Physics
Schrodinger Equation
The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is used to calculate the wave function of a particle and predict its behavior. The equation is essential for understanding the behavior of particles at the quantum level.
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12 Key excerpts on "Schrodinger Equation"
eBook - PDF- Henrik Smith(Author)
- 1991(Publication Date)
- WSPC(Publisher)
106 Introduction to Quantum Mechanics 4 THE Schrodinger Equation In the following we discuss some properties of the time-dependent Schrodinger Equation, which is the fundamental equation of motion in the quantum theory, corresponding to Newton's equations in classical mechanics. Like Newton's equations the validity of the Schrodinger Equation is limited to situations where relativistic effects may be neglected. We shall see how the time-dependent Schrodinger Equation is reduced to the eigenvalue equation for the energy under stationary conditions. The eigen-states that are solutions to this equation are called stationary states, since the associated probability densities are independent of time. By forming superpo-sitions of stationary states we are able to describe time-dependent phenomena and compare with the result of solving the classical equations of motion. Al-though a given state may not be an eigenstate for the energy operator, it may be an eigenstate for other operators associated with, say, momentum or an-gular momentum. As we shall see in Example 4 below, the state of a system may also be characterized by being an eigenstate for a non-Hermitian operator. The special feature of the eigenstates of the energy operator is the simplicity of their development in time, as demonstrated below. We consider a single particle with mass m moving in the potential V(r). As shown in Section 3.3.3 of the previous chapter, the time-dependent Schrodinger Equation has the form - ^V 2 V-(r,<) + VW(r,t) = ift^^M (4.1) where the left-hand side is Hip with H being the Hamiltonian. In general the wave function ip(r,t) is not necessarily an eigenstate of the Hamiltonian and may therefore not be labelled by an energy eigenvalue. Its physical interpretation is that of a probability amplitude. Thus |V(r,t)| 2 dr (4.2) is the probability that a measurement of the position of the particle yields a result in the volume element dr(= dxdydz) at r.
eBook - PDFQuantum Physics
An Introduction
- J Manners(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
In developing quantum mechanics, he was strongly influenced by de Broglie’s idea that a wave is associated with each free particle. Schrodinger’s attempts to extend this idea to confined particles led him to formulate his famous equation (Figure 2.15). He shared the 1933 Nobel Prize for physics with Paul Dirac for their important contributions to ‘new and productive forms of atomic theory’. He was a man of very broad scientific interests: in addition to his definitive work in atomic theory, he wrote papers in the fields of colour perception, X-ray diffraction, statistical mechanics and general relativity. Figure 2.1 5 The point in Schrodinger’s notes at which his celebrated equation first appears. 62 Quantum physics: an introduction The wavefunction *^that describes the behaviour of any particular quantum system is found by solving the appropriate form of Schrodinger’s equation. Now, there is an inherent difficulty in writing down Schrodinger’s equation and pursuing the ideas of quantum mechanics at the mathematical level of this course. It is an unfortunate fact that the mathematical language required is generally quite complicated. Nevertheless, by restricting our study of quantum mechanics to sufficiently simple situations, it is possible to develop an insight into the subject, and this is what we shall endeavour to do in this section. In particular, we shall mostly restrict the discussion to just one dimension — usually taken to be the jc-direction. Even so, you should still be able to see how Schrodinger’s equation provides a description of particle behaviour in terms of a wavefunction W{x , t), and why this description leads naturally to the quantization of the energy of particles in certain circumstances. In what follows we shall be concerned with essentially two types of waves, travelling waves and standing waves.
