# Creation and annihilation operator commutation confusion

• I
• 43arcsec
In summary, the commutator equation 3.14 in Quantum Field Theory by Lancaster states that the commutator between creation and annihilation operators is equal to the Dirac delta. This is proven in example 2.1, where the operators act on harmonic operator states. The trouble arises when considering the operators operating on a single particle in state |i>, as the result should be 1, but instead is 0. However, this discrepancy is resolved by taking into account the fact that the operators actually create or annihilate a state with two particles, resulting in a factor of √2.
43arcsec
In Quantum Field Theory by Lancaster, equation 3.14

$$[\hat{a_i},\hat{a_j}^\dagger]=\delta{ij}$$
is introduced as "we define". Yes, example 2.1, where the creation and annihilation operators applied to harmonic operator states, there is a nice simple proof that this is true (although instead of the dirac delta, it's just 1.)

The trouble I am having is that these are operators and they have to operate on something. The identity above says that the commutator must be true no matter what it operates on. Let's say the commutator acts on a single particle in state |i> and for this example, let i=j, so we should get 1. Then we'd have:

$$\hat{a_i}\hat{a_i}^\dagger|i> - \hat{a_i}^\dagger\hat{a_i}|i>$$
After the rightmost operators act we should get:

$$\hat{a_i}|2i> - \hat{a_i}^\dagger|0>$$

Here my |2i> represents a state with 2 particles in state i (there was one there to start, and the creation operator created another). Letting the remaining operators act we should get:

$$|i> - |i> = 0$$

I was expecting 1, not 0.

Thanks for looking.

43arcsec said:
After the rightmost operators act we should get:

Not quite: You are assuming that ##{a}^{\dagger}_i | i > = | 2i >##; but actually ##{a}^{\dagger}_i | i > = \sqrt{2} | 2i >##. (Make sure you understand why this is the case.) Similarly, ##a_i | 2i > = \sqrt{2} | i >##. The two factors of ##\sqrt{2}## applied to the first term should make things work out.

vanhees71
Got it. Thanks Peter!

## What is the concept of creation and annihilation operators?

The creation and annihilation operators are mathematical tools used in quantum mechanics to describe the creation and destruction of particles. They are represented by symbols, and their actions on a quantum state produce a new state with a different number of particles.

## What is the significance of the commutation relation between creation and annihilation operators?

The commutation relation between creation and annihilation operators is crucial in quantum mechanics as it determines the behavior of quantum systems. The commutation relation states that the order in which these operators are applied matters, and their outcome depends on the order. This relation also helps in defining the uncertainty principle.

## Why is there confusion surrounding the commutation relation between creation and annihilation operators?

The confusion surrounding the commutation relation between creation and annihilation operators arises because it is not always clear which operator should be applied first. This confusion is particularly significant when working with multi-particle systems, where different ordering of operators can lead to different results.

## How is the commutation relation between creation and annihilation operators resolved?

The commutation relation between creation and annihilation operators is resolved by using specific rules that define the order of operators. These rules are determined by the specific quantum system being studied and are crucial for correctly predicting the behavior of the system.

## What are some applications of creation and annihilation operators in physics?

Creation and annihilation operators have numerous applications in physics, particularly in quantum mechanics. They are used in calculations of particle interactions, quantum field theory, and the study of quantum systems. They also play a crucial role in the development of quantum computing and quantum information processing.

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