Infinite Square Well

12 Key excerpts on "Infinite Square Well"

• 

  eBook - PDF

  Quantum Mechanics I

  A Problem Text

  • David DeBruyne, Larry Sorensen(Authors)
  • 2018(Publication Date)
  • Sciendo

    (Publisher)

  Chapter 8 The Infinite Square Well Any wave function limited to an interval such as -a < x < a can be interpreted physically as being between infinitely thick, infinitely high potential energy “walls.” Intervals were introduced in chapter 4 so that methods of calculating probabilities, expectation values, and uncertainties for continuous systems could be addressed; wave functions that are not square integrable over all space generally are square integrable on an interval. This chapter introduces energy quantization for a continuous system, quantum numbers, the meaning of a linear combination of continuous eigenfunctions, and some common methods of treating boundary value problems. An electron in an atom is an example of a particle confined to a limited region. It demonstrates energy quantization while confined. The limited region can be considered to be a “box” with indistinct “soft walls” formed by electrical forces. The first step toward describing such a realistic system is to examine a one dimensional box with the simplest possible geometry and infinite “hard walls.” This bit of unrealism makes the mathematics most tractable yet reveals the same quantum mechanical phenomena of any bound particle, in particular, energy quantization. Any particle trapped in any potential energy well exhibits energy quantization. The “particle in a box” is an informal name for a square well. It is a nickname that sometimes masks phenomena of physical interest and seemingly becomes an end in itself. It is rather the second step (the free particle being the first step) in treating increasingly realistic potential energy functions describing increasingly sophisticated boundary conditions. It may be useful to picture an electron in a highly unusual atom or a proton in a highly unusual nucleus as you work through this precursor to and idealization of more realistic systems. An electron trapped in an atom is a particle in an electrostatic “box.” 8–1.

  Sign up to read

  Learn more about book
• 

  No longer available |Learn more

  Physics for Scientists and Engineers

  Foundations and Connections, Extended Version with Modern Physics

  • Debora Katz(Author)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  An infinitely wide well means that the particle can go anywhere. SOLVE Take the limit as L S ` . lim LS` E 1 5 lim LS` h 2 8mL 2 S 0 CHECK and THINK When a particle is free (not part of any system), it can have any energy value. So its minimum energy is zero and it may be at rest. Only a confined particle is required to have nonzero energy and to be in motion. Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 41-6 Special Case: A Particle in a Finite Square Well 1353 Unless otherwise noted, all content on this page is © Cengage Learning. FIGURE 41.9 The potential energy is zero inside the well and finite outside. U U 0 E E < U 0 L 0 x Schrödinger’s equation for the region inside the well is the same as it was in the case of an infinite well: d 2 c 1x 2 dx 2 1 k 2 c 1x 2 5 0 (41.13) where k 2 ; 8p 2 mE / h 2 . The general solution must be the same as in the infinite well: c 1x 2 5 A sin kx 1 B cos kx (41.14) But now the boundary conditions are different. We no longer require that c 1x 2 5 0 for x # 0 and for x $ L. Instead, we only need c 1x 2 and dc / dx to be continuous functions at the boundaries. (Later in this chapter we will show that these boundary conditions are sufficient to produce the same results as in the case of the Infinite Square Well when U 0 S ` .) With these more relaxed boundary conditions, B Z 0. Both terms in Equation 41.14 hold inside the well, and together they describe an oscillating function. Outside the well, Schrödinger’s equation is: 2 h 2 8p 2 m d 2 c 1x 2 dx 2 1 U 0 c 1x 2 5 Ec 1x 2 We group terms to write: d 2 c 1x 2 dx 2 2 8p 2 m h 2 1U 0 2 E 2 c 1x 2 5 0 and define k 2 ; 18p 2 m / h 2 21U 0 2 E 2 , so that we have: d 2 c 1x 2 dx 2 2 k 2 c 1x 2 5 0 (41.19) for Schrödinger’s equation outside the well. But we are only considering the case of a particle trapped in the well, such that E , U 0 or k 2 .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  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.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Quantum Physics for Beginners

  • Zbigniew Ficek(Author)
  • 2017(Publication Date)
  • Jenny Stanford Publishing

    (Publisher)

  Fig. 8.1 . The term “well” is a bit misleading since the particle is actually only trapped in one direction. It is still free to move in other two directions. However, the term “well” is commonly used in the literature and we will follow this terminology.

