{"id":91,"date":"2018-11-15T12:08:16","date_gmt":"2018-11-15T12:08:16","guid":{"rendered":"http:\/\/phy12.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=91"},"modified":"2022-01-07T05:41:11","modified_gmt":"2022-01-07T05:41:11","slug":"confinement-of-a-particle-in-a-box-and-in-3d-quantum-dot","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phy12\/chapter\/confinement-of-a-particle-in-a-box-and-in-3d-quantum-dot\/","title":{"rendered":"Confinement of a particle in a box and in 3D (Quantum Dot)"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\n \r\n\r\n3.1 Confinement of a particle\r\n\r\n3.1.1 Particle in 1D box\r\n\r\n3.1.1a Particle in 1D infinite box\r\n\r\n3.1.1b Particle in 1D finite box\r\n\r\n3.1.2 Particle in 2D box\r\n\r\n3.1.2a Particle in 2D rectangular infinite box\r\n\r\n3.1.2b Particle in 2D circular box\r\n\r\n<\/div>\r\n
\r\n\r\n \r\n\r\n3.1 CONFINEMENT<\/strong>\r\n\r\n \r\n

The properties of materials are strongly influenced by the size of the particle. For example Carbon is a non-metal but when considered at the nanoscale single layer allotrope of carbon i.e. Graphene is one of the best conductors. The phenomenon that can be used to explain the size dependent characteristics of materials is Quantum confinement.<\/p>\r\n \r\n

Quantum confinement effects are generally observed when the size of the particle is small and comparable to the de Broglie wavelength of the electron. The effect describes the phenomenon resulting due to squeezing the particles (electrons and holes) into a dimension that approaches a classical limit so that various quantum effects are manifested. In this regime, energy of the particle is no longer continuous, but discrete.<\/p>\r\n \r\n

Quantum confinement effects are observed in low dimension, two, one or zero dimension. When a particle is confined in two dimension (2D) or in a plane, then particle\u2019s motion is quantized along the direction of confinement, say z-axis and the particle motion is classical i.e. is free to move in the x-y plane. The systems in 2D is also known as \u201cquantum well\u201d. In case one dimension (1D), particle\u2019s motion is quantized due to confinement in 2D (say in x-y plane) and classical in 1D along z-axis. The system in 1D is also known as \u201cquantum wire\u201d. In case of zero dimension (0D), particle\u2019s motion is quantized in all three dimension due to confinement along x,y and z directions and known as \u201cquantum dot\u201d. In these systems, the spatial dimensions are of the order of the de Broglie wavelength of the carriers and therefore, the carrier energy states become quantised. As a result, the electronic, electrical and optical\u00a0behaviour of the carriers are governed by quantum mechanics along one dimension, two dimension and all three dimensions in quantum well (2D), quamtum wire (1D) and quantum dot (0D), respectively.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\nCalculation of eigenstates and eigenvalues for different system<\/strong>\r\n\r\n \r\n

The particle\u2019s motion in quantum wells, wires and dots can be described using the analogy to a particle in 1D box, a 2D box, and a 3D box, respectively. The energies of the carrier along the direction of confinement are no longer continuous as in the case where there is no confinement. The emergence of discrete states from continuum states is the most fundamental signature of nanomaterials or nanosystems. One has to solve the Schrodinger equation of the carrier to find out particle\u2019s eigenvalues and eigenfunctions. This is achieved by solving differential equations with particular boundary conditions which can be used to predict the actual shape of a quantum well, wire or dot.<\/p>\r\n \r\n\r\n3.1.1a Particle in 1D infinite box<\/strong>\r\n\r\n<\/div>\r\n \r\n\r\n\"\"\r\n\r\nFor this the boundary conditions are<\/span>\r\n\r\n\"\"\r\n\r\nThe Schrodinger equation to solve is<\/span>\r\n

