Finite Square Well

10 Key excerpts on "Finite Square Well"

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  A Student's Guide to the Schrödinger Equation

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

    (Publisher)

  These boundary conditions lead to a somewhat more complicated equation from which the allowed energy levels and wave-functions may be extracted. Another important difference between the finite and the infinite potential well is this: for a finite potential well, particles may be bound or free, depending on their energy and the characteristics of the well. Specifically, for the potential energy defined as in Fig. 5.9 , the particle will be bound if E < V 0 and free if E > V 0 . In this section, the energy will be taken as 0 < E < V 0 , so the wavefunctions and energy levels will be those of bound particles. The good news is that if you’ve worked through Chapter 4 , you’ve already seen the most important features of the finite potential well. That is, the wavefunction solutions are oscillatory inside the well, but they do not go to zero at the edges of the well. Instead, they decay exponentially in that region, often called the “evanescent” region. And just as in the case of an infinite rectangular well, the wavenumbers and energies of particles bound in a finite rectangular well are quantized (that is, they take on only certain discrete “allowed” values). But for a finite potential well, the number of allowed energy levels is not infinite, depending instead on the width and the “depth” of the well (that is, the difference in potential energy inside and outside of the well). In this section, you’ll find an explanation of why the energy levels are discrete in the finite potential well along with an elucidation of the meaning of the variables used in many quantum texts in the transcendental equation that arises from applying the boundary conditions of the finite rectangular well. If you’ve read Section 4.3 , you’ve already seen the basics of wavefunction behavior in a region of piecewise constant potential, in which the total energy E of the quantum particle may be greater than or less than the potential energy V in the region.

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  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.

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  Quantum Mechanics

  A Paradigms Approach

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

    (Publisher)

  154 Quantized Energies: Particle in a Box 5.10.3  General Potential Wells Given our approximate and numerical techniques, we can solve for the bound states in any potential well, in principle. A typical bound state solution is shown in Fig. 5.30. It exhibits the key features that we have mentioned above for bound state solutions: • Oscillatory in allowed region • Exponential decay in forbidden region • Oscillatory wave becomes less wiggly near classical turning point as kinetic energy decreases • Amplitude becomes larger near classical turning points Thus, though potential energy wells may appear quite different at first glance, they all can be called “particle-in-a-box” systems, albeit with differently shaped boxes. Some common boxes are shown in Fig. 5.31: (a) inFinite Square Well, (b) Finite Square Well, (c) harmonic oscillator (mass on a spring), and (d) linear potential (bouncing ball potential). SUMMARY In this chapter we learned the language of the wave function, which is the representation of the quan- tum state vector in position space. We express this as 0 c9  c1x2 c1x2 = 8 x 0 c9 . (5.157) The complex square of the wave function yields the spatial probability density P1x2 = 0 c1x20 2 . (5.158) The normalization condition is 1 = 8 c @ c9 = L  -  @ c1x2 @ 2 dx = 1. (5.159) The rules for translating bra-ket formulae to wave function formulae are: 1) Replace ket with wave function 0 c9 S c1x2 2) Replace bra with wave function conjugate 8 c 0 S c * 1x2 3) Replace bracket with integral over all space 8 @ 9 S L  -  dx 4) Replace operator with position representation A n S A1x2 . The probability of measuring the position of a particle to be in a finite spatial region is P a 6 x 6 b = L b a 0 c1x20 2 dx. (5.160) Summary 155 x E,Ψ FIGURE 5.30 Bound state in a generic potential energy well.

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  Fundamentals of Physics, Extended

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  39.3.4 For an electron trapped in a finite well with a given quantum number, sketch the probability density as a function of position across the well and into the walls. 39.3.5 Identify that a trapped electron can exist in only the allowed states and relate that energy of the state to the kinetic energy of the electron. 39.3.6 Calculate the energy that an electron must absorb or emit to move between the allowed states or between an allowed state and any value in the nonquantized region. 39.3.7 If a quantum jump involves light, apply the relationship between the energy change and the frequency and wavelength associated with the photon. 39.3.8 From a given allowed state in a finite well, calculate the minimum energy required for the electron to escape and the kinetic energy of the escaped electron if provided more than that minimal energy. 39.3.9 Identify the emission and absorption spectra of an electron in a one-dimensional finite potential well, including escaping the trap and falling into the trap. Key Ideas ● 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. An Electron in a Finite Well A potential energy well of infinite depth is an idealization. Figure 39.3.1 shows a realizable potential energy well—one in which the potential energy of an elec- tron outside the well is not infinitely great but has a finite positive value U 0 , 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.

