{"id":51,"date":"2018-12-06T05:36:27","date_gmt":"2018-12-06T05:36:27","guid":{"rendered":"http:\/\/msp08.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=51"},"modified":"2018-12-06T06:20:13","modified_gmt":"2018-12-06T06:20:13","slug":"quantum-confinement-of-electrons-in-semiconductor-nanostructures","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/chapter\/quantum-confinement-of-electrons-in-semiconductor-nanostructures\/","title":{"rendered":"Physics at Nanoscale"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n
\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\nIn this module we will learn the following:\r\n\r\n \r\n\r\n-\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Importance of particle-in-a-box systems\r\n\r\n-\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Wavefunctions and energies of particle in a one-dimensional system\r\n\r\n-\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Wavefunctions and energies of particle in a two-dimensional system\r\n\r\n-\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Wavefunctions and energies of particle in a three-dimensional system\r\n\r\n \r\n\r\n1.\u00a0\u00a0 Particle in a Box<\/strong>\r\n\r\n \r\n

Quantum mechanics describes the particle in a box model as a particle which is can freely move within a small space surrounded by impenetrable barriers. This model is also termed as infinite potential well or an infinite square well, mainly due to its structure. This model is specifically employed as in theoretical investigations to demonstrate the differences between the classical and quantum systems. For instance, in case of classical system, a ball entrapped within a large box is free to move inside the box at any speed, and has equal probability of being found at any point inside the box. As the well becomes narrow and reaches the nanoscale dimensions, quantum confinement effects assume importance. As a result, the particle is no longer to free to move anywhere within the box, but can only occupy specific energy levels. Additionally, it cannot have zero energy at any moment, implying that the particle is never at rest. Furthermore, depending upon the energy levels, the probability of finding the particle at certain positions is more than at other places. Thus, the particles may not be detectable at some specific positions. These positions which have zero probability of particle occupation are termed as spatial nodes.<\/p>\r\n \r\n

In quantum mechanics, very few problems are solvable analytically without using approximations. Particle in a box model is one such problem. It is a very simple model which provides essential insights into the quantum effects without going for complex mathematical derivations. It can be used to explain the appearance of energy quantization (or discrete energy levels) within complex quantum systems, e.g., atoms and molecules. This model is very frequently employed as an approximation for more complex quantum systems.<\/p>\r\n \r\n\r\n1.1 Particle in an One Dimensional System<\/strong>\r\n\r\n \r\n

One dimensional (1D) system is the simplest form of the particle in a box model. In this model, the particle can move only in backward and forward directions along a straight line and is surrounded by impenetrable barriers at either end. The walls of a 1D box can be visualized as regions of space which have infinitely large potential energy. On the contrary, a constant zero potential energy is assumed to present within the box. This implies that the particle is free to move within the box and no forces act upon it in this region. Further, owing to infinitely large potential energy at the walls, very large forces are applied on the particles by the walls of the box. These forces repel the particle when it touches the walls of the box. Thus the particle cannot escape the potential well. Potential energy, in case of 1D box, can be calculated as:\u00a0<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n

Figure 1 Schematics of a one dimensional particle in a box system. The barriers outside the box have infinitely large potential, whereas the interior of the box has a constant, zero potential.<\/strong><\/p>\r\n \r\n\r\nThe solution to Schrodinger equation in the region of interest (?=0) is:\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\nGeneral solutions to this are linear combinations of plane waves\r\n\r\n \r\n\r\n?(?)=?????<\/sup>+??\u2212???<\/sup>\r\n\r\n \r\n\r\nwhere ? and ? are constant coefficients to be determined.\r\n\r\n \r\n\r\nLet us now apply the boundary conditions to specify ? and ?. Namely, the first says that the wavefunction must gor to zero at the origin (i.e., ?(0)=0). Therefore, we get:\r\n\r\n \r\n\r\n?(0)=?+?=0\r\n\r\n \r\n\r\nOr ?=\u2212?. This gives the complex wavefunction:\r\n\r\n \r\n\r\n?(?)=?(????<\/sup>\u2212?\u2212???<\/sup>)<\/span>\r\n\r\n \r\n\r\nW<\/span>hich can be rewritten using Euler\u2019s relation: <\/span>\r\n\r\n \r\n\r\n?(?)=2??sin??<\/span>\r\n\r\n \r\n\r\nNext, we apply the second boundary condition, i.e., ?(?)=0 to get: <\/span>\r\n\r\n \r\n\r\n?(?)=2??sin??=0<\/span>\r\n\r\n \r\n\r\nAt this point, to obtain a solution, we have either ?=0 (the trivial solution) or ??=??, with ? an integer. We pursue the second solutions, since ?=0 just means that the wavefunction is zero. Consequently, ? values are restricted to: <\/span>\r\n\r\n \r\n\r\n?=??\/?<\/span>\r\n\r\n \r\n\r\nIt is interesting to note here a factor-of-two difference between the ? value in this case and that in the free particle case.<\/span>\r\n\r\n<\/div>\r\nTherefore,\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n

