Wave Function

11 Key excerpts on "Wave Function"

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  Quantum Dynamics

  Applications in Biological and Materials Systems

  • Eric R. Bittner(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  2 Waves and Wave Functions In the world of quantum physics, no phenomenon is a phenomenon until it is a recorded phenomenon. John Archibald Wheeler Bohr’s model of the hydrogen atom was successful in that it gave us a radically new way to look at atoms. However, it has serious shortcomings. It could not be used to explain the spectra of He or any multielectron atom. It could not predict the intensities of the H absorption and emission lines. With de Broglie’s hypothesis that matter was also wavelike, 1 there arose a question at the 1925 Solvey conference: What is the wave equation? De Broglie could not answer this; however, over the next year Erwin Schr¨ odinger, working in Vienna, published a series of papers in which he deduced the general form of the equation that bears his name and applied it successfully to the hydrogen atom. 2 , 3 What emerged was a new set of postulates, much like Newton’s, that laid the foundations of quantum theory. The physical basis of quantum mechanics is 1. That matter, such as electrons, always arrives at a point as a discrete chunk, but that the probibility of finding a chunk at a specified position is like the intensity distribution of a wave. 2. The “quantum state” of a system is described by a mathematical object called a “Wave Function” or state vector and is denoted | ψ . 3. The state | ψ can be expanded in terms of the basis states of a given vector space, {| φ i } , as | ψ = i | φ i φ i | ψ (2.1) where φ i | ψ denotes an inner product of the two vectors. 4. Observable quantities are associated with the expectation value of Hermi-tian operators and that the eigenvalues of such operators are always real. 5. If two operators commute, one can measure the two associated physical quantities simultaneously to arbitrary precision. 6. The result of a physical measurement projects | ψ onto an eigenstate of the associated operator | φ n yielding a measured value of a n with probability | φ n | ψ | 2 . 33

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  Quantum Mechanics, Volume 1

  Basic Concepts, Tools, and Applications

  • Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë(Authors)
  • 2020(Publication Date)
  • Wiley-VCH

    (Publisher)

  Moreover, the predictions of the measurement results are now only probabilistic (they yield only the probability of obtaining a given result in the measurement of a dynamical variable). The Wave Function is a solution of the Schrödinger equation, which enables us to calculate ψ (r, t) from ψ (r, 0). This equation implies a principle of superposition which leads to wave effects. This upheaval in our conception of mechanics was imposed by experiment. The structure and behavior of matter on an atomic level are incomprehensible in the framework of classical mechanics. The theory has thereby lost some of its simplicity, but it has gained a great deal of unity, since matter and radiation are described in terms of the same general scheme (wave-particle duality). We stress the fact that this general scheme, although it runs counter to our ideas and habits drawn from the study of the macroscopic domain, is perfectly consistent. No one has ever succeeded in imagining an experiment that could violate the uncertainty principle (cf. Complement D I of this chapter). In general, no observation has, to date, contradicted the fundamental principles of quantum mechanics. Nevertheless, at present, there is no global theory including quantum phenomena within general relativity (gravity) and, of course, nothing prevents the possibility of a new upheaval. References and suggestions for further reading: Description of physical phenomena which demonstrate the necessity of introducing quantum mechanical concepts: see the subsection “Introductory work – quantum physics” of section 1 of the bibliography; in particular, Wichmann (1.1) and Feynman III (1.2), Chaps

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  The Meaning of the Wave Function

  In Search of the Ontology of Quantum Mechanics

  • Shan Gao(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  7.3 The Wave Function as a Description 85 This description of the motion of a particle can be extended to the motion of many particles. At each instant a quantum system of N particles can be represented by a point in a 3N-dimensional configuration space. During an arbitrarily short time interval or an infinitesimal time interval around each instant, these particles perform random discontinuous motion in three-dimensional space, and, correspondingly, this point performs random discontinuous motion in the configuration space. Then, similar to the single-particle case, the state of the system can be described by the position density ρ(x 1 , x 2 , . . . , x N , t) and position flux density j(x 1 , x 2 , . . . , x N , t) defined in the configuration space. There is also the relation ρ(x 1 , x 2 , . . . , x N , t) = (x 1 , x 2 , . . . , x N , t), where (x 1 , x 2 , . . . , x N , t) is the probability density that particle 1 appears in position x 1 and particle 2 appears in position x 2 . . . and particle N appears in position x N . When these N particles are independent from each other, the position density can be reduced to the direct product of the position density for each particle, namely, ρ(x 1 , x 2 , . . . , x N , t) =  N i=1 ρ(x i , t). 7.3.3 Interpreting the Wave Function Although the motion of particles is essentially discontinuous and random, the dis- continuity and randomness of motion are absorbed into the state of motion, which is defined during an infinitesimal time interval around a given instant and described by the position density and position flux density. Therefore, the evolution of the state of random discontinuous motion of particles may obey a deterministic con- tinuous equation.