\r\n\r\n\"\"\r\n\r\nRearranging to yield in the box region where V=0<\/span>\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\nWhere k2<\/sup> = 2mE\/h2<\/sup> and general solution is\r\n\r\n\"\"\r\n\r\nApplying the boundary conditions\r\n\r\n\"\"\r\n\r\nThis leads to\r\n\r\n\"\"\r\n\r\nAnd by normalizing the wave function we get,\r\n\r\n\"\"\r\n\r\nParticle energies, wavefunctions, and probability densities for n = 0, 1 and 2.<\/span>\r\n\r\n \r\n\r\n3.1.1b Particle in a 1D finite box<\/strong>\r\n\r\n \r\n\r\nFrom quantum mechanics we know that as the potential is zero inside box, the solutions in the box region will be wavelike and in the barrier region the solution will be exponentially decaying. The potential is<\/span>\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\nThe solutions in the three regions indicated are\r\n\r\n\"\"\r\n

By applying the boundary conditions, B=0 and F=0 due to finiteness of the wavefunction outside the box, we get<\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\nNow by applying\u00a0 the boundary conditions at the interface, one gets\r\n\r\nHere either we get the trivial solution where A=B=C=D or that the determent of the large matrix is zero\r\n\r\n\"\"\r\n\r\nTo simplify the determent we have apply the following path as\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n\"\"\"\"\r\n
\r\n\r\n \r\n\r\n3.1.2\u00a0 Particle in 2D box<\/strong>\r\n\r\n \r\n\r\n3.1.2a Particle in a rectangular box<\/strong>\r\n\r\n \r\n\r\nThe potential is<\/span>\r\n\r\n<\/div>\r\n
<\/div>\r\n
\r\n\r\n\"\"\r\n\r\nThe Schrodinger equation to solve is\r\n\r\n\"\"\r\n\r\nLet us consider solution of above equation is \u00a0\u00a0( ,\u00a0 ) = ; where , are the functions of x and y only.<\/span>\r\n\r\nSubstituting the solution in eq. 3.15 and rearranging the terms to yield in the box region where V=0, the eq. 3.15 becomes<\/span>\r\n\r\n \r\n

Let us consider E= Ex<\/sub>+Ey<\/sub> and using variable separable method for a rectangular box eq. 3.16 transforms into two equations each consisting of one variable only.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\"\"\"\"\r\n\r\nAnd by normalizing the wavefunction we get,\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
\r\n\r\n \r\n\r\n3.1.2b Particle in circular box<\/strong>\r\n\r\n \r\n

Now, consider a circular box of radius R. The potential inside the box is 0 and outside of the box is \u221e such that the boundary conditions for the wave function would be<\/p>\r\n\"\"\r\n\r\n \r\n\r\nTransforming the Schrodinger question from Cartesian co-ordinates to polar coordinates, the transformation equations are\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\"\"\"\"\r\n\r\nSchrodinger equation for circular box becomes\r\n\r\n\"\"\r\n\r\nConsidering the solution for the above equation,<\/span>\r\n\r\n\"\"\r\n\r\nSubstituting into Schrodinger equation,<\/span>\r\n\r\n\"\"\r\n\r\nRearranging above equation,<\/span>\r\n\r\n\"\"\r\n\r\nThis is form of Bessel differential equation. To change into standard form, considering the variable<\/span>\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\nThe Schrodinger equation finally reduces to,\r\n\r\n\"\"\r\n\r\nTaking the solution of the above equation to be\r\n\r\n<\/div>\r\n\"\"\r\n
\"\"<\/div>\r\n
<\/div>\r\n
This Bessel Function\u2019s expansion remains an infinite series which never truncates. The quantization occurs only with the boundary conditions i.e. at the perimeter of circular ring. At the boundary (r=R) Bessel function should be zero, let us say it, 0. So,<\/span><\/div>\r\n
\"\"<\/div>\r\n
\r\n\r\n \r\n\r\n<\/div>\r\n
<\/div>\r\n3.2 Particle in 3D box<\/strong>\r\n3.2.1a Particle in an infinite cubical box<\/strong>\r\n\r\nLet us consider a general case i.e. a box with dimensions a, b and c:\r\n\r\nThe potential is\r\n
\r\n\r\n\"\"\r\n\r\nThe Schrodinger equation to solve is\r\n\r\n\"\"\r\n

