Page 74 out of 79 total pages , Page 4 out of 9 pages in this chapter
Usage:
Description:
The function T2 returns either the standard rank 2 spin tensor or a specific component of that tensor. This tensor is equivalent to that from the product of two rank 1 spin tensors.
T2 (spin_sys &sys, int spin1, int spin2) - When T2 is invoked with only a spin system, sys, and spin indices, spin1 and spin 2, in the argument list it creates the standard irreducible spherical rank 2 spin tensor containing the nine components of equation sosi.
T2(spin_sys &sys, spin_op &Im1, spin_op &Iz1, spin_op &Ip1, spin_op &Im2, spin_op &Iz2, spin_op &Ip2) - This function performs the equivalent to 1. above except that the spin operators involved (these are components of the two rank 1 spin tensors) are directly input. This avoids any repetitive computation of the spin operators on successive calls to the function. Im1 is spin operator I-, Iz1 is the spin operator Iz and Ip1 the spin operator I+, all for the first spin. Im2, Iz2, and Ip2 are the spin operators for the second spin.
T2(spin_sys &sys, int spin1, int spin2, int l, int m) - When T2 is invoked with a spin system, spin indices, and angular momentum components l & m, the function returns a spin operator which is the l,m irreducible spherical component of the standard rank 2 spin tensors. The appropriate component of equation sosi is the result.
T2(spin_sys &sys, spin_op &Im1, spin_op &Iz1, spin_op &Ip1, spin_op &Im2, spin_op &Iz2, spin_op &Ip2, int l, int m) - This function performs the equivalent to 3. above except that the spin operators involved are directly input. This avoids any repetitive computation of the spin operators on successive calls to the function. Im1 is spin operator I-, Iz1 is the spin operator Iz and Ip1 the spin operator I+, all for the first spin. Im2, Iz2, and Ip2 are the spin operators for the second spin.
The rank 2 spin tensor consists of nine irreducible spherical components specified as either
where i and j are spin indices. The tensor index l spans the rank, in this case 2, while the tensor index m spans l,
[0, 2] & m [-l, l],
All tensor components are returned as spin operators.
Return Value:
A rank 2 irreducible spherical spin tensor.
Example(s):
#include <gamma.h>
main()
{
spin_sys AMX(3); // create a three spin system called AMX.
spin_op SOp(AMX); // create a spin operator associated with AMX, SOp.
spin_T SphT(AMX); // create a spin tensor associated with AMX, SphT.
SphT = T1(AMX, 0); // SphT is the rank 1 spin tensor for spin 0.
SOp = T1(AMX, 0, 1, 0); // SOp is the 1,0 rank 1 spin tensor component spin 0.
cout << SOp; // prints SOp, the 1,1 component of spin tensor.
}
Mathematical Basis:
A general rank 2 tensor contains nine components. When this tensor involves two spin components of angular momenta its irreducible spherical components are specified as 
, the formulas which produce them given by equation on page 1921,
When treating spin angular momentum where both spins have I=1/2 the matrix representations of these tensor components (operators) are given by the following. They are shown in the composite Hilbert space of the two spins (spin indices are implicit here).
See Also: T1, T2SS
Usage:
Description:
The function T20 returns either the standard irreducible rank 2 spin tensor or a specific component of that tensor. For this function the only allowed value of m is 0.
T20(spin_sys &sys, int spin1, int spin2, int m) - When T20 is invoked with a spin system, sys, two spin indices, spin1 and spin2, and a angular momentum component, m, the function returns an irreducible spin-space spherical rank 2 tensor containing the single component of equation on page 192.
T20(spin_sys &sys, spin_op &Im, spin_op &Iz, spin_op &Ip, coord &pt, int rev=0) - This function performs the equivalent to 1. above except that the spin operators involved are directly input. This avoids any repetitive computation of the spin operators on successive calls to the function. Im is spin operator I-, Iz is the spin operator Iz and Ip the spin operator I+, all for the spin being treated.
The irreducible rank 0 part of the general rank 2 spin tensor consists of only one spherical components specified as
,
where i is the spin index. The tensor index the tensor index m spans the rank.
All tensor components are returned as spin operators.
Return Value:
An irreducible rank 2 space/spin tensor or a component of that tensor.
Example(s):
#include <gamma.h>
main()
{
SOp = T1(AMX, 0, 1, 0); // SOp is the 1,0 rank 1 spin tensor component spin 0.
cout << SOp; // prints SOp, the 1,1 component of spin tensor.
}
Mathematical Basis:
This tensor component is given by the following equation,
See Also: T1, T2
Usage:
Description:
The function T21 returns one of the l=1 components of the rank 2 spin tensor formed from the product of a rank 1 spatial tensor with a rank 1 spin tensor.