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  Solid State and Quantum Theory for Optoelectronics

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

    (Publisher)

  Before a measurement, they have limited information of the system. They know the probability P ( n ) ¼j b n j 2 that the particle occupies a given fundamental state (basis vector). Therefore, they know a Wave Function by the superposition of b n j f n i . Making a measurement naturally changes the wave func-tion because they then have more information on the actual state of the particle. After the measurement, they know for certain that the electron must be in state i for example. Therefore, they know b i ¼ 1 while all the other b must be zero. In effect, the Wave Function collapses from c to f i . With this fi rst view, they ascribe any wave motion of the electron to the probability amplitude while implicitly assuming that the electron occupies a single state and behaves as a point particle. Making a measurement removes their uncertainty. The collapse refers to probability and nothing more. As a second picture, and probably the most profound, let us view the collapse of the Wave Function as more related to physical phenomena. The Copenhagen interpretation (refer to Max Jammer ’ s book) of a quantum particle in a superposed state j c i¼ X n b n ( t ) j f n i views the particle as simultaneously existing in all of the fundamental states j f n i . In this case, we do not think of the particle as occupying a de fi nite state j f i i . Somehow the particle simultaneously has all of the attributes of all of the fundamental states. A measurement of the particle forces it to 258 Solid State and Quantum Theory for Optoelectronics ‘‘ decide ’’ on one particular state. This second point of view requires some explanation using examples and it produces one of the most profound theorems of modern times — Bell ’ s theorem. First let us consider the case of a particle described by the Wave Function c ( x ).

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  Principles of Engineering Physics 1

  • Md Nazoor Khan, Simanchala Panigrahi(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  The language of probability in quantum mechanics is same as that used in our day-to-day language. A surgeon who asserts that a patient has 70% probability of surviving an operation means that with a large number of similar cases, 70% of patients survive. The same language of probability is used in case of weather forecasting in our day-to-day life. In quantum physics, particles are represented by wave packets. Where is the particle in the wave packet? A wave packet can be considered to be the superposition of a large number of waves that which interfere constructively in the vicinity of the particle, giving a resultant wave of large amplitude and interfere destructively far way from the particle giving a resultant wave of small amplitude. Thus, the probability of finding the particle is high, where the amplitude of the resultant wave is large; the probability of finding the particle is less, where the amplitude of the resultant wave is small. When we say that the wave packet is spread out in space, it does not mean that the particle itself is spread out in space. If we say that the probability of the presence of an electron at a certain point in the wave packet is 30%, it does not mean that 30% of the electron is at that point and the rest 70% of the electrons is at some other points. Therefore, we conclude that the probability of finding the particle at any point depends on the amplitude of its de Broglie wave (matter wave) at that point. Intensity of a wave at a point µ |amlitude of the wave at that poin| 2 (Classical physics) Probability of finding a particle at a point µ |amlitude of matter wave at that poin| 2 (Quantum physics) This probabilistic description is the fundamental concept in quantum physics and is achieved by defining the Wave Function. 7.9 Wave Function y Matter waves are not real waves and therefore, cannot be represented by wave displacement. In electromagnetic waves, the electric and magnetic field vary periodically.

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  Conscious Mind in the Physical World

  • E.J Squires(Author)
  • 1990(Publication Date)
  • CRC Press

    (Publisher)

  To appreci-ate what this means we compare the situation in classical physics. There an electron is described by a position x(t). This states that, at time t, the particle is at x, and hence that we will observe it to be at that point. On the contrary, quantum theory describes the particle by a wavefunction; until an observation is made there is no position. It is important to realise that we are not saying here simply that we do not know the position until we measure it. This of course is often the situation in classical physics: the particle is at a particular point but we do not know which. The distinction is crucial for the quantum theory understanding of the double-slit experiment (for example). If we had to say that the electron really followed one trajectory, then there would be no way to explain interference, regardless of whether we know the route followed. Quantum theory and locality 167 In other words, classical probability theory just deals with our knowledge of a system. This is why interference does not happen. On the other hand, the quantum mechanical wavefunction is a real quantity, existing in the external world. Indeed in orthodox quantum theory the wavefunction is the external world. There are two questions that now arise. We have not said what we mean by an observation, and we have not said in what way an observation affects the physical reality, i.e.