Where is a normalization constant to be determined. This is the general form of wavefunction for a particle in a 1D box.<\/p>\r\n \r\n\r\nNormalization<\/strong>\r\n\r\n \r\n

The probabilistic interpretation of wavefunctions means that they must be normalized (i.e., the particle must be located somewhere within the box with unity probability):<\/p>\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\nWavefunctions<\/strong>\r\n\r\n \r\n\r\nNormalized wavefunctions for a particle in a 1D box are therefore:\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

Figure 2 The first three wavefunctions for a particle in a 1D box (a) and their associated probability densities (b).<\/strong><\/p>\r\n \r\n\r\nEnergies<\/strong>\r\n\r\n \r\n\r\nAssociated energies arise from the equivalence obtained earlier:\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n

where = 1,2,3, \u2026 Note that the quantum number is never zero, as this would violate the Heisenberg Uncertainty Principle. Finally, we can extend these results to two and three dimensions by simply varying the form of the Laplacian and assuming separable solutions.<\/p>\r\n \r\n

Figure 3 <\/strong>shows the first five energy levels of the particle in a box along with their corresponding quantum numbers. Note the absence of degeneracies (i.e., multiple levels having the same energy) in the one-dimensional box energy levels. Apparently, the states are more closely spaced together at lower energies but become more distant as increases.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

Figure 3 Energy levels for a particle in a one-dimensional box.<\/strong><\/p>\r\n \r\n\r\n1.2 Particle in a Two-Dimensional Box: Degeneracies<\/strong>\r\n\r\n \r\n

For simplicity, we shall solve the problem of a particle confined to a two-dimensional box in ( , ) plane of a Cartesian coordinate system. The wavefunctions and energies have and dependencies. Additionally, an important consequence of working in higher dimensions is increased likelihood of finding degeneracies. Implying that there will be increased propensity for finding states that have the same energy. This stems from built-in symmetries of the system.<\/p>\r\n \r\n\r\nThe Schrodinger equation in this case is:\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n

We assume the potentials and the boundary conditions as shown in Figure 4<\/strong>. Namely, we have a symmetric box with a dimension along each axis and with a potential that is zero inside the box but infinite outside it. The particle is therefore confined to the interior of the box.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

Figure 4 Potential for a particle in a two-dimensional box.<\/strong><\/p>\r\n \r\n\r\nAdditionally, its wavefunctions go to zero at the edges. We therefore have for the potential:\r\n\r\n\"\"\r\n\r\n \r\n\r\nwith the accompanying boundary conditions:\r\n\r\n \r\n\r\n?(0,?)=0 for 0\u2264?\u2264?,\r\n\r\n?(?,?)=0 for 0\u2264?\u2264?,\r\n\r\n?(?,0)=0 for 0\u2264?\u2264?,\r\n\r\n?(?,?)=0 for 0\u2264?\u2264?.\r\n\r\n \r\n\r\nWe now solve the Schrodinger equation in the region of interest, where ?(?,?)=0:\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\nFor simplicity, let us assume a separable solution of the form:\r\n\r\n \r\n\r\n?(?,?)=??<\/sub>(?)??<\/sub>(?)\u2261??<\/sub>??<\/sub>\r\n\r\n \r\n

which is basically a product of two independent wavefunctions, each with its own or dependence. There is a benefit in choosing this type of solution, because the result ultimately comprises two separate one-dimensional particle-in-a-box problems. Since we have already solved this model, the two-dimensional problem simplifies to a great extent.<\/p>\r\n \r\n\r\nIntroducing ?=??<\/sub>??<\/sub> into the Schrodinger equation gives:\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n

