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# Topic: Eigenstates

###### In the News (Sat 25 May 19)

 Perturbation theory (quantum mechanics) - Wikipedia, the free encyclopedia An example of this phenomenon may be found in conventional superconductivity, in which the phonon-mediated attraction between conduction electrons leads to the formation of correlated electron pairs known as Cooper pairs. Since the perturbation is weak, the energy levels and eigenstates should not deviate too much from their unperturbed values, and the terms should rapidly become smaller as we go to higher order. Imagine that we have two or more perturbed eigenstates with different energies, which are continuously generated from an equal number of unperturbed eigenstates that are degenerate. en.wikipedia.org /wiki/Perturbation_theory_(quantum_mechanics)   (2212 words)

 Berry's Phase Hints Note that the eigenstates m> would be independent of time if the parameters of the Hamiltonian did not change; their only time dependence is through the change of the parameters. The two eigenstates of the Hamiltonian represent the particle's spin aligned parallel or antiparallel to the magnetic field. Similarly, in the three-dimensional problem, you can define eigenstates that are continuous and differentiable everywhere in the parameter space except the origin and the negative z-axis (or, alternately, the positive z-axis). socrates.berkeley.edu /~kts/phys221A/ps3hints1.html   (698 words)

 [No title]   (Site not responding. Last check: 2007-09-18) On to physics: we reviewed the construction of the angular momentum eigenstates; discussed energy eigenstates and the Zeeman effect; and discussed representations of angular momentum operators within a basis of j,m> eigenstates for fixed j. Because a rotationally invariant Hamiltonian commutes with J^2 and J_z, the eigenstates j,m> that we have constructed {\em are} the energy eigenstates. Furthermore, we showed that eigenstates in the same representation --- that is, with the same j --- are degenerate in energy. www.physics.emory.edu /faculty/benson/380-96/notes/22.txt   (594 words)

 Expansion in Eigenstates   (Site not responding. Last check: 2007-09-18) The energy eigenstates are indicated on the upper right hand side of the window, on the same scale as the potential. A white curve is generated, which is a result of the expansion in terms of the eigenstates. Notice how the expansion coefficients change as you modify the curve, and also note that the fit is not perfect (and is worse for non-smooth functions) because only a finite nuber of eigen-states are used. www.fen.bilkent.edu.tr /~yalabik/applets/superp.html   (407 words)

 Compatible observables   (Site not responding. Last check: 2007-09-18) In principle, each of these eigenstates is associated with a different result of the measurement. Each eigenstate is again associated with a different possible result of the measurement. It follows that the simultaneous eigenstates of two observables must also form a complete set. farside.ph.utexas.edu /teaching/qm/fundamental/node17.html   (286 words)

 Schrödinger equation - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-09-18) Energy eigenstates are convenient to work with because their time-dependence is so simple; that is why the time-independent Schrödinger equation is so useful. Since the position eigenstates form a basis for the state space, the integral over all projection operators is the identity operator: We have previously shown that energy eigenstates vary only by a complex phase as time progresses. www.bexley.us /project/wikipedia/index.php/Schrodinger's_equation   (1509 words)

 Physics Help and Math Help - Physics Forums - degenerate perturbation theory question   (Site not responding. Last check: 2007-09-18) I guess the point is that you can take any linear combination of eigenstates of H corresponding to a degenerate eigenvalue and it will still be an eigenstate of H with the same eigenvalue. The new basis consists of functions which are eigenstates of H' and have eigenvalues E1 (which, may or may not be distinct). Furthermore, the eigenstates of L^2 and Lz are Y_lm. www.physicsforums.com /printthread.php?t=65773   (950 words)

 Neutrino oscillation To clarify the notion of mixing, let us consider (Figure 5) a system of Cartesian coordinates where x and y are the flavour eigenstates and another system of Cartesian coordinates x'-y' (where x' and y' are the mass eigenstates) rotated by a small angle with respect to the system x-y. For a given momentum[4] the energy associated to mass eigenstates is all the more higher as is their "rest" mass; indeed, according to the mass-energy equation, a particle "rest" mass contributes to its total energy, together with kinetic energy (associated to the momentum). If the masses of the eigenstates are different, the corresponding waves are propagated with a different rate, thus they would not reach the Earth with the same phase as at the start. choruswww.cern.ch /Public/textes/english/node4.html   (1097 words)

