# Second Quantization: Creation & Annihilation Operators

• Niles
In summary, the creation operators and annihilation operators in second quantization tell us which matrix element we are talking about. This is analogous to the ket and bra operators in first quantization, because they tell us which elements in the product are on the left and right sides, respectively.
Niles
Hi all

I am reading about second quantization. The kinetic energy operator T we write as

$$\hat T = \sum\limits_{i,j} {\left\langle i \right|T\left| j \right\rangle } \,c_i^\dag c_j^{}.$$

Now, the creation and annihilation operators really seem to be analogous (in some sense) to the ket and the bra in first quantization, since they tell us which matrix element we are talking about.

What is the reason for this? I understand that we have the new states in Fock space (the occupation number states), but my book never illuminates why the creation and annihilation operators designate the matrix elements just like the outer product in first quantization does.

Hi Niles,

The creation operator $$c_i^+$$ adds one particle to the system in the single particle state labeled by i. If $$|i\rangle$$ denotes the state where the system has one particle in single particle state i, then you know that $$|i\rangle = c_i^+ |\text{vac} \rangle$$. This means you can write $$c^+_i = |i \rangle \langle \text{vac} | + \text{...}$$ where ... consists of states with more than one particle. Thus in the single particle Hilbert space the creation operators and annihilation operators are literally the outer products you are more familiar with. For example, $$c_i^+ c_j = |i \rangle \langle \text{vac} | \text{vac} \rangle \langle j | + \text{...} = | i \rangle \langle j | + \text{...}$$.

Does this help at all?

Hi Physics_Monkey

Yes, that is a very good explanation. Although I do not quite get what you mean by: "where ... consists of states with more than one particle.". We have $|i \rangle \langle \text{vac} |$, which is an operator. To this operator we add multiple-particle states (i.e. vectors) - is that allowed?

Last edited:
What I mean is that because because $$c_i^+$$ adds one particle to any state, the expansion of $$c_i^+$$ in terms of outer products can't stop with $$| i \rangle \langle \text{vac} |$$. There must be other terms like $$|\text{2 particles} \rangle \langle \text{1 particle} |$$. However, each of these terms contains at least one bra or ket with more than one particle. This is what I mean by ... containing states with more than one particle.

## What is second quantization and why is it important in quantum mechanics?

Second quantization is a mathematical framework used to describe the behavior of many-particle quantum systems, such as atoms, molecules, and solids. It allows us to treat particles as indistinguishable objects and take into account their quantum mechanical properties, such as spin, in a more efficient way. This is essential for understanding the behavior of complex quantum systems and predicting their properties.

## What are creation and annihilation operators and how do they relate to second quantization?

Creation and annihilation operators are mathematical operators that are used to create or destroy particles in a quantum system. In second quantization, these operators act on a state vector to create or annihilate a particle in a specific quantum state. They allow us to describe the number of particles in a system and how they behave dynamically.

## How do creation and annihilation operators differ from each other?

Creation and annihilation operators are essentially each other's adjoints, meaning they are related by a mathematical operation called Hermitian conjugation. The main difference between them is that creation operators increase the number of particles in a system, while annihilation operators decrease the number of particles. This reflects the physical process of particle creation and annihilation in quantum systems.

## What is the commutation relationship between creation and annihilation operators?

The commutation relationship between creation and annihilation operators is a fundamental property of quantum systems. It states that the commutator of these operators is equal to the identity operator, meaning they do not commute with each other. This property is crucial for calculating the behavior of quantum systems and understanding the principles of quantum mechanics.

## How are creation and annihilation operators used in the calculation of physical observables?

Creation and annihilation operators are used in the calculation of physical observables by acting on the state vector of a quantum system. The expectation value of an observable can be calculated by taking the inner product of the state vector with the operator representing the observable. This allows us to calculate the probability of measuring a particular value for the observable in a given quantum state.

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