eBook - PDF- Junichiro Kono(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
2 Electron Waves and Schrödinger’s Equation Learning objectives: • Developing an understanding of when and how classical par- ticles start behaving as quan- tum mechanical waves. • Deriving the most fundamen- tal equation in quantum me- chanics, the Schrödinger equa- tion, which determines the states and dynamics of quan- tum particles. • Understanding the probabilis- tic meaning of the wavefunc- tion, ψ, and learn how to cal- culate the expectation values of observable quantities. • Solving the Schrödinger equa- tion for example problems in- volving electrons in simple po- tential energy landscapes. Quantum mechanics is currently the most fundamental theory in use in many disciplines of science and engineering. It is particu- larly important when one is dealing with nanoscale and atomic-scale systems. However, many phenomena and properties that occur at atomic scales are strange and nonintuitive. There are a number of concepts that simply do not exist in the macroscopic world where we live. Wave–particle duality is one of them. In this chapter, we exam- ine how and when classical particles start behaving as quantum me- chanical waves, derive the most important wave equation that quan- tum particles obey, Schrödinger’s equation, and solve it for the ele- mentary problems of electron waves in given potential energy land- scapes. We will also learn how to calculate the expectation values of observables when the wavefunction is known. Schrödinger’s equa- tion will be extensively used throughout the rest of this textbook. More complicated potential energy problems, particularly those rel- evant to materials and devices, will be dealt with in Chapters 5 and 7, building upon the formulations developed in this chapter. 2.1 Wave Equation and Wavefunction One of the striking consequences of quantum theory is that light be- haves as particles under certain circumstances and electrons behave as waves under certain circumstances.
- Daniel A. Fleisch(Author)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
3 The Schr¨ odinger Equation If you’ve worked through Chapters 1 and 2 , you’ve already seen several references to the Schr¨ odinger equation and its solutions. As you’ll learn in this chapter, the Schr¨ odinger equation describes how a quantum state evolves over time, and understanding the physical meaning of the terms of this powerful equation will prepare you to understand the behavior of quantum wavefunctions. So this chapter is all about the Schr¨ odinger equation, and you can read about the solutions to the Schr¨ odinger equation in Chapters 4 and 5 . In the first section of this chapter, you’ll see a “derivation” of several forms of the Schr¨ odinger equation, and you’ll learn why the word “derivation” is in quotes. Then, in Section 3.2 , you’ll find a description of the meaning of each term in the Schr¨ odinger equation as well as an explanation of exactly what the Schr¨ odinger equation tells you about the behavior of quantum wavefunctions. The subject of Section 3.3 is a time-independent version of the Schr¨ odinger equation that you’re sure to encounter if you read more advanced quantum books or take a course in quantum mechanics. To help you focus on the physics of the situation without getting too bogged down in mathematical notation, the Schr¨ odinger equation discussed in most of this chapter is a function of only one spatial variable ( x ). As you’ll see in later chapters, even this one-dimensional treatment will let you solve several interesting problems in quantum mechanics, but for certain situations you’re going to need the three-dimensional version of the Schr¨ odinger equation. So that’s the subject of the final section of this chapter ( Section 3.4 ). 63 64 3 The Schr¨ odinger Equation 3.1 Origin of the Schr¨ odinger Equation If you look at the introduction of the Schr¨ odinger equation in popular quantum texts, you’ll find that there are several ways to “derive” the Schr¨ odinger equation.
eBook - ePub- Steven A. Adelman(Author)
- 2021(Publication Date)
- CRC Press(Publisher)
3 The Schrodinger Equation and the Particle-in-a-BoxIn Chapters 4 –6 , we lay out the full formal basis of quantum mechanics, and in the succeeding chapters, we describe increasingly advanced applications of this basis. However, in order to convey some feeling for quantum mechanics before developing its full formal machinery in this chapter, we introduce the most useful quantum equation, the time-independent Schrodinger Equation, and then apply it to one of the simplest quantum systems, the instructive one-dimensional particle-in-a-box system.The time-independent Schrodinger Equation, as we will show in Section 3.2 , may be derived from the more fundamental time-dependent Schrodinger Equation, already touched on in Section 1.5 . So we begin with the time-dependent Schrodinger Equation. This equation cannot be derived from anything more fundamental. Rather, like Newton’s equation of motion, it is best viewed as a postulate that is accepted because it successfully predicts a vast range of phenomena.While the time-dependent Schrodinger Equation cannot be derived, several non-rigorous plausibility arguments for its form exist. We next give one of these.3.1 A Heuristic “Derivation” of the Time-Dependent Schrodinger Equation
