Erwin Schrödinger

11 Key excerpts on "Erwin Schrödinger"

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  Quantum 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.

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  A Student's Guide to the Schrödinger Equation

  • Daniel A. Fleisch(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  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. But as the authors of those texts invariably point out, none of those methods are rigorous derivations from first principles (hence the quotation marks). As the brilliant and always-entertaining physicist Richard Feynman said, “It’s not possible to derive it from anything you know. It came out of the mind of Schr¨ odinger.” So if Erwin Schr¨ odinger didn’t arrive at this equation from first principles, how exactly did he get there? The answer is that although his approach evolved over several papers, from the start Schr¨ odinger clearly recognized the need for a wave equation from the work of French physicist Louis de Broglie. But Schr¨ odinger also realized that unlike the classical wave equation, which is a second-order partial differential equation in both space and time, the form of the quantum wave equation should be first-order in time, for reasons explained later in this chapter. Importantly, he also saw that making the equation complex (that is, including a factor of √ − 1 in one of the coefficients) provided immense benefits. One approach to understanding the basis of the Schr¨ odinger equation is to begin with the classical equation relating total energy to the sum of kinetic energy and potential energy. To apply this principle to quantum wavefunctions 1 , begin with the equation developed by Max Planck and Albert Einstein in the early twentieth century relating the energy of a photon ( E ) to its frequency ( f ) or angular frequency ( ω = 2 π f ): E = hf = ¯ h ω , (3.1) in which h represents the Planck constant and ¯ h is the modified Planck constant ( ¯ h = h 2 π ).

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  Introduction to Quantum Mechanics

  • 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.

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  Quantum Mechanics for Tomorrow's Engineers

  • Junichiro Kono(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  (2.8) It is important to note that the Schrödinger equation can be written as ˆ Hψ = Eψ, 18 where 18 According to linear algebra, an equation of this form is known as an eigenvalue equation. See Chapter 3 for more details. ˆ H = − ¯ h 2 2m ∇ 2 + V(r ) (2.9) is known as the Hamiltonian operator. For a given potential energy V, and thus ˆ H, one can solve the Schrödinger equation to obtain a set of eigensolutions {ψ n } n=1,2, ... (or “eigenfunctions”) and the cor- responding eigenvalues {E n } n=1,2, ... (or “eigenenergies”). 19 19 Schrödinger solved this equation for a particular potential energy, V(r) = −e 2 /(4πε 0 r), i.e., an at- tractive Coulomb potential, to obtain eigenfunctions and eigenenergies that successfully explained the emission spectrum of the hydrogen atom. 2.1.3 Wavefunction The meaning of the wavefunction, ψ(r ), was initially a subject of much debate. 20 Schrödinger was trying to develop a theory that 20 This was just one of the many subjects of much debate among the founders of quantum mechanics. For more details about the early dis- putes about quantum theory (espe- cially those between Bohr and Ein- stein), see, e.g., Chapter IV of A. Pais, “Subtle is the Lord . . .”: The Science and the Life of Albert Einstein (Oxford Uni- versity Press, 1982); see also J. Bag- gott, The Meaning of Quantum The- ory (Oxford University Press, 1992), Chapter 3. explains atomic- and subatomic-scale phenomena entirely using the classical theory of waves. As such, he was seeking an interpreta- tion of the wavefunction purely from a wave point of view, with- out success. A correct interpretation of ψ(r ), embracing the concept of wave–particle duality, was proposed by Max Born: the square of the amplitude of the wavefunction in some specific region of space, |ψ(r )| 2 , is related to the probability of finding the associated quan- tum particle in that region of space.

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  Basic Molecular Quantum Mechanics

  • Steven A. Adelman(Author)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  3 The Schrodinger Equation and the Particle-in-a-Box

  In 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 wavelength

  λ =

  c υ

  ,

  where c is the speed of light, could also be viewed as a stream of particles called photons each with an energy E and momentum

  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 =

  h λ

  .

  (3.1)

  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 wavelength

  λ .