This relation remains valid for all values of and . Consequently, each term on the left-hand side must equal a constant; these constants turn out to be the energy contributions from and . We thus separate the total energy term into contributions, one from and one from :<\/p>\r\n \r\n

?=??<\/sub>+??<\/sub><\/p>\r\n \r\n\r\nThis in turn means that we can separate the full equation into two smaller ones:\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n

Which are two one-dimensional particle-in-a-box problems. This demonstrates how assuming a separable solution greatly simplifies the original two-dimensional problem. Now both these one-dimensional problems can be solved as shown above.<\/p>\r\n \r\n\r\nThe obtained independently normalized wavefunctions are:\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\nIn both cases, ??<\/sub> and ??<\/sub> are integers reflecting the quantization of the particle.\r\n\r\n \r\n\r\nWavefunctions<\/strong>\r\n\r\n \r\n\r\nWhen everything is put together, the total wavefunction for the particle in <\/span>two-dimensional square box is:<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

where and are independent integers. Figure 5<\/strong> shows the first few wavefunctions of a particle in a two-dimensional box. The associated probability densities are shown in Figure 6<\/strong>.<\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n

Figure 5 First few wavefunctions of a particle in a two-dimensional box.<\/strong><\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

\r\n

Figure 6 Probability densities associated with first few wavefunctions of a particle in a two-dimensional box.<\/strong><\/p>\r\n\r\n<\/div>\r\n\u00a0 \u00a0 Energies<\/strong>\r\n\r\n \r\n\r\nThe corresponding energies are:<\/span>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\nwhere ??<\/sub>=1,2,3,\u2026 and ??<\/sub>=1,2,3,\u2026\r\n\r\n \r\n

Figure 7 <\/strong>illustrate the energies of the particle in a two-dimensional box. Since the box is symmetric, there exist a number of degeneracies, where different states end up having the same energy.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

Figure 7 Energy levels for a particle in a two-dimensional box.<\/strong><\/p>\r\n \r\n

As can be seen from the above figure, the state characterized by the quantum numbers ??<\/sub>=1,??<\/sub>=2 possesses the same energy as the state with quantum numbers ??<\/sub>=2,??<\/sub>=1. Note that these degeneracies can be \u2018lifted\u2019 by making the box asymmetric. For instance, if we were to elongate one side of the box, (e.g., y direction), to a length ?, the particle\u2019s wavefunctions and energies would become:<\/p>\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n1.3<\/strong>\u00a0\u00a0 Particle in a Three-Dimensional Box: Degeneracies<\/strong>\r\n\r\n \r\n

Let us now consider the particle trapped inside a three-dimensional box having equal lengths on all sides (i.e., a cube). The relevant potential and boundary conditions are shown in Figure 8<\/strong>, where a box with a length on all three sides and a potential of zero (infinity) inside (outside).<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

Figure 8 Potential for a particle in a three-dimensional box.<\/strong><\/p>\r\n \r\n

The boundary conditions and the procedure explained for two-dimensional and one-dimensional systems can be applied in this case in similar manner to obtain the resulting particle wavefunctions and energies as:<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n

In both the expressions, ? is the width of the box, while ??<\/sub>=1,2,3,\u2026, ??<\/sub>=1,2,3,\u2026, and ??<\/sub>=1,2,3,\u2026 are quantum numbers reflecting the confinement of the particle.<\/p>\r\n \r\n

Figure 9 <\/strong>illustrates the first few energies of the particle in a three-dimensional box. Again, because of the system\u2019s high degree of symmetry, there are significant degeneracy, with multiple states having the same energy.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