In our discussion of the photoelectric effect in Section 1.1 , we noted that Einstein discovered that light exhibits a wave–particle duality, namely that a light wave of frequency υ or wavelengthλ =where c is the speed of light, could also be viewed as a stream of particles called photons each with an energy E and momentumc υ,p .Einstein postulated that the particle properties of light E and p were related to its wave properties υ and λ as follows:E = h υ a n d p =(3.1)h λ.We further noted in Section 1.4 that de Broglie later hypothesized that ordinary particles also exhibit a wave–particle duality. Namely, de Broglie hypothesized that associated with a particle is a wave with de Broglie wavelengthλ .In analogy to Einstein’s photon relationp =, de Broglie postulated that the wavelength of the matter wave associated with a particle of momentum ph λ
eBook - PDFElectrons, Neutrons and Protons in Engineering
A Study of Engineering Materials and Processes Whose Characteristics May Be Explained by Considering the Behavior of Small Particles When Grouped Into Systems Such as Nuclei, Atoms, Gases, and Crystals
- J. R. Eaton(Author)
- 2013(Publication Date)
- Pergamon(Publisher)
For example, in studying the hydrogen atom by this method, the results are finally presented as functions showing the probability density at various points in the near vicinity of the nucleus. Such information was shown without supporting proof in Chapter 5. As was indicated there, the probability density was expressible in three dimensional space co-ordinates, and this distribution changed markedly with changes in the energy of the system. All attempts to trace the path of moving particles is abandoned and no fixed orbits are implied. The pattern of motion is left entirely indeterminate with no loss in significance of the outcome. It might be expected that the mathematical equations applying to systems of particles would involve the variable U, the probability density, This, however, THE Schrodinger Equation 155 is not the case. The equations are written in terms of a new parameter ψ, called the wave function, which in itself has no physical significance. The wave function ψ serves as a mathematical intermediary in the process of calculating probability density and other characteristics of importance. It is possible to write equations in ψ which apply with great generality to particles systems. If it is possible then to make a quantitative determination of ψ, the probability den-sity U is then easily determined through the relation U=y> 2 . (11.1) This relation says Uis equal to the square of the scalar value of ψ, which itself is usually a complex number. An alternative form of this expression is ϋ=ψψ*, (11.1a) in which ψ* is the conjugate of ψ. For example, if ψ is a + jb, then ψ* is a — jb, and Uis a 2 + b 2 . 11.3. THE SCHRÖDINGER EQUATION Probably the most important single relation of wave mechanics is the Schrö-dinger equation, a relation which applies over a wide range of conditions ex-tending from particles in the microscopic system to the large scale objects of laboratory dimensions with which we are most familiar.
eBook - PDF- Don Shillady(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
11 The Schrödinger Wave Equation INTRODUCTION This chapter is primarily about ‘‘ wave mechanics ’’ since that is the most convenient way to introduce undergraduates to quantum mechanics using calculus. An equivalent form called ‘‘ matrix mechanics ’’ will be discussed brie fl y in a later chapter. Consider again the 1923 paper by De Broglie and the experiments that validated the particle-wave duality in Chapter 10. One might well ask that if there really is some ‘‘ wave ’’ that describes the behavior of particles, then is there an equation that the wave obeys? Even today it is dif fi cult to say what the ‘‘ wave ’’ is, but it may help to fi nd an equation it obeys. Step back a moment to some basic calculus: d 2 dx 2 sin ( ax ) ¼ d dx [ a cos ( ax )] ¼ ( a 2 ) sin ( ax ) : Perhaps you did not notice the pattern before but we can put this into a general form as ( Operator )( Eigenfunction ) ¼ ( Eigenvalue )( Eigenfunction ) : The word ‘‘ eigen ’’ in German means ‘‘ characteristic, unique, peculiar, special . . . ’’ and only certain functions satisfy this condition called an eigenfunction equation. An analogy that has been suc-cessful in explaining this to undergraduates is to consider an apple tree with ripe apples on it. If you hit the branches with a stout stick some apples will fall off the tree but the tree will still be there. The operator is the act or operation of hitting the tree with the stick, the tree is the eigenfunction and the apples are the eigenvalue(s). The eigen word comes from German because this relationship was fi rst linked to the De Broglie wave idea by Erwin Schrödinger in 1926 in a series of papers that are among the most important in modern science [1]. Schrödinger (1887 – 1961) was an Austrian physicist who received the Nobel Prize for his work in 1933 (Figure 11.1). We will now present a derivation of the Schrödinger equation that may not be the way he thought of it but that follows from limited use of calculus and simple algebra.