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  Quantum Mechanics

  • Arjun Berera, Luigi Del Debbio(Authors)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  5.5 Finite Potential Well The final example that we are going to discuss is the finite potential well, which corresponds to the potential V ( x ) = ⎧ ⎪ ⎨ ⎪ ⎩ −V 0 , for | x | < a/2, 0, otherwise. (5.45) 95 5.5 Finite Potential Well Once again we have a potential that is symmetric under parity, V (−x ) = V ( x ) . Therefore we can choose to look for solutions of the energy eigenvalue equation that are also parity eigenstates. As discussed previously, energy eigenvalues must be larger than −V 0 . This means that we can have states with −V 0 < E < 0; they are called bound states – the wave function for these states decays exponentially at large | x | , so that the probability of finding the particle outside the well becomes rapidly very small. On the other hand, states with E > 0 correspond to incident plane waves that are distorted by the potential. We will concentrate on the bound states here. The Schr¨ odinger equation reads ψ  = ⎧ ⎪ ⎨ ⎪ ⎩ − 2m  2 E ψ, − 2m  2 ( E + V 0 ) ψ. (5.46) The main difference between the infinite and the finite well comes from the fact that the wave function does not have to vanish outside the classically allowed region | x | < a/2. As we discussed before, the wave function for | x | > a/2 will decay exponentially. For −V 0 < E < 0, the even parity solutions are of the form ψ( x ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ A cos ( px /) , | x | < a/2, Ce − ¯ px/ , x > a/2, Ce ¯ px/ , x < −a/2, (5.47) while the odd parity solutions are ψ( x ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ A sin( px /) , | x | < a/2, Ce − ¯ px/ , x > a/2, −Ce ¯ px/ , x < −a/2. (5.48) As usual, we have introduced the momenta p =  2m( E + V 0 ) , ¯ p = √ −2mE. (5.49) Remember that we are looking for bound states, and hence −V 0 < E < 0 so the arguments of the square root in Eq. (5.49) are both positive. Note that the symmetry properties of the solutions have already been taken into account in the chosen parametrisations of the solutions.

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  How to Be a Quantum Mechanic

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

    (Publisher)

  If (in one dimension) V (x) is symmetric about some point, then the first, third, fifth, ... , eigenfunctions are symmetric about this point (have even or positive parity ); but the second, fourth, ... , eigenfunctions are antisymmetric (have odd or negative parity ). Thus in Fig. 2(a), ψ 1 (x) is symmetric about L/2, ψ 2 (x) is antisymmetric, and so on. It follows that the prob- ability densities |ψ n (x)| 2 of energy eigenstates in symmetric one-dimensional potential wells are symmetric about the point of symmetry; see Fig. 2(b). Figure 2(b) also shows the normalized classical probability densities. Classically, a parti- cle of given energy bouncing between the walls is moving at constant speed. If 1000 snapshots were taken at random times, the particle would with equal likelihood be found anywhere in the well. Thus the normalized probability density is 1/L for 0

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  Physics, Volume 2

  • David Halliday, Robert Resnick, Kenneth S. Krane(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  We refer to this region as the continuum. Here the energy of the electron can have any value. Probability Densities for an Electron in a Finite Well Figure 47-7 shows the probability densities for the three quantum states of an electron trapped in a finite well of width L  100 ppm and depth U 0  250 eV. The wave functions have been normalized (see Eq. 47-9), which means that the total area under each curve, including all three regions of interest, is unity. The electron must be somewhere — if not inside the well, then outside. We now call attention to three properties of the quantum states of an electron trapped in a finite well. First property: As it happens, the three states shown in Fig. 47-6 b are the only ones that can exist in this particular well. It can be shown that a fourth quantum state, corresponding to n  4, can exist in our well if, keeping the width of the well at L  100 ppm, we increase its depth from its present value of U 0  250 eV to a value somewhat above 340 eV. It is reasonable that a deeper well can accommodate more quantum states because the deeper the well the more it re- sembles an infinite well, which can accommodate all states, regardless of their quantum number. Alternatively, we can accommodate the n  4 quantum state in our well if, keeping the depth of the well at U 0  250 eV, we increase its width from its present value of L  100 pm to a value somewhat above 115 pm. It is reasonable that making the well wider should permit it to accommodate additional quantum states because the wider the well, the more the particle trapped within it resembles a free particle, which can possess any energy. Second property: As Fig. 47-7 shows, the probability of finding the electron outside the well increases as the quan- tum number increases. Table 47-1 shows that for n  1 (the ground state) this probability is 2%. For n  2 and n  3, it is 10% and 30%, respectively.