  For the infinite potential well centered at x = 0:

  V ( x ) = 0 for − a 2 ≤ x ≤ a 2 , V ( x ) = ∞     for x < − a 2         and           x > a 2 .

  (8.9)

  Figure 8.1 An infinite potential well. Outside the region −a/2 ≤ x ≤ a/2, the potential V (x) → ∞.

  Within the well, there is no potential energy, while outside the well the potential is infinite, so that the particle cannot exist there since it would have to have infinite energy. What classical physics and quantum physics tell us about the behavior of the particle inside the well?

  According to classical physics, the particle trapped between the potential walls will bounce back and forth indefinitely; its kinetic energy will be constant E = mv2 /2. Moreover, the probability of finding the particle at any point between the walls is constant and anywhere outside the walls is zero. In fact, if we know the initial momentum and position of the particle, we can specify the location of the particle at any time in the future. The classical case seems trivial.

  According to quantum physics, the particle is described by a wave function ϕ(x), which satisfies the Schrödinger equation and some boundary conditions. One of the boundary conditions says that the wave function ϕ(x) must be finite everywhere. Thus, in the regions x < −a/2 and x > a/2, the wave function ϕ(x) must be zero to satisfy this condition that V(x)ϕ(x) must be finite everywhere.

  In the region −a/2 ≤ x ≤ a/2, the potential V(x) = 0, and then the Schrödinger equation for the wave function takes the form

  d 2 ϕ ( x ) d x 2 = − k 2 ϕ ( x ) ,

  (8.10)

  where k2 = 2mE/ħ2 .

  Since k2 is positive, the Schrödinger equation (8.10)

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Applied Nanophotonics

  • Sergey V. Gaponenko, Hilmi Volkan Demir(Authors)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  Notably, though at least one localized state always exists in a rectangular one-dimensional well, this may not be the case for more complicated situations − e.g., a one-dimensional asymmetric well (left and right potential walls have different heights) may not contain a state inside. This means a quantum particle may not be captured by such a well. The same is true for two- and three-dimensional wells. Shallow and narrow wells may have no state inside. A relation between E and k in the case of a free particle has the form E k m / 2 2 2 =  (dashed curve in Figure 2.3(e)). In the case of the finite potential well, a part of the E(k) function relevant to confined states is replaced by discrete points. 2.1.5 Potential shape and energy spectrum Every time an electron or any other quantum particle experiences localization in space owing to the potential well, its energy spectrum becomes discrete. There are a number of important potential well profiles in quantum mechanics that are useful in practical problems and can provide intuitive insight on electron properties in various wells. From Figure 2.3(a) and Eq. (2.9) one can see that a rectangular potential well gives rise to an infinite set of expanding energy levels. Notably, wave function strictly vanishes to zero only at the border of an infinite poten- tial well. In all other cases wave function penetrates outside the potential barrier, i.e., a Figure 2.4 (a) Quantum harmonic oscillator, the first three wave functions and energy levels; (b) U-like and (c) V-like potential wells. The quantum harmonic oscillator represents a unique potential well that gives rise to equidistant energy levels of a quantum particle. U-like and V-like wells near the bottom can be approximated by parabolas (red dashes) and therefore a few lower states have energies close to those in a harmonic oscillator.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Quantum Mechanics

  A Paradigms Approach

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

    (Publisher)

