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Tilde90
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Please, can somebody show me why a Hamiltonian like [itex]\sum_nh(x_n)[/itex] can be written as [itex]\sum_{i,j}t_{i,j}a^+_ia_j[/itex], with [itex]t_{i,j}=\int f^*_i(x)h(x)f_j(x)dx[/itex]?
Thank you.
Thank you.
DimReg said:I think what you are looking for is that you can show that the raising and lowering operators are enough to create any operator on fock space. Basically, the proof of that, is by using raising and lowering operators as a "basis", you have enough freedom to make the operator matrix elements have any value you want. Weinberg's QFT book has a description of this in his "Cluster Decomposition Principle" chapter.
Tilde90 said:Thank you all for your replies. I apologise for the lack of details in my first post, but that evening I was at the brink of desperation trying to prove that result. :-) Finally, I found a proof online, but as you say it's not something which books usually demonstrate (despite its importance).
espen180 said:For the sake of claification, do you mean taking linear combinations of finite products of raising and lowering operators?
B-80 said:This is also relevant to my interests... Where did you find the proof if you don't mind me asking?
Second quantization is a mathematical formalism used in quantum mechanics to describe the behavior of a large number of identical particles. It allows for the creation and annihilation of particles in a quantum system, and is particularly useful for studying systems with an infinite number of particles.
Second quantization operators are mathematical operators that describe the creation and annihilation of particles in a second quantization formalism. They are represented as matrices or tensors, and act on the quantum states of the particles in a system.
In first quantization, the behavior of a system is described by a single wavefunction that represents the collective behavior of all the particles. In second quantization, the behavior of a system is described by a field of operators that create and annihilate particles, allowing for a more efficient and accurate description of systems with a large number of particles.
Second quantization is used in many areas of physics, including quantum mechanics, condensed matter physics, and particle physics. It allows for a more comprehensive description of systems with a large number of particles and is particularly useful for studying the behavior of many identical particles.
Second quantization has a wide range of applications, including the study of superconductivity, superfluidity, and quantum field theory. It is also used in the development of new materials, such as graphene, and in the understanding of fundamental particles and their interactions.