities or " municipios " at the secondary order.
A second-order low-pass filter with a very low quality factor has a nearly first-order step response ; the system's output responds to a step input by slowly rising toward an asymptote.
Second-and third-order winning percentage has been shown to predict future actual team winning percentage better than both actual winning percentage and first-order winning percentage.
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