  148 Quantized Energies: Particle in a Box an eigenstate can vary within the well, let’s make a slight modification to the Infinite Square Well. Consider the well shown in Fig. 5.25, which is commonly referred to as the asymmetric square well. By adding a “shelf” within the well, we now have two regions of constant but different poten- tial energy. The potential energy for this asymmetric square well is V1x2 = μ  , 0, V 0 ,  , x 6 0 0 6 x 6 L > 2 L > 2 6 x 6 L x 7 L . (5.140) We know that the infinite potential outside the well demands that the energy eigenstates are zero outside the well. Inside the well, we now have different energy eigenvalue equations in the left and right halves: a - U 2 2m d 2 dx 2 + 0b w E 1x2 = Ew E 1x2, left half a - U 2 2m d 2 dx 2 + V 0 b w E 1x2 = Ew E 1x2, right half . (5.141) For this discussion, let’s assume that the energy E is greater than the potential V 0 so that the solutions in each half of the well are sinusoidal. We then have different wave vectors in each half, defined by k 1 = B 2mE U 2 , left half k 2 = B 2m1E - V 0 2 U 2 , right half , (5.142) which yields a smaller wave vector 1k 2 6 k 1 2 and hence larger wavelength of the wave in the right half. We know that the left-half solution must be a sine function in order to match the zero wave func- tion outside the well, so the general solution is w E 1x2 = e A sin k 1 x , B sin k 2 x + C cos k 2 x , 0 6 x 6 L > 2 L > 2 6 x 6 L . (5.143) 0 L/2 L V 0 x V(x)  FIGURE 5.25 Asymmetric square well. 5.9 Asymmetric Square Well: Sneak Peek at Perturbations 149 Now we apply the boundary condition on the wave function continuity at the middle and right side of the well and the boundary condition on the continuity of the first derivative of the wave function at the middle of the well (recall that the infinite potential on the right means that the derivative condition is not applicable).

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Quantum Wells, Wires and Dots

  Theoretical and Computational Physics of Semiconductor Nanostructures

  • Paul Harrison, Alex Valavanis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  Chapter 2 Solutions to Schrödinger’s equation 2.1 The infinite well The infinitely deep one-dimensional potential well is the simplest confinement potential to treat in quantum mechanics. Virtually every introductory-level text on quantum mechanics considers this system, but nonetheless it is worth visiting again as some of the standard assumptions, often glossed over, do have important consequences for one-dimensional confinement potentials in general. The time-independent Schrödinger equation summarises the wave mechanics analogy to Hamilton’s formulation of classical mechanics [1] for time-independent potentials. In essence this states that the kinetic and potential energy components sum to the total energy; in wave mechanics, these quantities are the eigenvalues of linear operators, i.e. 2.1 where the eigenfunction ψ describes the state of the system. Again in analogy with classical mechanics, the kinetic energy operator for a particle of constant mass is given by: 2.2 where is the usual quantum mechanical linear momentum operator: 2.3 By using this form for the kinetic energy operator, the Schrödinger equation then becomes: 2.4 where the function V (x, y, z) represents the potential energy of the system as a function of the spatial coordinates. Restricting this to the one-dimensional potential of interest here, the Schrödinger equation for a particle of mass m in a potential well aligned along the z -axis (as in Fig. 2.1) would be: 2.5 Figure 2.1 The one-dimensional infinite well confining potential Outside of the well, V (z) = ∞, and hence the only possible solution is ψ (z) = 0, which in turn implies that all values of the energy E are allowed. Within the potential well, the Schrödinger equation simplifies to: 2.6 which implies that the solution for ψ is a linear combination of the functions f (z) which when differentiated twice give − f (z)

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Fundamentals of Quantum Mechanics

