If a particle and antiparticle are in the appropriate quantum states, then they can annihilate each other and produce other particles.

If denotes the quantum state of a particle ( n ) with momentum p, spin J whose component in the z-direction is σ, then one has

If a suitably sized quantum computer capable of running Grover's algorithm reliably becomes available, it would reduce a 128-bit key down to 64-bit security, roughly a DES equivalent.

If the quantum state is initially symmetric ( antisymmetric ), it will remain symmetric ( antisymmetric ) as the system evolves.

If S contains two elements that are not pairwise orthogonal ( in particular, the set of all quantum states includes such pairs ) then an argument like that given above shows that the answer is no.

If the error rate is small enough, it is thought to be possible to use quantum error correction, which corrects errors due to decoherence, thereby allowing the total calculation time to be longer than the decoherence time.

If the theory of quantum gravity also achieves a grand unification of the other known interactions, it is referred to as a theory of everything ( TOE ).

If one used Planck's energy quanta, and demanded that electromagnetic radiation at a given frequency could only transfer energy to matter in integer multiples of an energy quantum hν, then the photoelectric effect could be explained very simply.

If certain quantum inequalities conjectured by Ford and Roman hold, then the energy requirements for some warp drives may be absurdly gigantic as well as negative, for example the energy equivalent of − 10 < sup > 64 </ sup > kg might be required to transport a small spaceship across the Milky Way galaxy.

If the universe is described by an effective local quantum field theory down to the Planck scale, then we would expect a cosmological constant of the order of.

:" If the essential frequency range is limited to B cycles per second, 2B was given by Nyquist as the maximum number of code elements per second that could be unambiguously resolved, assuming the peak interference is less half a quantum step.

If a quantum computer with a sufficient number of qubits were to be constructed, Shor's algorithm could be used to break public-key cryptography schemes such as the widely used RSA scheme.

If true, sonoluminescence may be the first observable example of quantum vacuum radiation.

If quantum electrodynamics were an exact theory, the fine-structure constant would actually diverge at an energy known as the Landau pole.

If more than one quantum mechanical state is at the same energy, the energy levels are " degenerate ".

If quantum mechanics were to be applicable to macroscopic objects, there must be some limit in which quantum mechanics reduces to classical mechanics.

If the effects of the quantum mechanical nuclear motion are to be studied, for instance because a vibrational spectrum is required, this electronic computation must be in nuclear coordinates.

If a more general measurement is made that detects the system in a state then the system makes a " jump " or quantum leap from the original state to the final state with probability of.

If the factor i were absent, the H matrix would be antihermitian and would have purely imaginary eigenvalues, which is not the traditional way quantum mechanics represents observable quantities like the energy.

If the middle layer is made thin enough, it acts as a quantum well.

If parity transforms a pair of quantum states into each other, then the sum and difference of these two basis states are states of good parity.

If we allow the elements of the system to use quantum computation, the system is called a quantum interactive proof system, and the corresponding complexity class is called QIP.

If we introduce a vector then each quantum state corresponds to a point in ' n-space ' with energy

If a dancer is good, she suggests purely and superbly the fundamental mechanics of ancestry and progeny -- the continuum of mankind.

If the mechanics are similar, we can also infer that the thermodynamics will also be similar.

The evidence was compiled by W de Sitter ( 1927 ) who wrote " If we accept this hypothesis, then the ' astronomical time ', given by the earth's rotation, and used in all practical astronomical computations, differs from the ' uniform ' or ' Newtonian ' time, which is defined as the independent variable of the equations of celestial mechanics ".

If the legs are crossed above the knee, it is referred to as a wrap, and is considered poor technique, not only because it looks unattractive but because it interferes with the jump's mechanics.

Whether or not it is all-encompassing in so doing, Newtonian mechanics deals only with caused events, e. g.: If an object begins in a known position and is hit dead on by an object with some known velocity, then it will be pushed straight toward another predictable point.

If a non deterministic interpretation of quantum mechanics is correct then events at the microscopic level are necessarily indeterminate, where the degree of determinism increases as a function of the scale of the system ( see Quantum decoherence ).

If the sperm are of good quality and the mechanics of the woman's reproductive structures are good ( patent fallopian tubes, no adhesions or scarring ), physicians may start by prescribing a course of ovarian stimulating medication.

If so, what does statistical mechanics tell us about the arrow of time?