Let us consider solution of above equation is \u00a0\u00a0( ,\u00a0 ,\u00a0 ) = ; where , are the functions of only x, y and z respectively.<\/p>\r\n \r\n\r\nSubstituting the solution in eq. 3.29 and rearranging the terms to yield in the box region where V=0, the eq. 3.29 becomes\r\n\r\n\"\"\r\n

Let us consider = Ex<\/sub>+Ey<\/sub> +Ez and using variable separable method for a rectangular box eq. 3.30 transforms into two equations each consisting of one variable only.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\nFor this the boundary conditions are\r\n\r\n\"\"\r\n\r\nApplying the boundary conditions\r\n\r\n<\/div>\r\n
\r\n
<\/div>\r\n
<\/div>\r\n\"\"\r\n\r\nThis leads to\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
3.2.1b Particle in a spherical box<\/strong><\/div>\r\n
\r\n\r\n \r\n\r\nLet us consider a spherical box of radius R. The potential inside the box is 0 and outside of the box is \u221e\r\n\r\n\"\"\r\n\r\nsuch that the boundary conditions for the wave function would be\r\n\r\n\"\"\r\n

Transforming the Schrodinger question from Cartesian co-ordinates to polar coordinates, the transformation equation are<\/span><\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

<\/div>\r\n
<\/div>\r\n
\r\n\r\n\"\"\"\"\r\n\r\nEquations 3.38 and 3.39 are angular and radial equations respectively.\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\nEquations (3.39), (3.40) and (3.41) and the functions of only r,\u00a0 and respectively.\r\n\r\nTaking, Solution of eq 3.41 be \u00a0(\u00a0 ) \u221d\r\n\r\n\"\"\r\n\r\nWhich is form of Legendre\u2019s equation.<\/span>\r\n\r\n \r\n\r\nThis equation will have a solution only if<\/span>\r\n\r\n<\/div>\r\n
<\/div>\r\n\"\"\r\n\r\nThe solution Legendre\u2019s function will be<\/span>\r\n
\r\n\r\n \r\n\r\n\"\"\r\n\r\nThere angular part of the solution will be\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nThis equation is of the form of Bessel\u2019s equation. R(r) are the spherical Bessel functions then,\r\n\r\n\"\"\r\n

The spherical Bessel functions are oscillatory in nature and have zero many times. The functions ( ) are not square-integrable at r=0, whereas the functions ( ) are well defined in the entire region. Hence ( ) are unphysical, and that the radial wavefunction \u00a0, ( ) is thus only proportional to ( ). The general solution of the Schrodinger equation is<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

In order to satisfy the boundary condition at r=R, the value of k must be chosen such a way so that z=kR corresponds to one of the zeros of ( ) and given by<\/p>\r\n\"\"\r\n\r\nfor n=1,2,3.\r\n\r\n \r\n\r\nTherefore the allowed energy states will be\r\n\r\n\"\"\r\n\r\n\r\n\r\n
you can view video on Confinement of a particle in a box and in 3D (Quantum Dot)<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"
\"epgp<\/a>
\n<\/span><\/div>\n
\n

 <\/p>\n

3.1 Confinement of a particle<\/p>\n

3.1.1 Particle in 1D box<\/p>\n

3.1.1a Particle in 1D infinite box<\/p>\n

3.1.1b Particle in 1D finite box<\/p>\n

3.1.2 Particle in 2D box<\/p>\n

3.1.2a Particle in 2D rectangular infinite box<\/p>\n

3.1.2b Particle in 2D circular box<\/p>\n<\/div>\n

\n

 <\/p>\n

3.1 CONFINEMENT<\/strong><\/p>\n

 <\/p>\n

The properties of materials are strongly influenced by the size of the particle. For example Carbon is a non-metal but when considered at the nanoscale single layer allotrope of carbon i.e. Graphene is one of the best conductors. The phenomenon that can be used to explain the size dependent characteristics of materials is Quantum confinement.<\/p>\n

