Eigenstate

8 Key excerpts on "Eigenstate"

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  eBook - ePub

  The Universe

  A View from Classical and Quantum Gravity

  • Martin Bojowald(Author)
  • 2012(Publication Date)
  • Wiley-VCH

    (Publisher)

  n it takes. The probability of finding the same result in a second measurement right afterwards, leaving no time for further emission or absorption processes, must result in the same value, with certainty because we already know what the energy is. The wave function contains all information about the system, in a statistical way because our knowledge is never complete. However, if an observable has already been measured, we know its value right after the measurement, without qualifying statistics. After measuring an observable, the state must be one that fully belongs to the observable, or an Eigenstate with the eigenvalue measured.

  Energy Eigenstates are nothing but stationary states. The new terminology is useful because it applies to any observable O. If we know the operator Ô, we can compute eigenvalues and Eigenstates by the equation ÔΨ

  n

  = O

  n

  Ψ

  n

  . In analogy with energy Eigenstates, we say that a system takes on an Eigenstate Ψ

  n

  of Ô if the observable Ô has been measured with value O

  n

  . For momentum, for instance, we have the operator = −i Δ/Δx, and Eigenstates are Ψ

  k

  (x) = exp(i k x) with eigenvalues p

  k

  = k. As in this case, eigenvalues do not always form a discrete set.

  We know what states we can have after measuring an observable, and what measurement results are possible. The last piece, the actual value that we will measure, remains statistical. If the state is not an Eigenstate to begin with, after a measurement different Eigenstates can be reached with some probabilities. The usual statistical properties of the wave function appear: A general state can be written as Ψ (x, t) = Sum

  n

  C

  n

  Ψ

  n

  (x, t) in terms of Eigenstates Ψ

  n

  of the observable we measure, with complex numbers C

  n

  that tell us how each Eigenstate contributes. After the measurement, we find one state Ψ

  n

  , and the probability by which it appears is |C

  n

  |2 .

  Collapse of the wave function

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  Thinking about Physics

  • Roger G. Newton(Author)
  • 2021(Publication Date)
  • Princeton University Press

    (Publisher)

  It can be applied, however, only when we can imagine a long (in principle, infinitely long) sequence of repetitions of the same experiment, or an ensemble of identical systems subject to similar conditions. An immediate consequence 51 CHAPTER 2 of this interpretation is that the quantum-mechanical state vector never refers to, or completely describes, an individual physical sys-tem, but always refers to an ensemble (which is the reason why some physicists resist this interpretation). We should therefore say more accurately than before that in quantum mechanics, the state of a system is defined as the ensemble to which the system belongs, and this ensemble is represented by a vector in Hilbert space or by a density operator, depending upon the precision with which the ensemble is defined. The quantum state of a system is not a direct description of the individual system. ENTANGLEMENT The issue of entanglement of distant particles, another characteris-tic of quantum states, opens up a separate set of questions. When specifying a classical state of several particles that do not interact with one another, their individual dynamical variables can be inde-pendently assigned, even if in the past they interacted. In quantum mechanics this is not necessarily so; there are the notorious phase correlations, which are difficult to grasp intuitively. Recall Bohm's version of the EPR experiment. 8 Two spin 1/2 particles emerge as the decay products of a spin 0 parent, fly-ing off in opposite directions. Since their total angular momentum must be zero, measurement of any spin projection of one of them allows us to infer that of the other, no matter how far away: they are entangled. If the vertical projection of the spin of particle 1 is measured and found to be up, a measurement of the vertical pro-jection of the spin of particle 2 must yield down. But if it was a horizontal projection of the spin of particle 1 that was measured 8 A.

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  The Foundations of Quantum Theory

  • Sol Wieder(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  To illustrate the ideas developed in Chapter 3, we shall investigate the properties of one-dimen-sional, one-particle systems whose Hamiltonians are independent of time, with particular emphasis on the stationary Eigenstate solutions. The station-ary Eigenstates are of primary interest for the following reasons: (a) Each state is of characteristic energy e f . (b) If the system is initially in such a state, it remains in that state indefi-nitely. (c) Once the energy eigenvalues and eigenfunctions are obtained, the propagator G(r, r', r, r 0 ) may be constructed using (3-93), and the evolution of arbitrary states determined. 87 88 4 W A V E MECHANICS IN ONE DIMENSION I Classification of Stationary States in W a v e M e c h a n i c s The eigenfunctions and eigenvalues of stationary states are determined by the solutions of the Schroedinger (energy-eigenvalue) equation^ • ^ ( r ) = j - ^ + F (r))^(r) = ε, ^ ( r ) . (4-1) We classify a state as bound or unbound according to whether the cor-responding function (or probability density) vanishes at infinity. If the state function associated with |/?> has the property lim S> = lim Ψ,(Γ) - 0 (4-2) r -»-oo r -*oo then the state is bound; otherwise it is unbound. We shall see shortly that the following rule holds: Rule An energy Eigenstate will generally be part of a discrete spectrum if it is bound and part of a continuous spectrum if it is unbound. It is also important to have a rule which determines the type of spectrum associated with a given Hamiltonian. Rule A Hamiltonian operator will have bound (discrete) or unbound (continuous) Eigenstates according to whether or not its classical counterpart supports bound orbits. For example, classically a free particle governed by a Hamiltonian with V = 0 moves in a straight line at constant speed and eventually tends toward infinity. The corresponding quantum-mechanical Hamiltonian has only a continuous spectrum.

