The number operator

Adding together all ON op ators in the Fock space, we obtain the Hermitian operator 

In general, the application of the string X to a Fock-space vector increases the number of electrons by N.  

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The number operators N+,N. for the number of positively and negatively charged particles are then given by the following expressions... 

The number operators, when expressed in terms of the configuration space operators, assume the following form... 

Next we establish the connection of the previous formalism with the Fo< space description of photons. From the interpretation of a>(k) as the number operator for photons of momentum k polarization A, and of cA(k) and cA(k) as destruction and creation operators for... 

The new delightful book by Greenstein and Zajonc(9) contains several examples where the outcome of experiments was not what physicists expected. Careful analysis of the Schrddinger equation revealed what the intuitive argument had overlooked and showed that QM is correct. In Chapter 2, Photons , they tell the story that Einstein got the Nobel Prize in 1922 for the explaining the photoelectric effect with the concept of particle-like photons. In 1969 Crisp and Jaynes(IO) and Lamb and Scullyfl I) showed that the quantum nature of the photoelectric effect can be explained with a classical radiation field and a quantum description for the atom. Photons do exist, but they only show up when the EM field is in a state that is an eigenstate of the number operator, and they do not reveal themselves in the photoelectric effect. 

To determine the degeneracy of the energy levels or, equivalently, of the eigenvalues of the number operator N, we must first obtain the eigenvectors 0z) for the ground state. These eigenvectors are determined by equation (4.29). When equation (4.18a) is substituted for a, equation (4.29) takes the form... 

We next show that if the eigenvalue n of the number operator N is nondegenerate, then the eigenvalue n -h 1 is also non-degenerate. We begin with the assumption that there is only one eigenvector with the property that... 

Another interesting website, http //matti.usu.edu/nlvm/nav/index.html, has a game, Circle 0, for practicing one-digit integer arithmetic. Click on Virtual Library. Then, click on the 9-12 box in the Numbers Operations row. Click on Circle 0, and play the integer game. 

To complete our discussion of the formulas for the connected 3-RDM, we mention that the system of equations in Eq. (48) and the formula in Eq. (49), which is often called the Mazziotti correction [24, 26, 29] to distinguish it from the Nakatsuji-Yasuda correction [19, 24] for the 3-RDM, can also be derived from a contracted Schrodinger equation for the number operator ... 

Upon simplihcation it is not difficult to show that Eq. (57) is equivalent to Eqs. (48) and (55). Therefore the position of the number operator does not affect the relation that we have derived for the cumulant 3-RDM. More general relations for the cumulant p-RDM may similarly be derived by contracting the (p + 1)-RDM to the p-RDM. 

It is easier to construct Hamiltonians whose exact solutions are known when one uses the orbital representation. Within the orbital representation, the most fundamental operator is the number operator for a spin orbital, hj = ajaj. 

The number operator is clearly a projector hj = hj. The g-matrix can be converted to and from the orbital representation using... 

Because the eigenfunctions of the number operators are the Slater determinants, any polynomial of number operators will also have Slater determinant eigenfunctions. Starting with a basis set of K spin orbitals, 0] (x), > us... 

Any gth-degree polynomial of the number operators for these R orbitals... 

Pi > Pij- The probability that orbital i is occupied is greater than or equal to the probability that orbital i and orbital are both occupied. An alternative interpretation, based on the number operators representation in Eq. (43), is that the probability that orbital is occupied and orbital j" is empty is greater than or equal to zero. 

Among the simple linear operators that are commonly used to construct constraints, polynomials of the number operators (cf. Eq. (22)) are particularly useful. Polynomials of number operators are convenient because (i) the ground-state wavefunction of number-operator polynomials is a Slater determinant and (ii) the number-operator constraints depend only on the diagonal elements of... 

Because polynomials of orbital occupation number operators do not depend on the off-diagonal elements of the reduced density matrix, it is important to generalize the Q, R) constraints to off-diagonal elements. This can be done in two ways. First, because the Q, R) conditions must hold in any orbital basis, unitary transformations of the orbital basis set can be used to constrain the off-diagonal elements of the density matrix. (See Eq. (56) and the surrounding discussion.) Second, one can replace the number operators by creation and annihilation operators on different orbitals according to the rule... 

It can be easily checked by direct computation that we have really obtained a realization of the Lie algebra g in a Hilbert (Fock) space, [T a, T fc] = ifabc fc, in accordance with (11), where Ta = T f/aL . For an irreducible representation R, the second-order Casimir operator C2 is proportional to the identity operator I, which, in turn, is equal to the number operator N in our Fock representation, that is, if T" —> Ta, then I /V 5/,/a . Thus we obtain an important for our further considerations constant of motion N ... 

Let us confine our attention to the one-particle subspace of the Fock space. As the number operator N is conserved by virtue of Eq. (28), if we start from the one-particle subspace of the Fock space, we shall remain in this subspace during all the evolution. The transition amplitude Uki(t",t ) between the one-particle states ) d]k 0) and 11/) = n,+10) is given by the following scalar product in the holomorphic representation... 

For a complete one-electron basis, the canonical commutators become proportional to the number operators. For finite basis sets, the canonical commutator becomes a general one-body operator. 

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It is easy to see that the number operator is written as N a)a that are diagonal with respect to the eigenvalues N n) = n n) and also define the energy levels for the system since the Hamiltonian is... 

The operator Y aja where the sum runs over all elements of the shell, is called the number operator. When acting on a shell state it yields the corresponding occupation number, e.g. ... 

Here, the integration over X was performed in Eq. (63) to define W%a (X, ) which is the integrated value of the combination of the spectral density function with the time independent operator. This spectral density function contains the quantum equilibrium structure of the system. (X, t) is the time evolved matrix element of the number operator for the product state B. Thus, to calculate the rate, one samples initial configurations from the quantum equilibrium distribution, and then computes the evolution of the number operator for product state B. The QCL evolution of the species operator is accomplished using one of the algorithms discussed in Sec. 3.2. Alternative approaches to the dynamics may also be used such as the further approximations to the QCLE discussed in Sec. 4. 

The Coherent State as the Result of the Action of the Translation Operator on the Ground State of the Number Operator... 

Again, performing the trace on the eigenvectors of the number operator at a and inserting the closeness relation on the eigenvectors (214), the above expression of the ACF takes the final form ... 

Now, we may insert in front of a coherent state, the closeness relation on the eigenstates of the number operator at a of the quantum harmonic oscillator (with [a, at] = 1), in the following way ... 

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