eBook - PDF- Kenneth S. Krane(Author)
- 2020(Publication Date)
- Wiley(Publisher)
In practice, a feedback mechanism keeps the current constant by moving the tip up and down. The motion of the tip gives a map of the surface that reveals details smaller than 0.01 nm, about 1/100 the diameter of an atom! For the development of the scanning tunneling microscope, Gerd Binnig and Heinrich Rohrer were awarded the 1986 Nobel Prize in physics. Chapter Summary Section Time-independent Schrödinger equation − − h 2 2m d 2 dx 2 + U (x) (x) = E (x) 5.3 Time-dependent Schrödinger equation Ψ(x, t) = (x)e −it 5.3 Probability density P(x) = | (x)| 2 5.3 Normalization condition ∫ +∞ −∞ | (x)| 2 dx = 1 5.3 Section Probability in interval x 1 to x 2 P(x 1 ∶x 2 ) = ∫ x 2 x 1 | (x)| 2 dx 5.3 Average or expectation value of f (x) [f (x)] av = ∫ +∞ −∞ | (x)| 2 f (x) dx 5.3 Constant potential energy, E > U 0 (x) = A sin kx + B cos kx, k = √ 2m(E − U 0 )∕ − h 2 5.4 Questions 173 Section Constant potential energy, E < U 0 (x) = Ae k ′ x + Be −k ′ x , k ′ = √ 2m(U 0 − E)∕ − h 2 5.4 Infinite potential energy well n (x) = √ 2 L sin nx L , E n = h 2 n 2 8mL 2 (n = 1, 2, 3, … ) 5.4 Two-dimensional infinite well (x, y) = 2 L sin n x x L sin n y y L E = h 2 8mL 2 (n 2 x + n 2 y ) 5.4 Section Simple harmonic oscillator ground state (x) = (m 0 ∕ − h) 1∕4 e −( √ km∕2 − h)x 2 5.5 Simple harmonic oscillator energies E n = ( n + 1 2 ) − h 0 (n = 0, 1, 2, … ) 5.5 Potential energy step, E > U 0 0 (x < 0) = A sin k 0 x + B cos k 0 x 1 (x > 0) = C sin k 1 x + D cos k 1 x 5.6 Potential energy step, E < U 0 0 (x < 0) = A sin k 0 x + B cos k 0 x 1 (x > 0) = Ce k 1 x + De −k 1 x 5.6 Questions 1. Newton’s laws can be solved to give the future behavior of a particle. In what sense does the Schrödinger equation also do this? In what sense does it not? 2. Why is it important for a wave function to be normal- ized? Is an unnormalized wave function a solution to the Schrödinger equation? 3. What is the physical meaning of ∫ +∞ −∞ | | 2 dx = 1? 4.
eBook - PDF- 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).
eBook - ePub- Mark Julian Everitt, Kieran Niels Bjergstrom, Stephen Neil Alexander Duffus(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
key outcome that this introduction to quantum mechanics tries to make clear (although, as we shall see, this is not without its difficulties).Next, we use an examination of the energy of the system that arises from deriving the time-independent Schrödinger equation, to motivate a discussion of what it means to make a measurement in quantum mechanics. We discuss this in comparison to the classical analogy of measurement in statistical physics. The measurement process is the third key outcome of this introduction.Up until this point, a lot of the mathematical presentation may have appeared rather abstract. At some point we need to solve problems and make predictions. So we now give some specific examples of quantum states in terms of position, spin, and position and spin. The derivations here are long, but have been chosen as they make clear how few assumptions are needed to go from an abstract theory to some equations we can solve. We do not advise you try to remember all these long derivations – the key point is to understand how the few assumptions made at the beginning lead to the destination reached at the end of each section (such as the form of wave-function and the Schrödinger equation in position representation).We will close this chapter by presenting (i) a technical summary of what we have done, and (ii) a number of possible axiomatic foundations of quantum mechanics. These are important, as they are the theory.3.2 Motivating the Schrödinger Equation
Prerequisite Material: Section 1.2 : Generalising Vectors, Section 1.2.1 : Vector Spaces, Section 1.2.2 : Inner Product Section 1.2.3 : Dirac NotationIn developing a new theory, we do not want to abandon the best parts of an old theory. In the following discussion, we try to emphasise how much of classical physics is in fact retained in quantum physics, which is far less of a revolutionary theory than many people think. Note that arguments, such as the one that follows, are always speculative and based on having to reject some ideas thought evidently correct, and replace those with new ones that might look odd.
