Second quantization and creation/annih. operators

In summary: Similarly, we can calculate the norm of ##a|\alpha\rangle##, and by demanding that the state kets are normalized, we can obtain the expressions for ##|\alpha-1\rangle## and ##|\alpha+1\rangle##. In summary, the operators ##a## and ##a^\dagger## and the state ket ##|\alpha\rangle## are introduced in the context of second quantization, and it is shown that the state kets ##a|\alpha\rangle## and ##a^\dagger|\alpha\rangle## are eigenstates of the operator ##a^\dagger a## with eigenvalues ##\alpha-1## and ##\alpha+1##, respectively. By demanding that the state
  • #1
Swamp Thing
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I'd like some help with something in this introduction to second quantization ... http://yclept.ucdavis.edu/course/242/Class.html

They start by considering two operators ##a## and ##a^\dagger## such that ##[a,a^\dagger]=1## and they are adjoint to each other. Also introduced is a state ##|\alpha\rangle## that is an eigenvector of the (Hermitian) operator ##a^\dagger a##, so that
##a^\dagger a |\alpha\rangle = \alpha |\alpha\rangle##.

They go on to show that the state ##a|\alpha\rangle## is an eigenstate of ##a^\dagger a## with eignvalue ##\alpha-1##, and ##a^\dagger|\alpha\rangle## is an eigenstate of ##a^\dagger a## with eigenvalue ##\alpha-1##.

Up to this point I managed to follow the plot reasonably well.

But now they say that the above "implies" that

##|\alpha-1\rangle=\frac 1{\sqrt\alpha}\alpha|\alpha\rangle##
and
##|\alpha+1\rangle=\frac 1{\sqrt{\alpha+1}}\alpha|\alpha\rangle##

which I am absulutely unable to understand. How do they get this result? where do the square roots come from? Why is there an asymmetry in the square roots?
 
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  • #2
The square root come from the normalization, in order to have that the number of particle in the state ##\alpha##, so ##N_{\alpha}=a^{\dagger}a=n_{\alpha}##. I think that the asymmetry consist in that: in one case you have ##\sqrt{\alpha+1}## and in the other ##\sqrt{\alpha}## and not ##\sqrt{\alpha-1}## as the vector ##|\alpha-1\rangle##...
 
  • #3
Thanks for the reply.. One reason I am confused is that, up to this point, they are talking about ##a## and ##a^\dagger## as two abstract operators with certain assumed properties -- they have not yet brought in any physics. And they seem to be saying that the square roots follow from (are implied by) the discussion so far. So I'm trying to understand how to derive those terms from what precedes.
 
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  • #4
Swamp Thing said:
##a^\dagger|\alpha\rangle## is an eigenstate of ##a^\dagger a## with eigenvalue ##\alpha-1##
I'm pretty sure it should be ##\alpha+1##, and that
Swamp Thing said:
##|\alpha-1\rangle=\frac 1{\sqrt\alpha}\alpha|\alpha\rangle##
and
##|\alpha+1\rangle=\frac 1{\sqrt{\alpha+1}}\alpha|\alpha\rangle##
should be
##|\alpha-1\rangle=\frac 1{\sqrt\alpha}\hat{a}|\alpha \rangle## and ##|\alpha+1\rangle=\frac 1{\sqrt{\alpha+1}}\hat{a}^{\dagger}|\alpha\rangle##.
This result can be obtained by demanding that the state kets are normalised - for example, we know that ##a^\dagger|\alpha\rangle## is proportional to ##|\alpha+1\rangle##, and we can determine the constant of proportionality by calculating the norm on both sides.
 

What is second quantization?

Second quantization is a mathematical framework used in quantum mechanics to describe the behavior of many-particle systems. It involves the use of creation and annihilation operators to represent the creation and destruction of particles in a system.

What are creation and annihilation operators?

Creation and annihilation operators are mathematical operators used to describe the creation and annihilation of particles in a quantum system. These operators act on the quantum state of a particle and can create or destroy a particle in that state.

Why is second quantization important?

Second quantization allows us to describe the behavior of many-particle systems in quantum mechanics, which is crucial for understanding the behavior of atoms, molecules, and other complex systems. It also provides a powerful tool for calculating the properties of these systems.

How do creation and annihilation operators work?

Creation and annihilation operators act on the quantum state of a particle to create or destroy a particle in that state. The creation operator adds an additional particle to the system, while the annihilation operator removes a particle from the system. These operators follow specific commutation and anti-commutation rules, which dictate the behavior of the system.

What are some applications of second quantization?

Second quantization has many important applications in physics, including the description of electron behavior in solids, the study of superconductivity and superfluidity, and the behavior of quantum gases. It is also used in the development of quantum field theory, which is crucial for understanding particle physics and the behavior of the early universe.

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