 Electromagnetic Waves It can be shown that translational symmetry implies that linear momentum is conserved and is also responsible for the infinite degeneracy of the eigenstates of the free particle (recall that for a given energy there are an infinite number of momenta possible, differeing in their direction). As is described in the course pack notes on Angular Momentum, Spin, and the Pauli Principle, the constant of the motion associated with the spherical symmetry is angular momentum denoted by the symbol L. Thus each eigenstate of the hydrogen atom can be labeled uniquely in terms of its energy, magnitude of angular momentum, and z-component of angular momentum. www-personal.umich.edu /~pberman/20hyd.html   (1015 words)

 Introduction   (Site not responding. Last check: 2007-09-18) Third, the time-step size must be proportional to the spatial-grid size for numerical stability; thus, if you double the spatial resolution, you must double the number of time steps (the length of your simulation), even if you are looking at states with the same frequency as before. Fourth, you only get the frequencies of the states; to get the eigenstates themselves (so that you can see what the modes look like and do calculations with them), you must run the simulation again, once for each state that you want, and for a time inversely proportional to the frequency-spacing between adjacent states (i.e. A traditional disadvantage of frequency-domain methods was that you had to compute all of the lowest eigenstates, up to the desired one, even if you didn't care about the lower ones. ab-initio.mit.edu /mpb/doc/introduction.html   (2013 words)

 [No title]   (Site not responding. Last check: 2007-09-18) We then began a discussion of the role of rotational symmetry and its generators, the angular momentum operators, in quantum mechanics. Because of this, we can use a and a\adjoint as step operators on the eigenstates of N. We showed a n > to be an eigenstate of N with eigenvalue n-1, and a\adjoint n > to be an eigenstate of N with eigenvalue n+1. This implied that we could start with a single N eigenstate and construct a ladder of allowed eigenstates by repeated application of a and a\adjoint. www.physics.emory.edu /faculty/Benson/380-96/notes/18.txt   (497 words)

 [No title]   (Site not responding. Last check: 2007-09-18) We then proved the fact mentioned earlier: that commuting operators A and B have simultaneous eigenstates (that is, eigenstates of A are also eigenstates of B); and that, conversely, noncommuting operators cannot share all the same eigenstates. So when we look for angular momentum eigenstates, we must look for eigenstates of the operators J^2 and J_z, the largest set of incompatible angular momentum measurements we can make. To find the eigenstates, we will use our simple harmonic oscillator trick of finding step operators --- or eigenoperators under commutation with the operator whose eigenstates we seek --- to build a ladder of eigenstates. www.emory.edu /PHYSICS/Faculty/Benson/380-96/notes/20.txt   (275 words)

 Coherent States (Draft Version)   (Site not responding. Last check: 2007-09-18) Except for the ground state, the correspondence between the quantum energy eigenstates of the simple harmonic oscillator and the classical states in which a pendulum swings back and forth is not completely clear. But note that these eigenstates are not delta functions, like the eigenstates of the position operator—they are, like the ground state of the oscillator, states with the least possible uncertainty, That is emphatically not the case for states with nonzero l:  they correspond to the pendulum having been pulled to one side, and it will begin to swing. landau1.phys.virginia.edu /classes/751.mf1i.fall02/CoherentStatesDraft.htm   (1038 words)

 Replacement of Sums over All States with Sums over Eigenstates These energy eigenstates are a complete set of states. However, in quantum mechanics, the adjective complete assumes a specific technical meaning, namely that is a set of states is complete if an arbitrary state of the system can written as a sum over the complete set of states. Any other state of the system may be written as a linear combination of the energy eigenstates or as a linear combination of the helicity states. users.wpi.edu /~phillies/stat11/node2.html   (831 words)

 Choice of Basis Vectors A first question is whether or not we were justified in using the list of energy eigenstates of the system as a correct list of all states of the system. This Q may be rewritten as a double quantum and thermal average, using the notation of the previous Lecture. 11.5 and 11.6 are written in terms of energy eigenstates. users.wpi.edu /~phillies/stat11/node1.html   (872 words)