Figure 9 Energy levels for a particle in a three-dimensional box.<\/strong><\/p>\r\n \r\n\r\n<\/div>\r\n\r\n\r\n\r\n
you can view video on Physics at Nanoscale<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\u00a0 \u00a0 Questions<\/strong>\r\n
    \r\n \t
  1. Find the energies and normalized wavefunctions of a particle in a symmetric one-dimensional box with barriers at x = \u2212 1\/2a and x = 1\/2a . Assume the free electron mass m0<\/sub>.<\/li>\r\n \t
  2. Verify by spacing of energy levels shown in Figure 3 by using ?(?+1)\u2212?(?).<\/li>\r\n<\/ol>\r\n\u00a0 \u00a0 References & Suggested Reading:<\/strong>\r\n
      \r\n \t
    1. Maseru Kuno, Introductory Nanoscience: Physical and Chemical concepts. Garland Science, Taylor & Francis Group, 2012.<\/li>\r\n \t
    2. T. Pradeep, A Textbook of \u201cNanoscience and Nanotechnology\u201d, McGraw Hill Education, 2012.<\/li>\r\n \t
    3. https:\/\/chem.libretexts.org\/Core\/Physical_and_Theoretical_Chemistry\/Quantum_Mechan ics\/05.5%3A_Particle_in_Boxes\/Particle_in_a_1-Dimensional_box.<\/li>\r\n \t
    4. Chang, Raymond. Physical Chemistry for the Biosciences. Sansalito, CA: University Science, 2005.<\/li>\r\n \t
    5. nptel.ac.in\/courses\/122101001\/downloads\/lec-2.pdf<\/em><\/li>\r\n \t
    6. inside.mines.edu\/~dwu\/classes\/CH351\/...\/CH351%20class%20PIB%20problems.pdf<\/li>\r\n<\/ol>\r\n<\/div>","rendered":"
      \"epgp<\/a>
      \n<\/span><\/div>\n
      \n
      \n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      In this module we will learn the following:<\/p>\n

       <\/p>\n

      –\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Importance of particle-in-a-box systems<\/p>\n

      –\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Wavefunctions and energies of particle in a one-dimensional system<\/p>\n

      –\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Wavefunctions and energies of particle in a two-dimensional system<\/p>\n

      –\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Wavefunctions and energies of particle in a three-dimensional system<\/p>\n

       <\/p>\n

      1.\u00a0\u00a0 Particle in a Box<\/strong><\/p>\n

       <\/p>\n

      Quantum mechanics describes the particle in a box model as a particle which is can freely move within a small space surrounded by impenetrable barriers. This model is also termed as infinite potential well or an infinite square well, mainly due to its structure. This model is specifically employed as in theoretical investigations to demonstrate the differences between the classical and quantum systems. For instance, in case of classical system, a ball entrapped within a large box is free to move inside the box at any speed, and has equal probability of being found at any point inside the box. As the well becomes narrow and reaches the nanoscale dimensions, quantum confinement effects assume importance. As a result, the particle is no longer to free to move anywhere within the box, but can only occupy specific energy levels. Additionally, it cannot have zero energy at any moment, implying that the particle is never at rest. Furthermore, depending upon the energy levels, the probability of finding the particle at certain positions is more than at other places. Thus, the particles may not be detectable at some specific positions. These positions which have zero probability of particle occupation are termed as spatial nodes.<\/p>\n

       <\/p>\n

      In quantum mechanics, very few problems are solvable analytically without using approximations. Particle in a box model is one such problem. It is a very simple model which provides essential insights into the quantum effects without going for complex mathematical derivations. It can be used to explain the appearance of energy quantization (or discrete energy levels) within complex quantum systems, e.g., atoms and molecules. This model is very frequently employed as an approximation for more complex quantum systems.<\/p>\n

