Page 1 Module 1 : Quantum Mechanics Chapter 5 : Quantum mechanics in two dimensions Quantum mechanics in two dimensions For quantum mechanics in higher dimensions, the appropriate choice of coordinates is indicated by the form of the potential. For example, if the potential is a function of only the radial distance, , then it is useful to choose a set of coordinates with as one of the coordinates. For analysing the Schroedinger equation in curvilinear coordinates, we first write the equation in terms of cartesian coordinates and then go over to curvilinear coordinates. This is also the procedure followed in Lagrangian approach to classical mechanics. 5.1 Curvilinear coordinates We will specifically consider orthogonal coordinates for which the changes in due to changes are mutually orthogonal so that (5.1) where are the metric elements for . This implies that the corresponding volume element is (5.2) Now the gradient of a function is a vectorial rate of change of a scalar function, (5.3) which implies that the change in due to is (5.4) Now we consider a change in due to a change in the variable, for which is where is the vectorial change in due to . Then we have (5.5) which gives the component of the gradient along the direction corresponding to for the change . We can also obtain the divergence of a vector function by using Gauss' theorem. Consider the theorem for an infinitesimal volume resulting from the mutually orthogonal changes in , resulting from . With the volume given in Eq.(5.2), one gets (5.6) Page 2 Module 1 : Quantum Mechanics Chapter 5 : Quantum mechanics in two dimensions Quantum mechanics in two dimensions For quantum mechanics in higher dimensions, the appropriate choice of coordinates is indicated by the form of the potential. For example, if the potential is a function of only the radial distance, , then it is useful to choose a set of coordinates with as one of the coordinates. For analysing the Schroedinger equation in curvilinear coordinates, we first write the equation in terms of cartesian coordinates and then go over to curvilinear coordinates. This is also the procedure followed in Lagrangian approach to classical mechanics. 5.1 Curvilinear coordinates We will specifically consider orthogonal coordinates for which the changes in due to changes are mutually orthogonal so that (5.1) where are the metric elements for . This implies that the corresponding volume element is (5.2) Now the gradient of a function is a vectorial rate of change of a scalar function, (5.3) which implies that the change in due to is (5.4) Now we consider a change in due to a change in the variable, for which is where is the vectorial change in due to . Then we have (5.5) which gives the component of the gradient along the direction corresponding to for the change . We can also obtain the divergence of a vector function by using Gauss' theorem. Consider the theorem for an infinitesimal volume resulting from the mutually orthogonal changes in , resulting from . With the volume given in Eq.(5.2), one gets (5.6) with being the component of parallel to related to , and with additional contributions from surface elements parallel and anti-parallel to and resulting from changes and . Writing the differences of the terms in the form of derivatives, one gets (5.7) This leads to (5.8) Finally, taking , and using the relation in eq.(5.5), we get (5.9) For example, one obtains for cylindrical coordinates (5.10) We can leave out the -term and use the expression for polar coordinates in 2 dimensions, (5.11) A particularly important property is that (5.12) which relates the angular momentum to the derivative with respect to . To show this, we note that a rotation by about the axis leads to Page 3 Module 1 : Quantum Mechanics Chapter 5 : Quantum mechanics in two dimensions Quantum mechanics in two dimensions For quantum mechanics in higher dimensions, the appropriate choice of coordinates is indicated by the form of the potential. For example, if the potential is a function of only the radial distance, , then it is useful to choose a set of coordinates with as one of the coordinates. For analysing the Schroedinger equation in curvilinear coordinates, we first write the equation in terms of cartesian coordinates and then go over to curvilinear coordinates. This is also the procedure followed in Lagrangian approach to classical mechanics. 