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LostConjugate
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What does it mean for a creation or annihilation operator to act on the state <x|p>. For example:
[tex] a_p e^{ip \cdot x} [/tex]
[tex] a_p e^{ip \cdot x} [/tex]
This is not a state. It's just a number. In the second formula you have an operator multiplied by a number.<x|p>
haael said:This is not a state. It's just a number. In the second formula you have an operator multiplied by a number.
This is a functional transform, much just like the Fourier transform. You have a bunch of creation/annihilation operators that depend on a parameter p. You transform them into another operator set that depends on x.What does it mean to write a free field scalar as a linear sum of creation and annihilation operators like this?
The Creation/Annihilation operators are mathematical tools used to describe the creation and destruction of particles in quantum mechanics. They are essential in understanding the behavior of quantum systems and calculating physical quantities such as energy and momentum.
The Creation operator acts on the position state vector |x> and transforms it into a state vector of a particle at position x, while the Annihilation operator does the opposite and transforms the state vector into a state with no particle at position x. The operators also act on the momentum state vector |p> in a similar manner.
The commutation relationship between Creation/Ann operators and position/momentum operators is given by [a,a†] = 1, where a and a† are the Annihilation and Creation operators respectively. This relationship is crucial in quantum mechanics and is used to derive important properties of quantum systems.
Yes, Creation/Ann operators can act on any observable quantity in quantum mechanics. For example, they can be used to describe the creation and destruction of photons in the field of quantum optics. The operators can also act on spin states in quantum spin systems.
The number operator, N, is defined as the product of the Creation and Annihilation operators, N = a†a. This operator represents the number of particles in a given state and is an important quantity in quantum mechanics. The square of the number operator gives the probability of finding a certain number of particles in a specific state.