 <\/p>\n

Quantum confinement effects are generally observed when the size of the particle is small and comparable to the de Broglie wavelength of the electron. The effect describes the phenomenon resulting due to squeezing the particles (electrons and holes) into a dimension that approaches a classical limit so that various quantum effects are manifested. In this regime, energy of the particle is no longer continuous, but discrete.<\/p>\n

 <\/p>\n

Quantum confinement effects are observed in low dimension, two, one or zero dimension. When a particle is confined in two dimension (2D) or in a plane, then particle\u2019s motion is quantized along the direction of confinement, say z-axis and the particle motion is classical i.e. is free to move in the x-y plane. The systems in 2D is also known as \u201cquantum well\u201d. In case one dimension (1D), particle\u2019s motion is quantized due to confinement in 2D (say in x-y plane) and classical in 1D along z-axis. The system in 1D is also known as \u201cquantum wire\u201d. In case of zero dimension (0D), particle\u2019s motion is quantized in all three dimension due to confinement along x,y and z directions and known as \u201cquantum dot\u201d. In these systems, the spatial dimensions are of the order of the de Broglie wavelength of the carriers and therefore, the carrier energy states become quantised. As a result, the electronic, electrical and optical\u00a0behaviour of the carriers are governed by quantum mechanics along one dimension, two dimension and all three dimensions in quantum well (2D), quamtum wire (1D) and quantum dot (0D), respectively.<\/span><\/p>\n<\/div>\n

\n

 <\/p>\n

Calculation of eigenstates and eigenvalues for different system<\/strong><\/p>\n

 <\/p>\n

The particle\u2019s motion in quantum wells, wires and dots can be described using the analogy to a particle in 1D box, a 2D box, and a 3D box, respectively. The energies of the carrier along the direction of confinement are no longer continuous as in the case where there is no confinement. The emergence of discrete states from continuum states is the most fundamental signature of nanomaterials or nanosystems. One has to solve the Schrodinger equation of the carrier to find out particle\u2019s eigenvalues and eigenfunctions. This is achieved by solving differential equations with particular boundary conditions which can be used to predict the actual shape of a quantum well, wire or dot.<\/p>\n

 <\/p>\n

3.1.1a Particle in 1D infinite box<\/strong><\/p>\n<\/div>\n

 <\/p>\n

\"\"<\/p>\n

For this the boundary conditions are<\/span><\/p>\n

\"\"<\/p>\n

The Schrodinger equation to solve is<\/span><\/p>\n

\n

\"\"<\/p>\n

Rearranging to yield in the box region where V=0<\/span><\/p>\n<\/div>\n

\n

\"\"<\/p>\n

Where k2<\/sup> = 2mE\/h2<\/sup> and general solution is<\/p>\n

\"\"<\/p>\n

Applying the boundary conditions<\/p>\n

\"\"<\/p>\n

This leads to<\/p>\n

\"\"<\/p>\n

And by normalizing the wave function we get,<\/p>\n

\"\"<\/p>\n

Particle energies, wavefunctions, and probability densities for n = 0, 1 and 2.<\/span><\/p>\n

 <\/p>\n

3.1.1b Particle in a 1D finite box<\/strong><\/p>\n

 <\/p>\n

From quantum mechanics we know that as the potential is zero inside box, the solutions in the box region will be wavelike and in the barrier region the solution will be exponentially decaying. The potential is<\/span><\/p>\n<\/div>\n

\n

\"\"<\/p>\n

The solutions in the three regions indicated are<\/p>\n

\"\"<\/p>\n

By applying the boundary conditions, B=0 and F=0 due to finiteness of the wavefunction outside the box, we get<\/p>\n