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  Quantum Computing

  From Linear Algebra to Physical Realizations

  • Mikio Nakahara, Tetsuo Ohmi(Authors)
  • 2008(Publication Date)
  • CRC Press

    (Publisher)

  If we measure a in this state, then the state undergoes an abrupt change to one of the Eigenstates corresponding to the observed eigenvalue: If the observed eigenvalue is λ 1 ( λ 2 ), the system undergoes a wave function collapse as follows: c 1 | λ 1 + c 2 | λ 2 → | λ 1 ( | λ 2 ), and the state imme-diately after the measurement is | λ 1 ( | λ 2 ). Suppose we prepare many 29 30 QUANTUM COMPUTING copies of the state c 1 | λ 1 + c 2 | λ 2 . The probability of collapsing to the state | λ k is given by | c k | 2 ( k = 1 , 2). In this sense, the complex coeffi-cient c i is called the probability amplitude . It should be noted that a measurement produces one outcome λ i and the probability of obtaining it is experimentally evaluated only after repeating measurements with many copies of the same state. These statements are easily generalized to states in a superposition of more than two states. A 3 The time dependence of a state is governed by the Schr¨ odinger equa-tion i ∂ | ψ ∂t = H | ψ , (2.1) where is a physical constant known as the Planck constant and H is a Hermitian operator (matrix) corresponding to the energy of the system and is called the Hamiltonian . Several comments are in order. • In Axiom A 1, the phase of the vector may be chosen arbitrarily; | ψ in fact represents the “ray” { e iα | ψ | α ∈ R } . This is called the ray representation . In other words, we can totally igonore the phase of a vector since it has no observable consequence. Note, however, that the relative phase of two different states is meaningful. Although | φ | e iα ψ | 2 is independent of α , | φ | ψ 1 + e iα ψ 2 | 2 does depend on α . • Axiom A 2 may be formulated in a different but equivalent way as follows. Suppose we would like to measure an observable a . Let A = ∑ i λ i | λ i λ i | be the corresponding operator, where A | λ i = λ i | λ i . Then the expectation value A of a after measurements with respect to many copies of a state | ψ is A = ψ | A | ψ .

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  Physical Chemistry

  How Chemistry Works

  • Kurt W. Kolasinski(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  CHAPTER 20 The quantum mechanical description of nature

  PREVIEW OF IMPORTANT CONCEPTS

  • We can expect to observe quantum mechanical effects and the insufficiency of a classical mechanical description when at least one dimension of the system under consideration approaches the magnitude of its de Broglie wavelength.
  • A quantum mechanical description should be used when the thermal energy kB T is small compared to the energy level spacing ϵ
    i
    − ϵ
    j
    .
  • In the limit of high quantum numbers, systems behave classically. Equivalently, in the limit of large energies, large masses and large orbits, quantum calculations deliver the same results as classical calculations.
  • The postulates of quantum mechanics cannot be derived from more fundamental principles and form the basis of the theory from which its framework is derived.
  • The wavefunction describes everything there is to know about a system.
  • The absolute square of the wavefunction describes a probability distribution.
  • Operators act on wavefunctions to determine the values of observables.
  • If a system is in an Eigenstate of an operator, the corresponding observable has an exact value. If the system is not in an Eigenstate of the operator, the observable is described by an expectation value that is equal to the mean of many measurements.
  • The Eigenstates of a system form an orthonormal basis set.
  • Eigenstates can be superposed in linear combinations to create states of the system. Wavefunctions have relative phases, which can lead to interference effects when they are combined.