5.1 Curvilinear coordinates We will specifically consider orthogonal coordinates for which the changes in due to changes are mutually orthogonal so that (5.1) where are the metric elements for . This implies that the corresponding volume element is (5.2) Now the gradient of a function is a vectorial rate of change of a scalar function, (5.3) which implies that the change in due to is (5.4) Now we consider a change in due to a change in the variable, for which is where is the vectorial change in due to . Then we have (5.5) which gives the component of the gradient along the direction corresponding to for the change . We can also obtain the divergence of a vector function by using Gauss' theorem. Consider the theorem for an infinitesimal volume resulting from the mutually orthogonal changes in , resulting from . With the volume given in Eq.(5.2), one gets (5.6) with being the component of parallel to related to , and with additional contributions from surface elements parallel and anti-parallel to and resulting from changes and . Writing the differences of the terms in the form of derivatives, one gets (5.7) This leads to (5.8) Finally, taking , and using the relation in eq.(5.5), we get (5.9) For example, one obtains for cylindrical coordinates (5.10) We can leave out the -term and use the expression for polar coordinates in 2 dimensions, (5.11) A particularly important property is that (5.12) which relates the angular momentum to the derivative with respect to . To show this, we note that a rotation by about the axis leads to (5.13) which implies that (5.14) We can therefore write in 2 dimensions, as (5.15) where the second term is similar to the classical expression . 5.2 Simple harmonic oscillator We consider the s.h.o. potential in 2-d, (5.16) The corresponding Schroedinger equation is (5.17) Since the operator is separable in coordinates , we have special solutions of the form (5.18) The solutions to these equations are similar to the solutions for 1-d oscillator in Eq.(4.94), (5.19) The problem becomes very interesting when the oscillator is isotropic, i.e. . Then Page 4 Module 1 : Quantum Mechanics Chapter 5 : Quantum mechanics in two dimensions Quantum mechanics in two dimensions For quantum mechanics in higher dimensions, the appropriate choice of coordinates is indicated by the form of the potential. For example, if the potential is a function of only the radial distance, , then it is useful to choose a set of coordinates with as one of the coordinates. For analysing the Schroedinger equation in curvilinear coordinates, we first write the equation in terms of cartesian coordinates and then go over to curvilinear coordinates. This is also the procedure followed in Lagrangian approach to classical mechanics. 5.1 Curvilinear coordinates We will specifically consider orthogonal coordinates for which the changes in due to changes are mutually orthogonal so that (5.1) where are the metric elements for . This implies that the corresponding volume element is (5.2) Now the gradient of a function is a vectorial rate of change of a scalar function, (5.3) which implies that the change in due to is (5.4) Now we consider a change in due to a change in the variable, for which is where is the vectorial change in due to . Then we have (5.5) which gives the component of the gradient along the direction corresponding to for the change . We can also obtain the divergence of a vector function by using Gauss' theorem. Consider the theorem for an infinitesimal volume resulting from the mutually orthogonal changes in , resulting from . With the volume given in Eq.(5.2), one gets (5.6) with being the component of parallel to related to , and with additional contributions from surface elements parallel and anti-parallel to and resulting from changes and . Writing the differences of the terms in the form of derivatives, one gets (5.7) This leads to (5.8) Finally, taking , and using the relation in eq.(5.5), we get (5.9) For example, one obtains for cylindrical coordinates (5.10) We can leave out the -term and use the expression for polar coordinates in 2 dimensions, (5.11) A particularly important property is that (5.12) which relates the angular momentum to the derivative with respect to . To show this, we note that a rotation by about the axis leads to (5.13) which implies that (5.14) We can therefore write in 2 dimensions, as (5.15) where the second term is similar to the classical expression . 5.2 Simple harmonic oscillator We consider the s.h.o. potential in 2-d, (5.16) The corresponding Schroedinger equation is (5.17) Since the operator is separable in coordinates , we have special solutions of the form (5.18) The solutions to these equations are similar to the solutions for 1-d oscillator in Eq.