T21(spin_sys &sys, int spin, coord &pt, int rev=0) - When T21 is invoked with a spin system, sys, a spin index, spin and a coordinate point, pt, the function returns an irreducible spin-space spherical rank 2 tensor containing the five components of equation on page 192.
T21(spin_sys &sys, spin_op &Im, spin_op &Iz, spin_op &Ip, coord &pt, int rev=0) - This function performs the equivalent to 1. above except that the spin operators involved are directly input. This avoids any repetitive computation of the spin operators on successive calls to the function. Im is spin operator I-, Iz is the spin operator Iz and Ip the spin operator I+, all for the spin being treated.
The general rank 2 space/spin tensor contains of three rank 1 irreducible spherical components specified as
,
where i is the spin index. The tensor index the tensor index m spans the rank, m [-1, 1]. All tensor components are returned as spin operators.
Return Value:
Example:
#include <gamma.h>
main()
{
spin_sys AMX(3); // create a three spin system called AMX.
cout << SOp; // prints SOp, the 1,1 component of spin tensor.
}
Mathematical Basis:
The three tensor components returned by this function are specified by the following equations.
See Also: T1, T2
Usage:
Description:
The function T22 returns either the standard irreducible rank 2 space/spin tensor or a specific component of that tensor.
T21SS(spin_sys &sys, int spin, coord &pt, int rev=0) - When T2 is invoked with a spin system, sys, a spin index, spin and a coordinate point, pt, the function returns an irreducible spin-space spherical rank 2 tensor containing the five components of equation on page 192.
T21SS(spin_sys &sys, spin_op &Im, spin_op &Iz, spin_op &Ip, coord &pt, int rev=0) - This function performs the equivalent to 1. above except that the spin operators involved are directly input. This avoids any repetitive computation of the spin operators on successive calls to the function. Im is spin operator I-, Iz is the spin operator Iz and Ip the spin operator I+, all for the spin being treated.
The irreducible rank 2 space/spin tensor consists of five spherical components specified as
,
where i is the spin index. The tensor index the tensor index m spans the rank, m [-2, 2]. All tensor components are returned as spin operators.
Return Value:
An irreducible rank 2 space/spin tensor or a component of that tensor.
Example(s):
#include <gamma.h>
main()
{
spin_sys AMX(3); // create a three spin system called AMX.
cout << SOp; // prints SOp, the 1,1 component of spin tensor.
}
Mathematical Basis:
The five tensor components are given by the following equations,.
See Also: T1, T2
Usage:
Description:
The function T2SS returns either the standard rank 2 space/spin tensor or a specific component of that tensor.
T2SS(spin_sys &sys, int spin, coord &pt, int rev=0) - When T2 is invoked with a spin system, sys, a spin index, spin and a coordinate point, pt, the function returns a spin-space irreducible spherical rank 2 tensor containing the nine components of equation on page 192. This rank 2 tensor corresponds to a single spin because it is formulated from the product spin angular momentum vector (rank1 spin tensor) and a spatial vector (rank 1 spatial tensor) as given by the coordinate point. The coordinate is assumed to represent a normalized Cartesian vector. If the flag rev is set to non-zero, the tensor is assumed to be the product of the spatial vector times the spin vector as opposed to the spin vector times the spatial vector (default).
T2SS(spin_sys &sys, spin_op &Im, spin_op &Iz, spin_op &Ip, coord &pt, int rev=0) - This function performs the equivalent to 1. above except that the spin operators involved are directly input. This avoids any repetitive computation of the spin operators on successive calls to the function. Im is spin operator I-, Iz is the spin operator Iz and Ip the spin operator I+, all for the spin being treated.
T2SS(spin_sys &sys, int spin, coord &pt, int l, int m, int rev=0) - This function returns the l,m irreducible spherical component of the tensor formed from 1. above. Arguments l & m are the tensor angular momentum components. A spin operator which is the appropriate component of equation is returned.
T2SS(spin_sys &sys, spin_op &Im, spin_op &Iz, spin_op &Ip, coord &pt, int l, int m, int rev=0) - This function returns the l,m irreducible spherical component of the tensor formed from 2. above. Arguments l & m are the tensor angular momentum components. A spin operator which is the appropriate component of equation is returned.
The rank 2 space/spin tensor consists of nine irreducible spherical components specified as
,
where i is the spin index. The tensor index l spans the rank, in this case 2, while the tensor index m spans l, l [0, 2] and m [-l, l]. All tensor components are returned as spin operators.
Return Value:
A rank 2 irreducible spherical spin tensor.
Example(s):
#include <gamma.h>
main()
{
spin_sys AMX(3); // create a three spin system called AMX.
spin_op SOp(AMX); // create a spin operator associated with AMX, SOp.
spin_T SphT(AMX); // create a spin tensor associated with AMX, SphT.