 0011074 This is due to the principle of causality, which implies that different mass eigenstates with different momenta created in the production process determine their own kinematics in the detection process where they are detected. It is clear that the transition probability at the detector depends only on the neutrino mixing matrix and on the momentum and energy differences of the mass eigenstates, which are due to their mass differences. In this case the corresponding mass eigenstates cannot interfere and the oscillations are suppressed. www.nu.to.infn.it /pap/0011074   (3722 words)

 search.cpan.org: Quantum::Superpositions - QM-like superpositions in Perl Instead, the various comparison operators are redefined so that they form a superposition of those eigenstates of the left operand that cause the operation to be true. Providing an implicit conversion to numeric (for situations where superpositions are used as operands to an arithmetic operation, or as array indices) is more challenging than stringification, since there is no mechanism to capture the entire state of a superposition in a single non-superimposed number. Note that this is equivalent to the definition of an eigenstate. search.cpan.org /~lembark/Quantum-Superpositions-2.02/lib/Quantum/Superpositions.pm   (2233 words)

 Neutrino Oscillation Phenomenology   (Site not responding. Last check: 2007-09-18) By virtue of their charged-current weak interaction, neutrinos are known to exist as eigenstates of the electroweak interaction. If they are massive, then they will also exist as mass eigenstates, which are not necessarily identical to the weak eigenstates. If the mass and weak eigenstates were identical, then U would be a unit matrix. www.hep.umn.edu /~bspeak/writeup/node2.html   (172 words)

 Neutrino Oscillation Lab Squares of the matrix elements in a given row give the relative contribution of different mass eigenstates to a given physical states, whereas squares of elements in a given column give decomposition of a mass eigenstates in terms of the physical states. The produced ν state is a mixture of different mass eigenstates, and they evolve in time with their proper frequencies. Oscillation frequency of mass eigenstates depends on the beam energy, therefore the neutrino oscillations depend on the energy too. minos.phy.bnl.gov /nu-osc-lab/Superposition1.html   (520 words)

 harmonic(2)   (Site not responding. Last check: 2007-09-18) (0)> is an eigenstate of the lowering operator a and find the eigenvalue. The eigenstates of the harmonic oscillator, raising and lowering operators. The system is in a given linear superposition of eigenstates of the harmonic oscillator. electron6.phys.utk.edu /phys594/archives/quantum/Harmonic/harmonic2.htm   (451 words)

 LArTPC at Yale: project   (Site not responding. Last check: 2007-09-18) If they do have mass, however, then quantum mechanics allows a mixing of the mass eigenstates, and this mixing permits a single neutrino to switch between its component mass eigenstates as its wavefunction propagates. The mass eigenstates of neutrinos are considered the pure eigenstates, while the three flavors - electron, muon, and tao - are considered weak eigenstates, and are a combination of different mass eigenstates. In 1998, neutrino oscillations were observed in muon neutrinos produced by cosmic ray interactions with the atmosphere. hepwww.physics.yale.edu /nup/project.htm   (1822 words)

 Scarred Eigenstates for Quantum Cat Maps of Minimal Periods - Faure, Nonnenmacher, De Bievre (ResearchIndex)   (Site not responding. Last check: 2007-09-18) Scarred Eigenstates for Quantum Cat Maps of Minimal Periods (2002) Abstract: In this paper we construct a sequence of eigenfunctions of the "quantum Arnold's cat map" that, in the semiclassical limit, show a strong scarring phenomenon on the periodic orbits of the dynamics. Scarred eigenstates for quantum cat maps of minimal periods. citeseer.ist.psu.edu /faure02scarred.html   (454 words)

 No Title Consider a another Stern-Gerlach type experiment where, in the original coordinate system, a beam of deuterons (s=1) is prepared in pure eigenstates of S Find the first order correction to the energy of the lowest eigenstate. Suppose that the system is perturbed by an interaction of the form www.wfu.edu /~natalie/f97phy441/exam/f97ex2   (252 words)