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  Quantum Theory and Measurement

  • John Archibald Wheeler, Wojciech Hubert Zurek, John Archibald Wheeler, Wojciech Hubert Zurek, John Wheeler, Wojciech Zurek(Authors)
  • 2014(Publication Date)
  • Princeton University Press

    (Publisher)

  I did not succeed in doing this with the matrix form of quantum mechanics, but did with the Schrodinger formulation. According to Schrodinger, the atom in its nth quantum state is a vibration of a state function of fixed frequency W°/h spread over all of space. In particular, an electron moving in a straight line is such a vibratory phenomenon which corre- sponds to a plane wave. When two such waves interact, a complicated vibration arises. However, one sees immediately that one can determine it through its asymptotic behavior at infinity. Indeed one has nothing more than a "diffraction problem" in which an incoming plane wave is refracted or scattered at an atom. In place of the boundary conditions which one uses in optics for the description of the diffraction diaphragm, one has here the potential energy of interaction be- tween the atom and the electron. The task is clear. We have to solve the Schrodinger wave equation for the system atom-plus-electron subject to the boundary condition that the solution in a preselected direction of electron space goes over asymptotically into a plane wave with exactly this direction of propagation (the arriving electron). In a thus selected solution we are further interested principally in a behavior of the "scattered" wave at infinity, for it describes the behavior of the system after the collision. We spell this out a little further. Let ιt/*¾¾)* · · · be the eigenfunc- tions of the unperturbed atom (we assume that there is only a discrete spectrum). The unperturbed electron, in straight-line motion, corresponds to eigenfunctions sin (2π/λ)(αχ + βγ + yz + δ), a continuous manifold of plane waves. Their wave- length, according to de Broglie, is connected with the energy of translation τ by the relation τ = Ρ/(2μλ 2 ). The eigenfunction of the unperturbed state in which the electron arrives from the + ζ direction, is thus ^nAq k , Z) = Ά°(<Ζ)ί) sin (2πζ/λ).

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  An Introduction to Quantum Physics

  • A.P. French(Author)
  • 2018(Publication Date)
  • Routledge

    (Publisher)

  Notice that we have not needed to concern ourselves with the normalization of the quantum amplitudes. Our goal has been simply to find the form of the Wave Function for a possible state. Notice also that the sketch can be made without any need to consider the particle mass, or any other of the numerical magnitudes that are clearly relevant to a quantitative solution of the problem. The reason is that the indication of the energy [relative to V (x)], together with information about the state number, provide between them an implicit dimensional scale that allows us to ignore, for the purposes of a rough plot, the constants m and ħ in the Schrödinger equation (Eq. 3-13), along with numerical magnitudes of x, E, and V (x). In constructing such a Wave Function, you can make use of the following check list of properties of bound-state Wave Functions. Check that your Wave Function for quantized energy states has: 1. odd or even symmetry in the case of a symmetric potential; 2. the correct number of nodes for the specified number of the energy level; 3. the correct relative wavelengths (longer or shorter) for different values of the potential at different places inside the well; 4. the correct relative maximum values of amplitudes at adjacent points of different potential inside the well; 5. the correct relative rate of decrease (more or less gradual) of the Wave Function with distance outside the well for different values of potential at different places and for quantum states of different energy. In the problems at the end of this chapter you will find a number of potential wells for which you may sketch Wave Functions. The game of sketching Wave Functions for various potential wells can be fun. But beyond this, nature is so varied and prolific that almost any shape we choose for an exercise, no matter how bizarre, will resemble some real potential found in nature

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  Quantum Mechanics for Scientists and Engineers

  • David A. B. Miller(Author)
  • 2008(Publication Date)
  • Cambridge University Press

    (Publisher)