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  Fundamentals of Physics, Extended

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

    (Publisher)

  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. Figure 39-8 shows three results as graphs of ψ n 2 (x), the probability density, for a well with U 0 = 450 eV and L = 100 pm. The probability density ψ n 2 (x) for each graph in Fig. 39-8 satisfies Eq. 39-14, the normalization equation; so we know that the areas under all three probability density plots are numerically equal to 1. If you compare Fig. 39-8 for a finite well with Fig. 39-6 for an infinite well, you will see one striking difference: For a finite well, the electron matter wave penetrates the walls of the well — into a region in which Newtonian mechanics says the electron cannot exist. This penetration should not be surprising because we saw in Module 38-9 that an electron can tunnel through a potential energy barrier. “Leaking” into the walls of a finite potential energy well is a similar phenomenon. From the plots of ψ 2 in Fig. 39-8, we see that the leakage is greater for greater values of quantum number n. Because a matter wave does leak into the walls of a finite well, the wavelength λ for any given quantum state is greater when the electron is trapped in a finite well than when it is trapped in an infinite well of the same length L. Equation 39-3 (λ = h/ √ 2mE ) then tells us that the energy E for an electron in any given state is less in the finite well than in the infinite well. That fact allows us to approximate the energy-level diagram for an electron trapped in a finite well. As an example, we can approximate the diagram for the finite well of Fig. 39-8, which has width L = 100 pm and depth U 0 = 450 eV.

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  Modern Physics

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

    (Publisher)

  Why does quantum mechanics rule out such a solution? 7. How will the shape of the bound energy eigen- states of the Finite Square Well change as E increases? (This is asking about the limit as E → U 0 from below.) 8. A particle is in a Finite Square Well, in an energy eigenstate with E < U 0 . If you measure the particle’s position, is it possible to find it in a region where U = U 0 ? Discuss. Problems 9. Fill in the missing steps in Equation (5.11). (This equation is purely classical.) (a) First, show how the force law leads to the dif- ferential equation. (b) Then, show that the given solution solves that differential equation. 10. The ground state vibrational energy of a hydro- gen molecule is roughly 0.27 eV. Treating the molecule as a simple harmonic oscillator (an excel- lent approximation), what are its next two energy eigenvalues? 11. Figure 5.8 shows a function that solves Schrödinger’s equation for the simple harmonic oscillator with E = ¯ hω, but is not a valid wavefunc- tion. In this problem you’ll experiment to find an energy that does lead to a valid wavefunction. (a) To simplify your calculations, you can set ¯ h = ω = m = 1. (That is, effectively, choosing an unusual but useful set of units.) Write the Schrödinger equation for the simple harmonic oscillator using those values. (b) Have a computer solve the equation you wrote in Part (a) using the conditions ψ(1) = ψ(−1) = 1 for E = 0.1, E = 0.3, E = 0.5, E = 0.7, and E = 0.9. Plot the solution for each one in the domain −3 < x < 3 and describe its behavior. (c) The only energy eigenvalue in this range of energies is E = 0.5. What was wrong with all the other curves you drew? 12. This problem isn’t about any particular bound state, but it illustrates a key mathematical idea. (a) Find the general solution to the differential equation df /dx = 2kxf .

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  Quantum Chemistry Student Edition

  • John Lowe(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  In this apparatus, one can vary the potential difference between the metal dish and the collecting wire, and also the intensity and frequency of the incident light. Suppose that the potential difference is set at zero and a current is detected when light of a certain intensity and frequency strikes the dish. This means that electrons are being emitted from the dish with finite kinetic energy, enabling 2-3 THE BOX WITH ONE FINITE WALL 41 0 5 10 15 20 25 A FIG. 2-10 Solutions for particle in well with one finite wall (see Fig. 2-9 for details). Dotted lines correspond to energy levels which would exist if U = oo. we require that the wavefunction be normalized. A set of such solutions is shown in Fig. 2-10. Before solving for the case where E > U, let us discuss in detail the results just obtained. In the first place, the energies are quantized, much as they were in the infinitely deep square well. There is some difference, however. In the infinitely deep well or box, the energy levels increased with the square of the quantum number n. Here they increase less rapidly (the dotted lines in Fig. 2-10 show the allowed energy levels which result when U = oo) because the barrier becomes 42 2. QUANTUM MECHANICS OF SOME SIMPLE SYSTEMS effectively less restrictive for particles with higher energies (see the following). For the lowest solution, for example, slightly less than one-half a sine wave is needed in one box width of distance. Thus, the wavelength here is slightly longer than in an infinitely deep well of equal width, and so, by de Broglie's relation the energy is slightly lower. Notice that the effect of lowering the height of one wall is least for the levels lying deepest in the well. The solutions sketched in Fig. 2-10 indicate that there is a finite probability for finding the particle in the region x > L even though it must have a negative kinetic energy there.

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