SphT = T1(AMX, 0); // SphT is the rank 1 spin tensor for spin 0.
SOp = T1(AMX, 0, 1, 0); // SOp is the 1,0 rank 1 spin tensor component spin 0.
cout << SOp; // prints SOp, the 1,1 component of spin tensor.
}
Mathematical Basis:
There are occasions when it is convenient to utilize a rank 2 "spin" tensor which actually contains both spatial and spin terms. In this case the tensor will involve only one spin, its nine components are specified as
.
This tensor if formed from the product of a rank 1 spin tensor with a rank 1 spatial tensor. The nine formulas which produce these components are given by2 equations
where 
is a normalized vector 
. Of course, setting the vector 
equal to the angular momentum (vector) operator 
simply regenerates equations (18-3). The simplest situation mathematically occurs when 
points along the positive z-axis, 
. Put another way, the tensor simplifies when placed in a coordinates system whose z-axis is aligned with the z-axis of the spatial tensor (vector). The applicable equations in this case are shown in the following figure.
For a spin 1/2 particle and 
along the positive z-axis the matrix form of these tensor components are shown in the following figure3 (in the single spin Hilbert space).
One must be very careful in using single spin rank 2 tensors of this type because they contain both spatial and spin components. It is improper to rotate this tensor in spin space because it also rotates spatial variables. If some Hamiltonian is a product of a spatial tensor and this tensor, the spatial tensor cannot be rotated as it rotates only part of the spatial components4.
See Also: T1, T2
Usage:
Description:
The function T20SS returns either the standard irreducible rank 2 space/spin tensor or a specific component of that tensor. For this function the only allowed value of m is 0. Furthermore the argument rev is not utilized. Both of these arguments are included for consistency with the functions T21SS and T22SS.
T20SS(spin_sys &sys, int spin, coord &pt, int rev=0) - When T2 is invoked with a spin system, sys, a spin index, spin and a coordinate point, pt, the function returns an irreducible spin-space spherical rank 2 tensor containing the five components of equation on page 192. This irreducible rank 2 tensor corresponds to a single spin because it is formulated from the product spin angular momentum vector (rank1 spin tensor) and a spatial vector (rank 1 spatial tensor) as given by the coordinate point. The coordinate is assumed to represent a normalized Cartesian vector. If the flag rev is set to non-zero, the tensor is assumed to be the product of the spatial vector times the spin vector as opposed to the spin vector times the spatial vector (default).
T20SS(spin_sys &sys, spin_op &Im, spin_op &Iz, spin_op &Ip, coord &pt, int rev=0) - This function performs the equivalent to 1. above except that the spin operators involved are directly input. This avoids any repetitive computation of the spin operators on successive calls to the function. Im is spin operator I-, Iz is the spin operator Iz and Ip the spin operator I+, all for the spin being treated.
The irreducible rank 2 space/spin tensor consists of five spherical components specified as
,
where i is the spin index. The tensor index the tensor index m spans the rank, m [-0, 0]. All tensor components are returned as spin operators.
Return Value:
An irreducible rank 2 space/spin tensor or a component of that tensor.
Example(s):
#include <gamma.h>
main()
{
spin_sys AMX(3); // create a three spin system called AMX.
spin_op SOp(AMX); // create a spin operator associated with AMX, SOp.
spin_T SphT(AMX); // create a spin tensor associated with AMX, SphT.
SphT = T1(AMX, 0); // SphT is the rank 1 spin tensor for spin 0.
SOp = T1(AMX, 0, 1, 0); // SOp is the 1,0 rank 1 spin tensor component spin 0.
cout << SOp; // prints SOp, the 1,1 component of spin tensor.
}
Mathematical Basis:
This tensor component is given by the following equation,
where 
is a normalized vector 
.
See Also: T1, T2
Usage:
Description:
The function T21SS returns one of the 
components of the rank 2 space/spin tensor formed from the product of a rank 1 spatial tensor with a rank 1 spin tensor.
T21SS(spin_sys &sys, int spin, coord &pt, int rev=0) - When T21SS is invoked with a spin system, sys, a spin index, spin and a coordinate point, pt, the function returns an irreducible spin-space spherical rank 2 tensor containing the five components of equation on page 192. This irreducible rank 2 tensor corresponds to a single spin because it is formulated from the product spin angular momentum vector (rank1 spin tensor) and a spatial vector (rank 1 spatial tensor) as given by the coordinate point. The coordinate is assumed to represent a normalized Cartesian vector. If the flag rev is set to non-zero, the tensor is assumed to be the product of the spatial vector times the spin vector as opposed to the spin vector times the spatial vector (default).