  This would appear to be meaningless classically; an oscillator that has energy ought to oscillate. To understand how we recover oscillating behavior and, indeed, to understand the true meaning of the stationary eigenstates we have calculated, we need first to understand the time-dependent Schrödinger equation, which is the subject of the next chapter. 42 Chapter 2 Waves and quantum mechanics – Schrödinger’s equation Problem 2.10.1 Suppose we have a “half harmonic oscillator” potential, for example, exactly half of a parabolic potential on the right of the “center” and an infinitely high potential barrier to the left of the center. Compared to the normal harmonic oscillator, what are the (normalized) energy eigenfunctions and eigenvalues? (Hint: There is very little you have to solve here; this problem mostly requires thought, not mathematics.) 2.11 Particle in a linearly varying potential This topic can be omitted on a first reading, though it does give some very useful insights into wave mechanics and is useful for several practical problems. Another situation that occurs frequently in quantum mechanics is that we have applied a uniform electric field, E , in some direction – say, the z direction. For a charged particle, this leads to a potential that varies linearly in distance. For example, an electron, which is negatively charged with a charge of magnitude e , will see a potential energy, relative to that at 0 z = , of V e z = E (2.90) In practice, we find this kind of potential in many semiconductor devices. We use it when we are calculating the quantum mechanical penetration (tunneling) through the gate oxide in Metal-Oxide-Semiconductor (MOS) transistors, for example. We see it in semiconductor optical modulators, 35 which use optical absorption changes that result from electric fields. It is of basic interest also if we want to understand how an electron is accelerated by a field, a point to which we return in the next chapter.

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  Band Theory of Metals

  The Elements

  • Simon L. Altmann(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  Since each measurement of a variable such as χ involves an unavoidable perturbation of the system measured, the results of a large number of measurements must be spread, i.e. different results x t will appear, each of them repeated a different number of times w f . We shall call ω ( χ .) = mjn (η = total number of measurements) the frequency or probability of the value x f . When we plot ω(χ) as a function of χ we expect to get the usual gaussian probability distribution curve of Fig. la, where, as in the theory of errors, the error Δ χ is measured by the distance between the turning points of the gaussian. Similarly, a probability distribution for ρ will be obtained, such as that represented in Fig. lb. It follows from Heisenberg's principle that these two curves 4 Band Theory of Metals [Ch. 1] are not independent. Rather, Δρ and Δχ are related by (3), which requires that, if one of the distributions is very narrow, the other one must be very flat. 3. The state function We define in classical mechanics the state Φ of a system as some function of the values of ρ and JC for all the particles of a system: Φ = Φ(ρ, χ). This is so because we know from Newton's equations that, given ρ and χ at the time /, their values at a later time t' are determined. Hence, given Φ, (the value of Φ at the time t), we can always calculate Φ Γ . | In practice, the definition of a state function is always arrived at by a process of trial and error: one has to find the smallest number of variables which describe a system and that are self-predicting, that is such that equations can be found which permit the calculation of the values of all the chosen variables at a time t' from the known values of the same variables at the time t. These variables are called state variables and a function Φ of them which appears in the equations just mentioned is a state function, the basic property of which is that Φ ί determines Φ Γ , a relation which we shall denote symbolically as follows: Φ ί -+Φ ί ..

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  Modern Physics for Scientists and Engineers

  • Stephen Thornton, Andrew Rex, Carol Hood, , Stephen Thornton, Stephen Thornton, Andrew Rex, Carol Hood(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  In the process of doing this we will find that some observables, including energy, have quan- tized values. We begin by exploring the simplest such system—that of a particle trapped in a box with infinitely hard walls that the particle cannot penetrate. This is the same physical system as the particle in a box we presented in Section 5.8, but now we present the full quantum-mechanical solution. The potential, called an infinite square well, is shown in Figure 6.2 and is given by V sxd 5 5 ` x # 0, x $ L 0 0 , x , L (6.30) The particle is constrained to move only between x 5 0 and x 5 L, where the particle experiences no forces. Although the infinite square-well potential is simple, we will see that it is useful because many physical situations can be ap- proximated by it. We will also see that requiring the Wave Function to satisfy certain boundary conditions leads to energy quantization. We will use this fact to explore energy levels of simple atomic and nuclear systems. As we stated previously, most of the situations we encounter allow us to use the time-independent Schrödinger wave equation. Such is the case here. If we insert V 5 ` in Equation (6.14), we see that the only possible solution for the Wave Function is c(x) 5 0. Therefore, there is zero probability for the particle to be located at x # 0 or x $ L. Because the kinetic energy of the particle must be finite, the particle can never penetrate into the region of infinite potential. How- ever, when V 5 0, Equation (6.14) becomes, after rearranging, d 2 c dx 2 5 2 2m E " 2 c 5 2k 2 c where we have used Equation (6.14) with V 5 0 and let the wave number k5 Ï2mE y " 2 . A suitable solution to this equation that satisfies the properties given in Section 6.1 is csxd 5 A sin kx 1 B cos kx (6.31) V( x) x 0 Position ∞ ∞ L Figure 6.2 Infinite square- well potential. The potential is V 5 ∞ everywhere except the region 0 , x , L, where V 5 0. Copyright 2021 Cengage Learning. All Rights Reserved.

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