eBook - PDFQuantum Mechanics
Foundations and Applications
- Donald Gary Swanson(Author)
- 2006(Publication Date)
- CRC Press(Publisher)
2 The Schr¨ odinger Equation in One Dimension In the previous chapter, the postulates helped us to use the wave function to find the expected results of experimental measurements by letting a dy-namical quantity F ( x, p, t ) → F (ˆ x, ˆ p, t ) and integrating over space. In the examples, however, the wave functions were typically given and not derived from first principles. We now endeavor to find the appropriate wave functions for a variety of physical systems in one dimension by solving the Schr¨ odinger equation where for time-independent cases, the total energy is an eigenvalue. In subsequent chapters, we will extend the method to three dimensions and develop the appropriate operators, eigenvalues, and eigenfunctions for angu-lar momentum. We will also develop methods for examining time-dependent systems. The latter chapters address special systems of interest that may be referred to as applications of quantum mechanics. 2.1 The Free Particle As our first example of solving the Schr¨ odinger equation, we choose the free particle in one dimension, which means that the particle is free of any forces. It thus has a constant potential, so the time-independent Schr¨ odinger equation in one dimension is written as ˆ H ψ = - 2 2 m d 2 ψ d x 2 + V ψ = Eψ , (2.1) where ψ = ψ ( x ) is the spatial part of the wave function, and the complete wave function is Ψ( x, t ) = ψ ( x ) exp ( -i Et/ ). Just as the time-dependent equation was easily solved in the previous chapter, so is the solution of Equation (2.1) easily solved, with solution ψ ( x ) = A exp ± i 2 m ( E -V ) x/ . (2.2) Problem 2.1 Show that the solution of the 1-dimensional Schr¨ odinger equa-tion for a free particle can be written as ψ ( x ) = A exp ( ± i kx ) where k is the classical momentum of a free particle of energy E . 41
eBook - PDF- Paul Bracken(Author)
- 2013(Publication Date)
- IntechOpen(Publisher)
At much higher energy values many other particles may be, of course, formed. In such a case the subspace Θ is to be substituted by a more complex system of products (or sums) of corresponding subspaces describing further evolution of separated physical systems. While the simple subspaces (in which the interacting particles remain stable) may be in principle described with the help of corresponding Schrödinger equations the other processes require to be characterized by additional probabilities between concrete states in individual subspaces. There is not any interference between amplitudes from different orthogonal subspaces. The individual (stable as well as unstable) objects represent closed physical systems that are characterized by some quantum physical values. Each object has some internal dynamics (eventually, exists in some different internal states -stable or unstable). And just these questions represent evidently one of the main problems of the future quantum physics. 3. Closed systems and quantum states The idea of quantum states has been based on experimental data concerning the measured light spectra emitted by excited atoms, as it was formulated in two phenomenological postulates of N. Bohr [25]. These spectra have been correlated to transitions between different quantum energy levels. The existence of quantum states have been then derived with the help of Schrödinger equation containing Coulomb potential. However, it is necessary to call attention to the fact that the Schrödinger equation provides an approximative phenomenological description of quantum phenomena only. It predicts and admits the existence of quantum states but it cannot explain at all how such a quantum state may arise when two corresponding objects (forming then the closed system) are mutually attracted and go always nearer one to the other.
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