       <\/p>\n

      1.1 Particle in an One Dimensional System<\/strong><\/p>\n

       <\/p>\n

      One dimensional (1D) system is the simplest form of the particle in a box model. In this model, the particle can move only in backward and forward directions along a straight line and is surrounded by impenetrable barriers at either end. The walls of a 1D box can be visualized as regions of space which have infinitely large potential energy. On the contrary, a constant zero potential energy is assumed to present within the box. This implies that the particle is free to move within the box and no forces act upon it in this region. Further, owing to infinitely large potential energy at the walls, very large forces are applied on the particles by the walls of the box. These forces repel the particle when it touches the walls of the box. Thus the particle cannot escape the potential well. Potential energy, in case of 1D box, can be calculated as:\u00a0<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      Figure 1 Schematics of a one dimensional particle in a box system. The barriers outside the box have infinitely large potential, whereas the interior of the box has a constant, zero potential.<\/strong><\/p>\n

       <\/p>\n

      The solution to Schrodinger equation in the region of interest (?=0) is:<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      General solutions to this are linear combinations of plane waves<\/p>\n

       <\/p>\n

      ?(?)=?????<\/sup>+??\u2212???<\/sup><\/p>\n

       <\/p>\n

      where ? and ? are constant coefficients to be determined.<\/p>\n

       <\/p>\n

      Let us now apply the boundary conditions to specify ? and ?. Namely, the first says that the wavefunction must gor to zero at the origin (i.e., ?(0)=0). Therefore, we get:<\/p>\n

       <\/p>\n

      ?(0)=?+?=0<\/p>\n

       <\/p>\n

      Or ?=\u2212?. This gives the complex wavefunction:<\/p>\n

       <\/p>\n

      ?(?)=?(????<\/sup>\u2212?\u2212???<\/sup>)<\/span><\/p>\n

       <\/p>\n

      W<\/span>hich can be rewritten using Euler\u2019s relation: <\/span><\/p>\n

       <\/p>\n

      ?(?)=2??sin??<\/span><\/p>\n

       <\/p>\n

      Next, we apply the second boundary condition, i.e., ?(?)=0 to get: <\/span><\/p>\n

       <\/p>\n

      ?(?)=2??sin??=0<\/span><\/p>\n

       <\/p>\n

      At this point, to obtain a solution, we have either ?=0 (the trivial solution) or ??=??, with ? an integer. We pursue the second solutions, since ?=0 just means that the wavefunction is zero. Consequently, ? values are restricted to: <\/span><\/p>\n

       <\/p>\n

      ?=??\/?<\/span><\/p>\n

       <\/p>\n

      It is interesting to note here a factor-of-two difference between the ? value in this case and that in the free particle case.<\/span><\/p>\n<\/div>\n

      Therefore,<\/p>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      Where is a normalization constant to be determined. This is the general form of wavefunction for a particle in a 1D box.<\/p>\n

       <\/p>\n

      Normalization<\/strong><\/p>\n

       <\/p>\n

      The probabilistic interpretation of wavefunctions means that they must be normalized (i.e., the particle must be located somewhere within the box with unity probability):<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      Wavefunctions<\/strong><\/p>\n

       <\/p>\n

      Normalized wavefunctions for a particle in a 1D box are therefore:<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

      Figure 2 The first three wavefunctions for a particle in a 1D box (a) and their associated probability densities (b).<\/strong><\/p>\n

       <\/p>\n

      Energies<\/strong><\/p>\n

       <\/p>\n

      Associated energies arise from the equivalence obtained earlier:<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      where = 1,2,3, \u2026 Note that the quantum number is never zero, as this would violate the Heisenberg Uncertainty Principle. Finally, we can extend these results to two and three dimensions by simply varying the form of the Laplacian and assuming separable solutions.<\/p>\n

       <\/p>\n

      Figure 3 <\/strong>shows the first five energy levels of the particle in a box along with their corresponding quantum numbers. Note the absence of degeneracies (i.e., multiple levels having the same energy) in the one-dimensional box energy levels. Apparently, the states are more closely spaced together at lower energies but become more distant as increases.<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

      Figure 3 Energy levels for a particle in a one-dimensional box.<\/strong><\/p>\n

       <\/p>\n

      1.2 Particle in a Two-Dimensional Box: Degeneracies<\/strong><\/p>\n

       <\/p>\n

      For simplicity, we shall solve the problem of a particle confined to a two-dimensional box in ( , ) plane of a Cartesian coordinate system. The wavefunctions and energies have and dependencies. Additionally, an important consequence of working in higher dimensions is increased likelihood of finding degeneracies. Implying that there will be increased propensity for finding states that have the same energy. This stems from built-in symmetries of the system.<\/p>\n