  • Ajit Kumar(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  Since the motion of the particle is confined inside the well, quantum mechanically, it corresponds to the case of a bound state problem. In order to find the bound state energies and wave functions, we must solve the TISE with appropriate boundary conditions. Since the particle cannot penetrate the regions x < 0 and x > a, the wave function of the particle must be zero in these regions: ψ = 0 for x < 0 and x > a. The TISE d 2 φ dx 2 + 2m ¯ h 2 (E - V ) φ = 0 (3.3.2) for the given case can be written as φ 00 φ = - 2m ¯ h 2 (E - V ) , (3.3.3) One-dimensional Problems 63 where the prime stands for ordinary derivative with respect to x. Inside the well, V = 0, and the solution is given by the linear combination φ (x) = A sin(kx) + B cos(kx), (3.3.4) where A and B are arbitrary constants and k 2 = 2mE ¯ h 2 . (3.3.5) 0 – 2 f ( ) x n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f 1 f 2 f 3 x ® 2 Figure 3.4 Spatial parts of the wave functions for the first three stationary states of a particle in the Infinite Square Well potential with a = 1. According to the standard conditions, the wave function has to be continuous across the boundaries and we must have φ ≡ 0 for x = 0 and x = a. The first boundary condition φ (x = 0) = 0 leads to B = 0. So, we are left with φ (x) = A sin(kx). The second boundary condition yields sin(ka) = 0, ⇒ k n = nπ a , n = 1,2,3,... (3.3.6) Taking into account (3.3.6), we conclude that the boundary conditions can be satisfied only for the discrete values of energy 64 Fundamentals of Quantum Mechanics E n = n 2 ¯ h 2 π 2 2ma 2 , n = 1,2,3, ... , (3.3.7) where we have omitted n = 0 because it leads to an uninteresting result: φ 0 (x) = 0 and E 0 = 0. Thus, a particle, trapped inside an infinite potential well, can have only discrete set of energy eigenvalues given by (3.3.7). The corresponding eigenfunctions are φ n (x) = B n sin  nπ a x  .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  How to Be a Quantum Mechanic

  • Charles G. Wohl(Author)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  The quantum-mechanical “components” then are the constants c 1 , c 2 , c 3 , ... , in the expansion Ψ(x,t) = ∑ c n Ψ n (x,t). We develop this geometrical analogy in Chap. 5; it makes matters more intuitive. We need all such help we can get. Some of the deepest consequences of quantum mechanics arise from the superposition of states. For the rest of this chapter, however, we simply solve the time-independent Schr¨ odinger equation, ˆ Hψ = Eψ, for several different forms of V (x). The examples are important in their own right; and we need some familiarity with the methods and results of quantum mechanics before we address the general formalism. 25 2·4. THE Infinite Square Well A particle with mass m is confined to a one-dimensional box of length L; see Fig. 1. The potential energy is V (x) = braceleftBig 0 for 0

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Fundamentals of Physics, Extended

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  We can use integral 11 in Appendix E to evaluate the inte- gral, obtaining the equation A 2 L nπ [ y 2 − sin 2y 4 ] 0 nπ = 1. Evaluating at the limits yields A 2 L nπ nπ 2 = 1; thus A = √ 2 L . (Answer) (39-17) This result tells us that the dimension for A 2 , and thus for ψ n 2 (x), is an inverse length. This is appropriate because the probability density of Eq. 39-12 is a probability per unit length. An Electron in a Finite Well A potential energy well of infinite depth is an idealization. Figure 39-7 shows a realizable potential energy well — one in which the potential energy of an electron outside the well is not infinitely great but has a finite positive value U 0 , Figure 39-7 A finite potential energy well. The depth of the well is U 0 and its width is L. As in the infinite potential well of Fig. 39-2, the motion of the trapped electron is restricted to the x direction. x U 0 U U(x) 0 0 L ● The wave function for an electron in a finite, one- dimensional potential well extends into the walls, where the wave function decreases exponentially with depth. ● Compared to the states in an infinite well of the same size, the states in a finite well have a limited number, longer de Broglie wavelengths, and lower energies. Key Ideas called the well depth. The analogy between waves on a stretched string and mat- ter waves fails us for wells of finite depth because we can no longer be sure that matter wave nodes exist at x = 0 and at x = L. (As we shall see, they don’t.) To find the wave functions describing the quantum states of an electron in the finite well of Fig. 39-7, we must resort to Schrödinger’s equation, the basic equation of quantum physics. From Module 38-6 recall that, for motion in one dimension, we use Schrödinger’s equation in the form of Eq. 38-19: d 2 ψ dx 2 + 8π 2 m h 2 [E − U(x)]ψ = 0. (39-18) Rather than attempting to solve this equation for the finite well, we simply state the results for particular numerical values of U 0 and L.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Modern Physics