\"\"<\/p>\n<\/div>\n

\n

\"\"<\/p>\n

Now by applying\u00a0 the boundary conditions at the interface, one gets<\/p>\n

Here either we get the trivial solution where A=B=C=D or that the determent of the large matrix is zero<\/p>\n

\"\"<\/p>\n

To simplify the determent we have apply the following path as<\/p>\n<\/div>\n

\n

\"\"<\/p>\n

\"\"<\/p>\n<\/div>\n

\"\"\"\"<\/p>\n

\n

 <\/p>\n

3.1.2\u00a0 Particle in 2D box<\/strong><\/p>\n

 <\/p>\n

3.1.2a Particle in a rectangular box<\/strong><\/p>\n

 <\/p>\n

The potential is<\/span><\/p>\n<\/div>\n

<\/div>\n
\n

\"\"<\/p>\n

The Schrodinger equation to solve is<\/p>\n

\"\"<\/p>\n

Let us consider solution of above equation is \u00a0\u00a0( ,\u00a0 ) = ; where , are the functions of x and y only.<\/span><\/p>\n

Substituting the solution in eq. 3.15 and rearranging the terms to yield in the box region where V=0, the eq. 3.15 becomes<\/span><\/p>\n

 <\/p>\n

Let us consider E= Ex<\/sub>+Ey<\/sub> and using variable separable method for a rectangular box eq. 3.16 transforms into two equations each consisting of one variable only.<\/span><\/p>\n<\/div>\n

\n

\"\"\"\"\"\"<\/p>\n

And by normalizing the wavefunction we get,<\/p>\n

\"\"<\/p>\n<\/div>\n

\n

 <\/p>\n

3.1.2b Particle in circular box<\/strong><\/p>\n

 <\/p>\n

Now, consider a circular box of radius R. The potential inside the box is 0 and outside of the box is \u221e such that the boundary conditions for the wave function would be<\/p>\n

\"\"<\/p>\n

 <\/p>\n

Transforming the Schrodinger question from Cartesian co-ordinates to polar coordinates, the transformation equations are<\/p>\n<\/div>\n

\n

\"\"\"\"\"\"<\/p>\n

Schrodinger equation for circular box becomes<\/p>\n

\"\"<\/p>\n

Considering the solution for the above equation,<\/span><\/p>\n

\"\"<\/p>\n

Substituting into Schrodinger equation,<\/span><\/p>\n

\"\"<\/p>\n

Rearranging above equation,<\/span><\/p>\n

\"\"<\/p>\n

This is form of Bessel differential equation. To change into standard form, considering the variable<\/span><\/p>\n<\/div>\n

\n

\"\"<\/p>\n

The Schrodinger equation finally reduces to,<\/p>\n

\"\"<\/p>\n

Taking the solution of the above equation to be<\/p>\n<\/div>\n

\"\"<\/p>\n

\"\"<\/div>\n
<\/div>\n
This Bessel Function\u2019s expansion remains an infinite series which never truncates. The quantization occurs only with the boundary conditions i.e. at the perimeter of circular ring. At the boundary (r=R) Bessel function should be zero, let us say it, 0. So,<\/span><\/div>\n
\"\"<\/div>\n
\n

 <\/p>\n<\/div>\n

<\/div>\n

3.2 Particle in 3D box<\/strong>
\n3.2.1a Particle in an infinite cubical box<\/strong><\/p>\n

Let us consider a general case i.e. a box with dimensions a, b and c:<\/p>\n

The potential is<\/p>\n

\n

\"\"<\/p>\n

The Schrodinger equation to solve is<\/p>\n

\"\"<\/p>\n

Let us consider solution of above equation is \u00a0\u00a0( ,\u00a0 ,\u00a0 ) = ; where , are the functions of only x, y and z respectively.<\/p>\n

 <\/p>\n

Substituting the solution in eq. 3.29 and rearranging the terms to yield in the box region where V=0, the eq. 3.29 becomes<\/p>\n