       <\/p>\n

      The Schrodinger equation in this case is:<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      We assume the potentials and the boundary conditions as shown in Figure 4<\/strong>. Namely, we have a symmetric box with a dimension along each axis and with a potential that is zero inside the box but infinite outside it. The particle is therefore confined to the interior of the box.<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

      Figure 4 Potential for a particle in a two-dimensional box.<\/strong><\/p>\n

       <\/p>\n

      Additionally, its wavefunctions go to zero at the edges. We therefore have for the potential:<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      with the accompanying boundary conditions:<\/p>\n

       <\/p>\n

      ?(0,?)=0 for 0\u2264?\u2264?,<\/p>\n

      ?(?,?)=0 for 0\u2264?\u2264?,<\/p>\n

      ?(?,0)=0 for 0\u2264?\u2264?,<\/p>\n

      ?(?,?)=0 for 0\u2264?\u2264?.<\/p>\n

       <\/p>\n

      We now solve the Schrodinger equation in the region of interest, where ?(?,?)=0:<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      For simplicity, let us assume a separable solution of the form:<\/p>\n

       <\/p>\n

      ?(?,?)=??<\/sub>(?)??<\/sub>(?)\u2261??<\/sub>??<\/sub><\/p>\n

       <\/p>\n

      which is basically a product of two independent wavefunctions, each with its own or dependence. There is a benefit in choosing this type of solution, because the result ultimately comprises two separate one-dimensional particle-in-a-box problems. Since we have already solved this model, the two-dimensional problem simplifies to a great extent.<\/p>\n

       <\/p>\n

      Introducing ?=??<\/sub>??<\/sub> into the Schrodinger equation gives:<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      This relation remains valid for all values of and . Consequently, each term on the left-hand side must equal a constant; these constants turn out to be the energy contributions from and . We thus separate the total energy term into contributions, one from and one from :<\/p>\n

       <\/p>\n

      ?=??<\/sub>+??<\/sub><\/p>\n

       <\/p>\n

      This in turn means that we can separate the full equation into two smaller ones:<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      Which are two one-dimensional particle-in-a-box problems. This demonstrates how assuming a separable solution greatly simplifies the original two-dimensional problem. Now both these one-dimensional problems can be solved as shown above.<\/p>\n

       <\/p>\n

      The obtained independently normalized wavefunctions are:<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      In both cases, ??<\/sub> and ??<\/sub> are integers reflecting the quantization of the particle.<\/p>\n

       <\/p>\n

      Wavefunctions<\/strong><\/p>\n

       <\/p>\n

      When everything is put together, the total wavefunction for the particle in <\/span>two-dimensional square box is:<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

      where and are independent integers. Figure 5<\/strong> shows the first few wavefunctions of a particle in a two-dimensional box. The associated probability densities are shown in Figure 6<\/strong>.<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      Figure 5 First few wavefunctions of a particle in a two-dimensional box.<\/strong><\/p>\n<\/div>\n

      \"\"<\/p>\n

      \n

      Figure 6 Probability densities associated with first few wavefunctions of a particle in a two-dimensional box.<\/strong><\/p>\n<\/div>\n

      \u00a0 \u00a0 Energies<\/strong><\/p>\n

       <\/p>\n

      The corresponding energies are:<\/span><\/p>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

      where ??<\/sub>=1,2,3,\u2026 and ??<\/sub>=1,2,3,\u2026<\/p>\n

       <\/p>\n

      Figure 7 <\/strong>illustrate the energies of the particle in a two-dimensional box. Since the box is symmetric, there exist a number of degeneracies, where different states end up having the same energy.<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

