Operations on Vectors

A vector is defined by length and direction. The length F of a generic vector V is given by its modulus  

With F being the modulus of vector V, its unit vector U is such that 

The vectorial product a X b of vectors a and b is a vector c whose modulus c is 

The direction of c is orthogonal to a and b and depends on the rotation required for a to overlap on b (the right-hand rule ). 

By decomposing the vector into its directional components, it can be shown that the modulus of the vectorial product of two vectors a and b is given by the determinant 

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Both in Eq. (8-149) and Eq. (8-147), we have written the function in the center of the integrand simply for ease of visual memory in fact both /(q) and F(q,q ) commute with all the B-operators and their positions are immaterial. The B-operators operate on vectors > in occupation number space, so that we can evaluate the matrix elements of F in occupation number representation, viz., Eq. (8-145), either from Eq. (8-147) or from Eq. (8-149). 

Beginning with the vector form of the vorticity-transport equation (Eq. 6.47), derive the steady-state, axisymmetric, scaled vorticity transport equation (Eq. 6.50). Be careful in the evaluation of derivative operators on vector fields. 

The operator V can operate on vector functions as well as on scalar functions. An example of a vector function is the velocity of a compressible flowing fluid... 

Many students with engineering or physics backgrounds are already familiar with the stress tensor. They may skip ahead to the next section. The key concepts in this section are understanding (1) that tensors can operate on vectors (eq. 1.2.10), (2) standard index notation (eq. 1.2.21), (3) symmetry of the stress tensor (eq. 1.2.37), (4) the concept of pressure (eq. 1.2.44), and(5)normalstressdifferences(eq. 1.2.45). 

Arithmetical operations on complex numbers are performed much as for vectors. Thus, if a j hi and y = c + di, then ... 

Note that both /> and L/> are vectors in the same space Jf. The operator L is said to be linear if operating on a sum equals the sum of the individual operators ... 

If we operate on the vacuum state with any annihilation operator the result is the null vector ... 

These expressions can be thought of as matrix expressions for the operators B(q,f) and B (q,t) respectively. We note that the number of particles represented on the left differs by unity from that on the right. The operators connect vectors having different populations. 

To rationalize the name projection operator, we take a general vector f> in and operate on it with P n ... 

Thus the current operator indeed transforms like a vector. This must be the case in order that the equation Qdu(x) = ju(x) transform properly, assuming the transformation property (11-267) for Au(x). We now inquire briefly into tike question of the uniqueness of the U(ia) operator, in particular into the question of the phase associated with the fermion field operator. Note that the phase of the photon field operator is uniquely determined (Eq. (11-267)) by the fact that An is a hermitian field which commutes with the total charge operator Q. The negaton-positon field operator on the other hand does not commute with the total charge operator, in fact... 

The details of the operation Rr can be further speeified by the 3x3 matrix which represents the operation R in a suitably chosen coordinate system [2], in which also the vector r is expressed. For the operation on a function of r we need the inverse of the space group operation,... 

For compactness of notation, we introduce the 4A-dimensional vector Q with components q, for / = 1, 2,. .., N. The permutation operators P are allowed to operate on Q directly rather than on the wave functions. Thus, the expression P P(1, 2, N) is identical to I (.PQ). In this notation, equation (8.32) takes the form... 

The space inversion transformation is x —> —x and the corresponding operator on state vector space is called the parity operator (P). The parity operator reverses... 

Other operations on a vector include reflections. Consider, for example, the operation of reflection in the x,y plane. The result of this operation is to change the sign of the z component of the vector. Thus, a reflection in the x, y plane can be represented by... 

Before going on to calculate the energy levels it is necessary to digress and briefly describe the wavefunction. The spin Hamiltonian only operates on the spin part of the wavefunction. Every unpaired electron has a spin vector /S = with spin quantum numbers ms = + and mB = — f. The wavefunctions for these two spin states are denoted by ae) and d ), respectively. The proton likewise has I = with spin wavefunctions an) and dn)- In the present example these will be used as the basis functions in our calculation of energy levels, although it is sometimes convenient to use a linear combination of these spin states. 

In the vector space L defined over the field of real numbers, every operator acting on L does not necessarily have eigenvalues and eigenvectors. Thus for the operation of 7t/2 rotation on a two-dimensional vector space of (real) position vectors, the operator has no eigenvectors since there is no non-zero vector in this space which transforms into a real multiple of itself. However, if L is a vector space over the field of complex numbers, every operator on L has eigenvectors. The number of eigenvalues is equal to the dimension of the space L. The set of eigenvalues of an operator is called its spectrum. 

Solution. How should we start to convert the words of the problem into mathematical statements First, let us define the variables. There will be four of them (tAVtA2, tBl, and tB2, designated as a set by the vector t) representing, respectively, the number of days per year each plant operates on each material as indicated by the subscripts. 

Here H is the unperturbed Hamiltonian of the system and h denotes a complete set of excitation and de-excitation operators, arranged as column vector h or as row vector h. Completeness of the set of operators h means that all possible excited states of the system must be generated by operating on... 

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