Generalized creation and annihilation operators: Physical interpretation and reordering properties.
Also, the maximum of the module of amplitude in interference of probability for Fermi-Dirac particles will be reached when. This set of projectors we identified to the set of pure states P U.
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From what has been said, it follows that the operators d n and d n are operators of creation and annihilation, respectively. Journals by Subject. Journals by Title. Author s Malkhaz Mumladze. Affiliation s Gori University, Gori, Georgia. ABSTRACT In this article we proposed derivation of the wave function of a quantum system from the functional representation of the state of this system. There is considered representation of creation and annihilation operators by boundary and co-boundary operators of chain and co-chain complexes on the physical space.
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Cite this paper Mumladze, M. Open Access Library Journal , 6 , Nauka, Moscow. Oxford Science Publication, Oxford. Using the formula above for the Hamiltonian, one obtains. This gives the ground state energy. This allows one to identify the energy eigenvalue of any eigenstate as . The ground state of the quantum harmonic oscillator can be found by imposing the condition that. Explicit formulas for all the eigenfunctions can now be found by repeated application of to.
Creation and annihilation operators
This, and further operator formalism, can be found in Glimm and Jaffe, Quantum Physics, pp. The matrix expression of the creation and annihilation operators of the quantum harmonic oscillator with respect to the above orthonormal basis is.
These can be obtained via the relationships and. The eigenvectors are those of the quantum harmonic oscillator, and are sometimes called the "number basis". The operators derived above are actually a specific instance of a more generalized notion of creation and annihilation operators. The more abstract form of the operators are constructed as follows. Let H be a one-particle Hilbert space that is, any Hilbert space, viewed as representing the state of a single particle.
Thus H embeds as a complex vector subspace of its own CCR algebra. In general, the CCR algebra is infinite dimensional. For fermions, the fermionic CAR algebra over H is constructed similarly, but using anticommutator relations instead, namely. The CAR algebra is finite dimensional only if H is finite dimensional. The CAR algebra is closely related to, but not identical to, a Clifford algebra.
Physically speaking, a f removes i. To see how this kind of reaction can be described by the annihilation and creation operator formalism, consider particles at a site on a 1-d lattice. Each particle moves to the right or left with a certain probability, and each pair of particles at the same site annihilates each other with a certain other probability. The probability that one particle leaves the site during the short time period is proportional to , let us say a probability to hop left and to hop right.
All particles will stay put with a probability. Since is so short, the probability that two or more will leave during is very small and will be ignored.
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It represents the juxtaposition or conjunction, or tensor product of the number states A slight modification of the annihilation and creation operators is needed so that. This modification preserves the commutation relation. The reaction term can be deduced by noting that particles can interact in different ways, so that the probability that a pair annihilates is , yielding a term. Thus the state evolves by. This kind of notation allows the use of quantum field theoretic techniques to be used in the analysis of reaction diffusion systems.
In quantum field theories and many-body problems one works with creation and annihilation operators of quantum states, and. These operators change the eigenvalues of the number operator ,. The indices such as represent quantum numbers that label the single-particle states of the system; hence, they are not necessarily single numbers. For example, a tuple of quantum numbers is used to label states in the hydrogen atom. The commutation relations of creation and annihilation operators in a multiple- boson system are,.
For fermions , the commutator is replaced by the anticommutator ,. Therefore, exchanging disjoint i. If the states labelled by i are an orthonormal basis of a Hilbert space H , then the result of this construction coincides with the CCR algebra and CAR algebra construction in the previous section but one. If they represent "eigenvectors" corresponding to the continuous spectrum of some operator, as for unbound particles in QFT, then the interpretation is more subtle.
quantum mechanics - Annihilation and Creation operators not hermitian - Physics Stack Exchange
Creation and annihilation operators Creation and annihilation operators are mathematical operators that have widespread applications in quantum mechanics , notably in the study of quantum harmonic oscillators and many-particle systems. Note that the quantity is the same energy as that found for light quanta and that the parenthesis in the Hamiltonian can be written as The last two terms can be simplified by considering their effect on an arbitrary differentiable function f q , which implies, , coinciding with the usual canonical commutation relation in position space representation.
Letting , where "p" is the nondimensionalized momentum operator then we have and Note that these imply that The operators and may be contrasted with normal operators , which commute with their adjoints. Using the commutation relations given above, the Hamiltonian operator can be expressed as One can compute the commutation relations between the and operators and the Hamiltonian:  These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator.
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Using these commutation relations, it follows that  This shows that and are also eigenstates of the Hamiltonian, with eigenvalues and respectively. Using the formula above for the Hamiltonian, one obtains so is an eigenfunction of the Hamiltonian.
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