 Class minutes---week 6 a is not hermitian, so there need not be more eigenstates, and, if there are, the eigenvalue z need not be real. = I. # Evolution: The number eigenstates are energy eigenstates and evolve trivially: n,t> = e^(-i(n+1/2)wt) n>. Orthonormality of the number eigenstates =delta(n,m) implies for the Hermite polynomials www.glue.umd.edu /~tajac/622c/622minutes6.html   (607 words)

 [No title]   (Site not responding. Last check: 2007-09-18) is an eigenstate of observable Q with eigenvalue q1, then when the operator Q-hat acts on that state, the result is Examples: eikx and p e-iwt and E¡ÎG’a@’a ’a’açÿ4çÿ@  ª,‘ð ² ð  s ð*A Ì?¿Àÿÿÿ?"ñ¿`ðà ®o ð8Á Ì\$ñD  0ðH ð   ð0“”Þ½h¿ÿ ?ð fÿÿ———fÿÿÌ33ÌÌÌÿ²²²ÈÀº___PPT10 +xzpÑDñL'ñ ¹ÿÿÿÿ=ñ @Bñ Dñ'ñ ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ  is an eigenstate of observable Q with eigenvalue q1, then the expectation value of operator Q-hat If every measurement of that observable will yield the eigenvalue, then the average will be the eigenvalue. is an eigenstate of observable Q with eigenvalue q1, then the uncertainty of operator Q-hat If every measurement of that observable will yield the eigenvalue, then there will be no deviation. helios.augustana.edu /~cv/401/week4-m.ppt   (363 words)

 T-signature or T-simplex, and signature or simplex We may then only work either in the basis of eigenstates of 3.1.2, or in the basis of eigenstates of It is easy to check that after multiplying eigenstates listed in Table 4 by the following phase factors: info.fuw.edu.pl /~dobaczew/symhfy57w/node14.html   (151 words)

 Perturbation theory (quantum mechanics)   (Site not responding. Last check: 2007-09-18) '' and ''n''> in terms of the energy levels and eigenstates of the old Hamiltonian. However, the true energy shift is slightly different, because the perturbed eigenstate is not exactly the same as ''n If the unperturbed system is in eigenstate ''j''> at time ''t = 0'', its state at subsequent times varies only by a phase (waves)phase (we are following the Schrödinger picture/, where state vectors evolve in time and operators are constant): www.infothis.com /find/Perturbation_theory_(quantum_mechanics)   (2394 words)

 Neutrino oscillations   (Site not responding. Last check: 2007-09-18) A particular neutrino state can be expressed as a superposition of different mass flavor eigenstates by analogy with quark mixing and the CKM matrix. Since there are only three flavor eigenstates possible, the matrix connecting mass eigenstates is likely a 3 X 3 unitary matrix relating the flavor eigenstates There are other possibilities, such as sterile neutrinos that would modify our observations of this and the MINOS experiment will examine them as well. www.hep.umn.edu /~ebeall/neutrinos/node4.html   (162 words)

 Neutrino Mass Eigenstates Before talking about the experimental limits on neutrino masses and the reasons why we believe neutrinos are probably massive, it is important to define the neutrino mass eigenstate lest the whole discussion make very little sense. where n is the number of mass eigenstates and U is an orthonormal mixing matrix. nearly diagonal) as in the quark sector, then each of the flavor eigenstates nearly overlaps with one of the mass eigenstates and only involves minor quantities of the others. www.fynu.ucl.ac.be /librairie/theses/gustaaf.brooijmans/node10.html   (198 words)

 Citebase - Accuracy of Approximate Eigenstates   (Site not responding. Last check: 2007-09-18) Accordingly, various measures for the accuracy of the approximate eigenstates derived by variational techniques are scrutinized. In particular, the matrix elements of the commutator of the operator H and (suitably chosen) different operators, with respect to degenerate approximate eigenstates of H obtained by some variational method, are proposed here as new criteria for the accuracy of variational eigenstates. These considerations are applied to that Hamiltonian the eigenvalue problem of which defines the "spinless Salpeter equation." This (bound-state) wave equation may be regarded as the most straightforward relativistic generalization of the usual nonrelativistic Schroedinger formalism, and is frequently used to describe, e.g., spin-averaged mass spectra of bound states of quarks. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-ph/9909451   (1406 words)

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