\"\"<\/p>\n

Let us consider = Ex<\/sub>+Ey<\/sub> +Ez and using variable separable method for a rectangular box eq. 3.30 transforms into two equations each consisting of one variable only.<\/p>\n<\/div>\n

\n

\"\"<\/p>\n

For this the boundary conditions are<\/p>\n

\"\"<\/p>\n

Applying the boundary conditions<\/p>\n<\/div>\n

\n
<\/div>\n
<\/div>\n

\"\"<\/p>\n

This leads to<\/p>\n

\"\"<\/p>\n<\/div>\n

3.2.1b Particle in a spherical box<\/strong><\/div>\n
\n

 <\/p>\n

Let us consider a spherical box of radius R. The potential inside the box is 0 and outside of the box is \u221e<\/p>\n

\"\"<\/p>\n

such that the boundary conditions for the wave function would be<\/p>\n

\"\"<\/p>\n

Transforming the Schrodinger question from Cartesian co-ordinates to polar coordinates, the transformation equation are<\/span><\/p>\n<\/div>\n

\"\"<\/p>\n

<\/div>\n
<\/div>\n
\n

\"\"\"\"<\/p>\n

Equations 3.38 and 3.39 are angular and radial equations respectively.<\/p>\n<\/div>\n

\n

\"\"<\/p>\n

Equations (3.39), (3.40) and (3.41) and the functions of only r,\u00a0 and respectively.<\/p>\n

Taking, Solution of eq 3.41 be \u00a0(\u00a0 ) \u221d<\/p>\n

\"\"<\/p>\n

Which is form of Legendre\u2019s equation.<\/span><\/p>\n

 <\/p>\n

This equation will have a solution only if<\/span><\/p>\n<\/div>\n

<\/div>\n

\"\"<\/p>\n

The solution Legendre\u2019s function will be<\/span><\/p>\n

\n

 <\/p>\n

\"\"<\/p>\n

There angular part of the solution will be<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

This equation is of the form of Bessel\u2019s equation. R(r) are the spherical Bessel functions then,<\/p>\n

\"\"<\/p>\n

The spherical Bessel functions are oscillatory in nature and have zero many times. The functions ( ) are not square-integrable at r=0, whereas the functions ( ) are well defined in the entire region. Hence ( ) are unphysical, and that the radial wavefunction \u00a0, ( ) is thus only proportional to ( ). The general solution of the Schrodinger equation is<\/p>\n<\/div>\n

\"\"<\/p>\n

In order to satisfy the boundary condition at r=R, the value of k must be chosen such a way so that z=kR corresponds to one of the zeros of ( ) and given by<\/p>\n

\"\"<\/p>\n

for n=1,2,3.<\/p>\n

 <\/p>\n

Therefore the allowed energy states will be<\/p>\n

\"\"<\/p>\n\n\n\n
you can view video on Confinement of a particle in a box and in 3D (Quantum Dot)<\/strong><\/td>\n\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":3,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["prof-subhasis-ghosh"],"pb_section_license":""},"chapter-type":[],"contributor":[58],"license":[],"class_list":["post-91","chapter","type-chapter","status-publish","hentry","contributor-prof-subhasis-ghosh"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phy12\/wp-json\/pressbooks\/v2\/chapters\/91","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phy12\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phy12\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phy12\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":10,"href":"https:\/\/ebooks.inflibnet.ac.in\/phy12\/wp-json\/pressbooks\/v2\/chapters\/91\/revisions"}],"predecessor-version":[{"id":946,"href":"https:\/\/ebooks.inflibnet.ac.in\/phy12\/wp-json\/pressbooks\/v2\/chapters\/91\/revisions\/946"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phy12\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phy12\/wp-json\/pressbooks\/v2\/chapters\/91\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phy12\/wp-json\/wp\/v2\/media?parent=91"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phy12\/wp-json\/pressbooks\/v2\/chapter-type?post=91"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phy12\/wp-json\/wp\/v2\/contributor?post=91"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phy12\/wp-json\/wp\/v2\/license?post=91"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}