Chapter 5 - Quantum mechanics in two dimensions Notes | EduRev

: Chapter 5 - Quantum mechanics in two dimensions Notes | EduRev

 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 obtain
 
  
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