If we want to ignore the physics inside the conductor and only describe the physics in the outside region, it becomes natural to mathematically describe the quantum electron by a section in a complex line bundle with an " external " connection rather than an external EM field ( by incorporating local gauge transformations we have already acknowledged that quantum mechanics defines the notion of a ( locally ) flat wavefunction ( zero momentum density ) but not that of unit wavefunction ).

If one only considers black holes classically, then they have zero temperature and, by the no hair theorem, zero entropy, and the laws of black hole mechanics remain an analogy.

If one wants to consider antipodal points as identified, one passes to projective space ( see also projective Hilbert space, for this idea as applied in quantum mechanics ).

The Knudsen number is useful for determining whether statistical mechanics or the continuum mechanics formulation of fluid dynamics should be used: If the Knudsen number is near or greater than one, the mean free path of a molecule is comparable to a length scale of the problem, and the continuum assumption of fluid mechanics is no longer a good approximation.

If one prefers to have the and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by X < sub > i </ sub > and ( as is frequently done in quantum mechanics ).

# If Fredkin's second fundamental law is correct then the Copenhagen interpretation of quantum mechanics is not entirely correct, because quantum randomness lacks a digitally deterministic explanation.

If " econophysics " is taken to denote the principle of applying statistical mechanics to economic analysis, as opposed to a particular literature or network, priority of innovation is probably due to Farjoun and Machover ( 1983 ).

If quantum mechanics can be made complete in Einstein's sense, it cannot be done locally ; this fact was demonstrated by John Stewart Bell with the formulation of Bell's inequality in 1964 ; however, should we live in a superdeterminist universe, that demonstration would not be valid, as admitted by Bell himself.

If the number of objects ( e. g. particles ) in a system is small and calculations are possible, the mechanics show amazing accuracy.

If this new Bible does not increase in significance by repeated readings throughout the years, it will not survive the ages as has the King James Version.

If birds don't belong in a Square or Park, what does??

If private brand competition hasn't been felt in your product field as yet, have you thought how you will cope with it if and when it does appear??

If Af leads to an address Af that is equal to the address computed from Af, even though Af does not match Af, the chain of information cells is extended from Af by storing the address of the next available cell in the Y-region, Af, in Af.

If both George and a piece of information George does not have are prominent in the context, but the idea of telling George is new, then dominant stress will probably be on tell in why not tell George??

If, on the other hand, it is not settled, or while it is being debated and resolved, does the Federal government have a responsibility in situations like that in Prince Edward County??

If this threefold division of costs were to have its counterpart in the actual rates of charge for service, as it actually does have in some rates, there would result a three-part rate for any one class of service.

If this attitude is seriously questioned in the Soviet Union, it does not necessarily follow that the majority of the society in which I live is too aware of the necessity for clarity on this ethical as well as aesthetic point of view.

If a continuation card follows a statement that does not permit continuation cards, the compiler will generate a NOP and issue an error message.

If the compiler does encounter such statements, a warning message will be issued.

If the fourth dimension is a physical concept and not purely metaphysical, through what medium does it extend??

If a child does lose his first teeth prematurely because of decay -- and if no preventive steps are taken -- the other teeth may shift out of position, become overcrowded and malformed.

If he does, it's still better than an even chance he won't notice the transposition of the numbers, and if he should notice it, the thing can be passed off as an honest mistake.

If the administration does not succeed in passing the sales tax bill, or any other tax bill, it could very well be faced this spring at the fiscal session of the Legislature with an interesting dilemma.

If Bultmann's own definition of myth is strictly adhered to ( and it is interesting that this is almost never done by those who make such pronouncements ), the evidence is overwhelming that he does not at all exaggerate the extent to which the mythological concepts of traditional theology have become incredible and irrelevant.

If you know that he ( God ) is just ( righteous ), know that everyone also who does what is just ( righteous ) has been born of him.

If I get a low M. R., I do so because I want it that way or my dark self does.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

If the appellate court does find a legal defect in the decision " below " ( i. e., in the lower court ), it may " modify " the ruling to correct the defect, or it may nullify (" reverse " or " vacate ") the whole decision or any part of it.

If it does, ocean levels would rise by a few metres in a very short period of time.

If, however, there are no shares left when the chain is founded, then the founding player does not receive the free share.

If Anne formed an attachment to Weightman it does not imply that he was attracted to her.

If the numbering does not start at 0, the constant B may not be the address of any element.

If abiotic stress does not have a strong presence in an area, the biodiversity will be much higher.

If light moves instantaneously, the telescope does not move, and the true direction of the star relative to the observer can be found by following the line ES.