  In analogy to Einstein’s photon relation

  p =

  h λ

  , de Broglie postulated that the wavelength of the matter wave associated with a particle of momentum p

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  Electrons, 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, 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. When applying to ob-jects of large dimensions the Schrödinger equation, as might be expected, leads to relations which are equivalent to Newton's Laws. The Schrödinger equation in rectangular co-ordinance is J^(^ + ^ + £ï_ E JL^ÊÏ. (n. 2)t %n 2 m dx 2 dy 2 dz 2 ) j 2π dt where E p is the potential energy of the particle under study. The behavior of ψ is limited by the following restrictions: ψ must be finite, continuous, and single-valued at all points in space and time; (11.3) the rate of change of ψ must be finite and continuous at all points in space. (11.4) The reasons for these restrictions will be pointed out later in regard to a one-dimensional problem. The Schrödinger equation in general form is a second-order partial differen-tial equation involving space co-ordinates and time, and contains both real and imaginary terms. The potential energy E p may also vary in time and space. t In this text j = yj — 1. 156 ELECTRONS, NEUTRONS AND PROTONS IN ENGINEERING In attempting to apply the Schrödinger equation to a practical problem one finds, as would be expected, that the simplest situation is that of the hydrogen atom. In attempting to apply the Schrödinger equation to more complicated atoms, to molecules, and to crystals, exact mathematical solutions are, in many cases, impossible and the investigator is forced to resort to methods of ap-proximation. 11.4. A P P L I C A T I O N TO A ONE-DIMENSIONAL PROBLEM.

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  Modern Physics

  • 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 −it 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 nx 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.

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  Quantum Mechanics

  • 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 Notation

  In 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.

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  Modern Physics

  • Gary N. Felder, Kenny M. Felder(Authors)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  With those two skills in mind, you can see where we are in this chapter and where we need to go. In Section 5.2 (and most of the rest of this chapter) the time-independent Schrödinger equation is used to find the energy eigenstates associated with a given potential function. In this section we will present a general rule that you can use to predict how any wavefunction will evolve over time, based on knowing its energy eigenstates. In Chapter 6 we will present the time-dependent Schrödinger equation, the fundamental axiom of quantum mechanics, and show how it leads to that time-evolution rule. This section relies on the math from Section 5.5. Make sure you are comfortable with complex numbers, modulus, complex conjugates, complex exponentials, and Euler’s formula, or you’ll miss all the fun. 5.6.1 Discovery Exercise: Time Evolution of a Wavefunction In each question below we’ll give you a complex number and ask you to calculate its modulus squared. Assume x is real, and simplify your answers as much as possible. You should be able to write each answer in terms of all real quantities (no i). 1. z 1 = e ix , so |z 1 | 2 = 2. z 2 = e 2ix so |z 2 | 2 = 3. z 3 = z 1 + z 2 = e ix + e 2ix so |z 1 + z 2 | 2 = Hint : The answer is not the sum of the previous two answers, |z 1 | 2 + |z 2 | 2 . 5.6.2 Explanation: Time Evolution of a Wavefunction You have a particle with a known wavefunction in a known potential energy field. How will that wavefunction evolve over time? This section will build up a process for answering that question through three scenarios of increasing complexity. First, though, we need to introduce some notation. Some Notation It’s common in quantum mechanics to write a wavefunction at one particular time as ψ(x), and to write the wavefunction at all times as (x, t ). (Those symbols are the lowercase and uppercase versions of the Greek letter psi.) So if we say a particle’s wavefunction started out

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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)

  As far as I remember these discus- sions took place in Copenhagen around September 1926 and in particular they left me with a very strong impression of Bohr's personality. For though Bohr was an unusually considerate and oblig- ing person, he was able in such a discus- sion, which concerned epistemological problems which he considered to be of vital importance, to insist fanatically and with almost terrifying relentlessness on complete clarity in all arguments. He would not give up, even after hours of struggling, before Schrodinger had ad- mitted that this interpretation was in- sufficient, and could not even explain Planck's law. Every attempt from Schrodinger's side to get round this bitter result was slowly refuted point by point in infinitely laborious discussions. It was perhaps from over-exertion that after a few days Schrodinger became ill and had to lie abed as a guest in Bohr's home. Even here it was hard to get Bohr away from Schrodinger's bed and the phrase, "But, Schrodinger, you must at least admit that..." could be heard again and again. Once Schrodinger burst out almost desperately, "If one has to go on with these damned quan- tum jumps, then I'm sorry that I ever started to work on atomic theory." To which Bohr answered, "But the rest of us are so grateful that you did, for you have thus brought atomic physics a decisive step forward." Schrodinger fi- nally left Copenhagen rather discour- aged, while we at Bohr's Institute felt that at least Schrodinger's interpreta- tion of quantum theory, an interpreta- tion rather too hastily arrived at using the classical wave-theories as models, was now disposed of, but that we still 1.3 COMMENTARY 57 lacked some important ideas before we could really reach a full understanding of quantum mechanics.

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  Quantum 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

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