Dot products

Recall that L contains the frequency or (equation (B2.4.8)). To trace out a spectrum, equation (B2.4.11)) is solved for each frequency. In order to obtain the observed signal v, the sum of the two individual magnetizations can be written as the dot product of two vectors, equation (B2.4.12)). 

In Liouville space, both the density matrix and the operator are vectors. The dot product of these Liouville space... 

APPENDIX - SUMMARY OF VECTOR AND TENSOR ANALYSIS The scalar (dot) product of two vectors is a number found as... 

The tensor (single-dot) product of two tensors is found as follows... 

The matrix A in Eq. (7-21) is comprised of orthogonal vectors. Orthogonal vectors have a dot product of zero. The mutually perpendicular (and independent) Cartesian coordinates of 3-space are orthogonal. An orthogonal n x n such as matr ix A may be thought of as n columns of n-element vectors that are mutually perpendicular in an n-dimensional vector space. 

For vectors as we have been dealing with, the scalar or dot product is defined as we have seen as follows ... 

The easiest way to proceed is to use vectors to describe this part of the problem. We represent the distance between the pair of scattering sites by the vector OP the length of which is simply r. To express di and d2 in terms of OP we construct the unit vectors a and b which are parallel to the incident and scattered directions, respectively. The projection of OP into direction a, given by the dot product of these two vectors, equals dj. Likewise, the projection of OP into direction b gives d2. Therefore we can write... 

In the final stage of this involved derivation, we have to free Eq. (10.78) from the dependence it contains on the geometry of Fig. 10.11. The problem lies in the dot product of the vector rj, -which replaces OP in Fig. 10.11-and... 

The energy conservation equation is not normally solved as given in (9.4). Instead, an evolution equation for internal energy is used [9]. First an evolution equation for the kinetic energy is derived by taking the dot product of the momentum balance equation with the velocity and integrating the resulting differential equation. The differential equation is... 

Here q is the net charge (monopole), p, is the (electric) dipole moment, Q is the quadrupole moment, and F and F are the field and field gradient d /dr), respectively. The dipole moment and electric field are vectors, and the pF term should be interpreted as the dot product (p F = + EyPy + Ez z)- "I e quadrupole moment and field... 

Here (r - Rc) (r - Rq) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r - RG)(r — Rg) is a 3x3 matrix containing the products of the x,y,z components, analogous to the quadrupole moment, eq. (10.4). Note that both the L and P operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave funetion (for a singlet state) eqs. (10.18)/(10.23). 

Gradient operator Laplace operator Dot product Cross product Divergence operator Curl operator Vector transposition Complex conjugate... 

We have used bold characters to show how the C32 was calculated. C32 is the dot product of the second column of matrix B with the third row of matrix A ... 

If the dot product of two vectors is equal to zero, those vectors are orthogonal (perpendicular) to each other. For example, the dot product of the vectors ... 

Equation (7-19) has only formal significance it can be used in calculations only if the dot product in the exponent need not be expanded. Otherwise, the equation would read... 

The force / and displacement s are vector quantities, and equation (2.5) indicates that the vector dot product of the two gives a scalar quantity. The result of this operation is equation (2.6)... 

Note that in Section 9.2.23 the dot product x y is used as an equivalent notation for the scalar product x y. 

The x-component, px, of the momentum now needs to carry a subscript, whereas before it was denoted simply as p. The scalar or dot product of r and p is... 

The scalar product, often called the dot product , obeys the commutative and distributive laws of ordinary multiplication, viz. 

Thus, the vector C represents the product of the. vectors A and B such Jhat its length is given by C = ABsin 6. In the usual notation C = A x B Tnis operation is referred to as the vector product of the two vectors and in the jargon used in this application it is called die "cross product". It must then lie carefully distinguished from the dot product defined by Eq. (9). 

It is noteworthy that the induced energy is the dot product of the induced dipole and the static field and not the total field. The interpretation of Eq. (9-11) is that a static field is required to originate induced dipoles. 

When we include the Zeeman interaction term, gpBB-S, in the spin Hamiltonian a complication arises. We have been accustomed to evaluating the dot product by simply taking the direction of the magnetic field to define the z-axis (the axis of quantization). When we have a strong dipolar interaction, the... 

Here, fp is the field due to pp P is a polarization density, which is related to the electric field E produced by the remaining charge density p E = -47tP. Finally, a dot represents the dot product between functions, / fp(r) P(r)dr, where the integration is over all space. We also introduce ... 

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