(4.94), (5.19) The problem becomes very interesting when the oscillator is isotropic, i.e. . Then (5.20) Now we have degeneracy, different states with the same energy, of order : . he presence of the extra degeneracy suggests that there is an operator which commutes with . To observe this, we note that for , (5.21) which implies that also is an energy eigenstate with the same energy. In the present case, apart from the parity operator, angular momentum operator also commutes with the Hamiltonian: (5.22) It is therefore possible to write our states as simultaneous eigenstates of and . This is facilitated by considering the Schroedinger equation in terms of polar coordinates, (5.23) Taking leads to (5.24) For incorporating the appropriate asymptotic behaviour, we note that (5.25) We then take (5.26) Page 5 Module 1 : Quantum Mechanics Chapter 5 : Quantum mechanics in two dimensions Quantum mechanics in two dimensions For quantum mechanics in higher dimensions, the appropriate choice of coordinates is indicated by the form of the potential. For example, if the potential is a function of only the radial distance, , then it is useful to choose a set of coordinates with as one of the coordinates. For analysing the Schroedinger equation in curvilinear coordinates, we first write the equation in terms of cartesian coordinates and then go over to curvilinear coordinates. This is also the procedure followed in Lagrangian approach to classical mechanics. 5.1 Curvilinear coordinates We will specifically consider orthogonal coordinates for which the changes in due to changes are mutually orthogonal so that (5.1) where are the metric elements for . This implies that the corresponding volume element is (5.2) Now the gradient of a function is a vectorial rate of change of a scalar function, (5.3) which implies that the change in due to is (5.4) Now we consider a change in due to a change in the variable, for which is where is the vectorial change in due to . Then we have (5.5) which gives the component of the gradient along the direction corresponding to for the change . We can also obtain the divergence of a vector function by using Gauss' theorem. Consider the theorem for an infinitesimal volume resulting from the mutually orthogonal changes in , resulting from . With the volume given in Eq.(5.2), one gets (5.6) with being the component of parallel to related to , and with additional contributions from surface elements parallel and anti-parallel to and resulting from changes and . Writing the differences of the terms in the form of derivatives, one gets (5.7) This leads to (5.8) Finally, taking , and using the relation in eq.(5.5), we get (5.9) For example, one obtains for cylindrical coordinates (5.10) We can leave out the -term and use the expression for polar coordinates in 2 dimensions, (5.11) A particularly important property is that (5.12) which relates the angular momentum to the derivative with respect to . To show this, we note that a rotation by about the axis leads to (5.13) which implies that (5.14) We can therefore write in 2 dimensions, as (5.15) where the second term is similar to the classical expression . 5.2 Simple harmonic oscillator We consider the s.h.o. potential in 2-d, (5.16) The corresponding Schroedinger equation is (5.17) Since the operator is separable in coordinates , we have special solutions of the form (5.18) The solutions to these equations are similar to the solutions for 1-d oscillator in Eq.(4.94), (5.19) The problem becomes very interesting when the oscillator is isotropic, i.e. . Then (5.20) Now we have degeneracy, different states with the same energy, of order : . he presence of the extra degeneracy suggests that there is an operator which commutes with . To observe this, we note that for , (5.21) which implies that also is an energy eigenstate with the same energy. In the present case, apart from the parity operator, angular momentum operator also commutes with the Hamiltonian: (5.22) It is therefore possible to write our states as simultaneous eigenstates of and . This is facilitated by considering the Schroedinger equation in terms of polar coordinates, (5.23) Taking leads to (5.24) For incorporating the appropriate asymptotic behaviour, we note that (5.25) We then take (5.26) Using these in Eq.(5.24), we get (5.27) Since the equations are invariant under , the solutions can be taken to be odd or even functions of and we consider (5.28) Use of this in Eq.(5.27) leads to (5.29) Taking the coefficients of leads to (5.30) Taking and noting that implies , and we take . We then have (5.31) Comparing this with the ratio in Eq.(4.90) for the confluent hypergeometric function in Eq.(4.89), we obtainRead More

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