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  Quantum Mechanics

  • Gennaro Auletta, Mauro Fortunato, Giorgio Parisi(Authors)
  • 2009(Publication Date)
  • Cambridge University Press

    (Publisher)

  In conclusion, this proposal aims at measuring the state of a quantum system and there-fore to consider it as an observable. In other words, Aharonov and co-workers’ try to consider the quantum state in classical terms. Aharonov and co-workers’ proposal has been criticized 2 (see Prob. 15.1 ) by pointing out that it only proved that, if the wave vector of a system S is known beforehand to be the Eigenstate of the unknown Hamiltonian of S , then it is possible to determine the properties of that Eigenstate. In other words, one can determine some of the properties of an unknown Hamiltonian of S , if one knows that S is in an Eigenstate of that observable. In fact, the main condition of their model is a protective measurement, i.e. the system S interacts with an apparatus A or with the rest of the world in such a way that its wave function remains unchanged after the measurement but affects A , so that a succession of measurements can completely determine it. This in turn means that, if S is in an energy Eigenstate and if the interaction between S and the rest of the world is adiabatic, then the state vector after the measurement would still represent the same energy Eigenstate. While the state vector is unchanged, the rest of the world has been changed in a manner dependent on the specific state of S . However, if so, what we have obtained is only the measurement of an observable (the Hamiltonian) and not of the wave function as such. The problem is that we can force the wave function to be the Eigenstate of an observable, but we cannot force the observable to have the unknown wave function as its Eigenstate. On the other hand, we know that the density matrix describing a pure state is a projector, i.e. it is an observable (see Subsec. 2.1.1 and Eq. ( 5.26 )). So, why one can measure a pro-jector but cannot obtain information about the state? The question is, what are the possible values that we would obtain by measuring a projector? Obviously, 0 or 1.

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  Solid State and Quantum Theory for Optoelectronics

  • Michael A. Parker(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  The particle can therefore reside in the direct product space (see Chapters 2 and 3) given by V ¼ V 1 V 2 where V 1 might describe the energy V 2 might describe the spin, and so on 262 Solid State and Quantum Theory for Optoelectronics The basis set for the direct product space consists of the combination of the basis vectors for the individual spaces such as j C i¼j f , h , . . . i¼j f ij h i . . . where we assume, for example, that the space spanned by { j f i } refers to the energy content and { j h i } refers to spin, etc. The basis states can be most conveniently labeled by the eigenvalues of the commuting Hermitian operators. For example, j E i , p j i represents the state of the particle with energy E i and momentum p j assuming, of course, that the Hamiltonian and momentum commute. These two operators might represent all we care to know about the system. 5.2 FUNDAMENTAL OPERATORS AND PROCEDURES FOR QUANTUM MECHANICS Quantum mechanics represents physical objects in terms of mathematics. As such, there must be well-de fi ned symbols and procedures established to fi rst translate the physical situation into the mathematics, provide for manipulation of the symbols, and then to interpret the results back in terms of the physical world. The Hilbert spaces have a close symbiotic relation with the quantum mechanics. The present section discusses usable forms of the operators and shows the Schrödinger wave equation (SWE) as the primary quantity of interest for determining the time evolution of quantum level particles and systems. The next section applies the formalism to examples of a 1-D in fi nitely deep and fi nitely deep quantum well. 5.2.1 S UMMARY OF E LEMENTARY F ACTS Electrons, holes, photons, and phonons can be pictured as particles or waves. Momentum and energy usually apply to particles while wavelength and frequency apply to waves.

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  Quantum Mechanics

  A Paradigms Approach

  • David H. McIntyre(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  In classical mechanics, waves and particles are clearly distinct, whereas in quantum mechanics a system exhibits properties that remind us of classical particles but also exhibits properties of classical waves. This is often referred to as wave-particle duality. We will see more of this in the next chapter when we discuss free particles. Example 5.3 It is useful to put some numbers into these expressions to get a sense of scale. For example, if we confine an electron 1m e = 511 k eV > c 2 2 in a box of size 0.2 nm (about the size of an atom), the ground state (n  1) energy is E 1 = p 2 U 2 2m e L 2 = p 2 16.58 * 10 -16 eV s2 2 210.511 * 10 6 eV > c 2 210.2 * 10 -9 m2 2 (5.69) = 9.4 eV. This is comparable to typical atomic binding energies. The spectrum of this system will include the transition between the ground state and the first excited state. The first excited state has energy E 2 = 2 2 E 1 = 4E 1 , so the wavelength of light for this transition is l 21 = hc E 2 - E 1 = hc 3E 1 = 1240 eV nm 319.4 eV2 = 44 nm . (5.70) Note that l 21 is the wavelength of the photon emitted or absorbed in the transition, not the wave- length of the bound particle that is associated with the wave vector of the wave function, which is 0.4 nm for the ground state and 0.2 nm for the excited state, in agreement with Eq. (5.68). Now that we have found the energy Eigenstates, we have what we need to calculate probabilities and expectation values to compare with experiments. The square of the wave function gives us the probability density P n 1x2 = 0 w n 1x2 0 2 = 2 L sin 2 npx L , (5.71) which is shown in Fig. 5.12 for the first three states. Note that the probability density is zero outside the well, so the probability of finding the particle anywhere outside the well is zero, just as in the clas- sical case. However, in the quantum system there are positions within the well where the probability of finding the particle is zero, which does not happen in the classical case.

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