T20SS(spin_sys &sys, spin_op &Im, spin_op &Iz, spin_op &Ip, coord &pt, int rev=0) - This function performs the equivalent to 1. above except that the spin operators involved are directly input. This avoids any repetitive computation of the spin operators on successive calls to the function. Im is spin operator I-, Iz is the spin operator Iz and Ip the spin operator I+, all for the spin being treated.
The general rank 2 space/spin tensor contains of three rank 1 irreducible spherical components specified as
,
where i is the spin index. The tensor index the tensor index m spans the rank, m [-1, 1]. All tensor components are returned as spin operators.
Return Value:
Example:
#include <gamma.h>
main()
{
spin_sys AMX(3); // create a three spin system called AMX.
cout << SOp; // prints SOp, the 1,1 component of spin tensor.
}
Mathematical Basis:
The three tensor components returned by this function are specified by the following equations,
where 
is a normalized vector 
. If the tensor product ordering is reversed (using the function rev flag) there will be a sign change on all returned components.
See Also: T1, T2
Usage:
Description:
The function T22SS returns either the standard irreducible rank 2 space/spin tensor or a specific component of that tensor. The argument rev is not utilized, it is included for consistency with the function T21SS.
T21SS(spin_sys &sys, int spin, coord &pt, int rev=0) - When T2 is invoked with a spin system, sys, a spin index, spin and a coordinate point, pt, the function returns an irreducible spin-space spherical rank 2 tensor containing the five components of equation on page 192. This irreducible rank 2 tensor corresponds to a single spin because it is formulated from the product spin angular momentum vector (rank1 spin tensor) and a spatial vector (rank 1 spatial tensor) as given by the coordinate point. The coordinate is assumed to represent a normalized Cartesian vector. If the flag rev is set to non-zero, the tensor is assumed to be the product of the spatial vector times the spin vector as opposed to the spin vector times the spatial vector (default).
T21SS(spin_sys &sys, spin_op &Im, spin_op &Iz, spin_op &Ip, coord &pt, int rev=0) - This function performs the equivalent to 1. above except that the spin operators involved are directly input. This avoids any repetitive computation of the spin operators on successive calls to the function. Im is spin operator I-, Iz is the spin operator Iz and Ip the spin operator I+, all for the spin being treated.
The irreducible rank 2 space/spin tensor consists of five spherical components specified as
,
where i is the spin index. The tensor index the tensor index m spans the rank, m [-2, 2]. All tensor components are returned as spin operators.
Return Value:
An irreducible rank 2 space/spin tensor or a component of that tensor.
Example(s):
#include <gamma.h>
main()
{
spin_sys AMX(3); // create a three spin system called AMX.
cout << SOp; // prints SOp, the 1,1 component of spin tensor.
}
Mathematical Basis:
The five tensor components are given by the following equations,
where 
is a normalized vector 
.
See Also: T1, T2
Readers should note that there are several different conventions found in the literature used in expressing these tensor components. Therefore these formulas may vary from those given by other sources. The following reasons will likely explain any discrepancies found between the formulae here and other references:
1.) All components differ by a constant - this is mathematically valid because a constant multiple does not change the tensor properties, it is just a normalization factor difference. Equation has a normalization in which these components result from a product of two of rank 1 tensors given in equation (10-12) as shown in the discussion Section at the end of this Chapter.
2.) The spin operators are different - most differences result either from use of Ix & Iy versus I+ & I- or from the explicit expansion of the product of Ii·Ij. For example, the T0,0 component formula given shows three common variations due to these differences. Another explanations is that some authors (like Haeberlen) use specially defined operators I+ & I- which make their formulations look different. A third more direct reason for a formula discrepancy is that the components being compared are not the same, i.e. one may accidentally be comparing T1,0 above with the T1,0 obtained directly from a rank 1 treatment or the single spin variant of the rank 2 treatment. While similar in appearance, they are scaled differently: compare equation (10-12) and equation .)
3.) The components differ by several constants - if one is always paring up these tensor components with spatial tensor components it is allowable to scale one as long as compensation is made in the other (See EBW, the bottom of page 47 for an example). This is not recommended. One reason is that the tensor (either space or spin) will no longer rotate properly. Another reason is that the components no longer relate to one another via the ladder operators. Equation (10-11), which defines the irreducible spherical tensor, is invalid in such a treatment.2
Bear in mind that these formulae have the 2nd tensor as the spatial one. A sign change will occur for the l=1 components if3is set to the 1st and the spin tensor to the second.
The GAMMA program which produced these matrix representations can be found at the end of this Chapter, Rank2SS_SpinT.cc.4
See the discussion in Mehring.
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GAMMA Support Provided by the National High Magnetic Field Laboratory
© 1996 Scott A. Smith, The NHMFL, and The Florida State University. All Rights Reserved. |