      Figure 7 Energy levels for a particle in a two-dimensional box.<\/strong><\/p>\n

       <\/p>\n

      As can be seen from the above figure, the state characterized by the quantum numbers ??<\/sub>=1,??<\/sub>=2 possesses the same energy as the state with quantum numbers ??<\/sub>=2,??<\/sub>=1. Note that these degeneracies can be \u2018lifted\u2019 by making the box asymmetric. For instance, if we were to elongate one side of the box, (e.g., y direction), to a length ?, the particle\u2019s wavefunctions and energies would become:<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      1.3<\/strong>\u00a0\u00a0 Particle in a Three-Dimensional Box: Degeneracies<\/strong><\/p>\n

       <\/p>\n

      Let us now consider the particle trapped inside a three-dimensional box having equal lengths on all sides (i.e., a cube). The relevant potential and boundary conditions are shown in Figure 8<\/strong>, where a box with a length on all three sides and a potential of zero (infinity) inside (outside).<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      Figure 8 Potential for a particle in a three-dimensional box.<\/strong><\/p>\n

       <\/p>\n

      The boundary conditions and the procedure explained for two-dimensional and one-dimensional systems can be applied in this case in similar manner to obtain the resulting particle wavefunctions and energies as:<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n

      In both the expressions, ? is the width of the box, while ??<\/sub>=1,2,3,\u2026, ??<\/sub>=1,2,3,\u2026, and ??<\/sub>=1,2,3,\u2026 are quantum numbers reflecting the confinement of the particle.<\/p>\n

       <\/p>\n

      Figure 9 <\/strong>illustrates the first few energies of the particle in a three-dimensional box. Again, because of the system\u2019s high degree of symmetry, there are significant degeneracy, with multiple states having the same energy.<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      Figure 9 Energy levels for a particle in a three-dimensional box.<\/strong><\/p>\n

       <\/p>\n<\/div>\n\n\n\n
      you can view video on Physics at Nanoscale<\/strong><\/td>\n\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      \u00a0 \u00a0 Questions<\/strong><\/p>\n

        \n
      1. Find the energies and normalized wavefunctions of a particle in a symmetric one-dimensional box with barriers at x = \u2212 1\/2a and x = 1\/2a . Assume the free electron mass m0<\/sub>.<\/li>\n
      2. Verify by spacing of energy levels shown in Figure 3 by using ?(?+1)\u2212?(?).<\/li>\n<\/ol>\n

        \u00a0 \u00a0 References & Suggested Reading:<\/strong><\/p>\n

          \n
        1. Maseru Kuno, Introductory Nanoscience: Physical and Chemical concepts. Garland Science, Taylor & Francis Group, 2012.<\/li>\n
        2. T. Pradeep, A Textbook of \u201cNanoscience and Nanotechnology\u201d, McGraw Hill Education, 2012.<\/li>\n
        3. https:\/\/chem.libretexts.org\/Core\/Physical_and_Theoretical_Chemistry\/Quantum_Mechan ics\/05.5%3A_Particle_in_Boxes\/Particle_in_a_1-Dimensional_box.<\/li>\n
        4. Chang, Raymond. Physical Chemistry for the Biosciences. Sansalito, CA: University Science, 2005.<\/li>\n
        5. nptel.ac.in\/courses\/122101001\/downloads\/lec-2.pdf<\/em><\/li>\n
        6. inside.mines.edu\/~dwu\/classes\/CH351\/…\/CH351%20class%20PIB%20problems.pdf<\/li>\n<\/ol>\n<\/div>\n","protected":false},"author":3,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["dr-s-s-islam"],"pb_section_license":""},"chapter-type":[],"contributor":[58],"license":[],"class_list":["post-51","chapter","type-chapter","status-publish","hentry","contributor-dr-s-s-islam"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/pressbooks\/v2\/chapters\/51","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":6,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/pressbooks\/v2\/chapters\/51\/revisions"}],"predecessor-version":[{"id":85,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/pressbooks\/v2\/chapters\/51\/revisions\/85"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/pressbooks\/v2\/chapters\/51\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/wp\/v2\/media?parent=51"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/pressbooks\/v2\/chapter-type?post=51"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/wp\/v2\/contributor?post=51"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp08\/wp-json\/wp\/v2\/license?post=51"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}