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

    (Publisher)

  (c) Plot the function f (E) over a large enough domain to see how many roots it has and to estimate their values. (d) What do your results in Part (c) tell you about this finite square well? (e) Repeat Parts (b) and (c) for ¯ h = m = L = 1, U 0 = 30. How are the results different? 15. On pp. 239–240. we listed the steps involved in solving Schrödinger’s equation for a bound state. Identify each of those steps in the solution for the Infinite Square Well. This includes deciding which of those steps don’t apply to the Infinite Square Well and why. 242 5 The Schrödinger Equation 16. A particle has the potential energy function U(x) = U 0 for x ≤ 0, U(x) = 0 for 0 < x < L, and U(x) = 2U 0 for x ≥ L (where U 0 > 0). (a) Write Schrödinger’s equation in each of these three regions. (b) Write the general solution to Schrödinger’s equation in each region, assuming 0 < E < U 0 . Your answer should have a total of six arbitrary constants. Use only real formulas, so make sure your square roots are all of positive quantities. (c) Which two of those arbitrary constants must equal 0 and why? (d) Figure 5.11 shows an energy eigenstate for a symmetric finite square well. How do the eigenstates of this problem’s asymmetric well differ from that one, and what does that differ- ence imply about the probability distribution for position? 17. A Half Finite Well A particle has the potential energy function U(x) = ∞ for x ≤ 0, U(x) = 0 for 0 < x < L, and U(x) = U 0 for x ≥ L. (a) What is ψ(x) in the region x ≤ 0? (b) Write the general solution to the Schrödinger equation using real functions in each of the two remaining regions, assuming 0 < E < U 0 . Your answer should have a total of four arbi- trary constants. (c) Two of the arbitrary constants from Part (b) must be zero. Which ones, and why? (d) Write an equation relating the two remaining arbitrary constants, and explain how you know it is true. Use it to rewrite the wavefunction in all of space with only one arbitrary constant.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Physics of Electronic Materials

  Principles and Applications

  • Jørgen Rammer(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  6 Quantum Wells In atoms, electrons are confined by the attractive Coulomb force of the atomic nucleus to mostly roam a region on the order of 10 − 8 cm. Atoms, consisting of opposite charges, gen-erally attract each other, say, through electric dipole interaction, and when atoms are close together, electrons can tunnel from one to the other. The corresponding negative electron charge probability density in between the atoms counteracts the repulsive force between the positively charged nuclei, thereby binding atoms into molecules. The same mechanism, chemical bonding, is at play when a gigantic number of atoms form semiconductor crys-tal structures. A qualitative understanding of why matter lumps together is sufficient for our purpose, and this can be provided by solving simple one-dimensional models where the potentials are stepwise constant. Nowadays, these potentials are not only of academic interest, since by modern technology, for example, molecular beam epitaxy, such potentials can be created for electrons in so-called heterostructures, as discussed in Chapter 12 . 6.1 Symmetric Well Consider a particle of mass m in a one-dimensional symmetric well of extension a , i.e. the potential V ( x ) = ⎧ ⎪ ⎨ ⎪ ⎩ V , x < − 1 2 a , 0, − 1 2 a < x < 1 2 a , V , x > 1 2 a , (6.1) where V > 0, i.e. the potential depicted in Figure 6.1 . This potential could be a crude model of an electron bound in an atom, or a quite realistic model of a quantum well created inside a heterostructure, as discussed in Chapter 12 . The solutions to the time-independent Schrödinger equation corresponding to the con-tinuous part of the energy spectrum, E > V , are plane waves, corresponding to different constant potential energy inside and outside the well, however, and thereby different wave numbers, due to the jump in potential at the two points, x = ± a / 2, defining the well.

  Sign up to read

  Learn more about book