Creation and annihilation operators acting on energy eigenstates

TL;DR

Creation and annihilation operators change states by either increasing or decreasing the energy level, while maintaining the same state normalization.

Transcript

PROFESSOR: Important thing to do is to just try to understand one more thing. The creation and annihilation operators-- what do they do to those states? You see, a creation operator will I add one more a dagger, so somehow must change phi n into phi n plus 1. A destruction operator with an a will kill one of these factors, and therefore it will giv... Read More

Key Insights

• 🎚️ Creation and annihilation operators change states by modifying the energy levels.
• 🆘 Commutation relations help determine the precise relationships between the operators and the states.
• 👨‍🏭 The factor of square root of n in the lowering operator relation ensures state normalization.

Questions & Answers

Q: What do creation and annihilation operators do to quantum states?

Creation operators add one more factor to the state, increasing the energy level by 1, while annihilation operators remove a factor, reducing the energy level by 1.

Q: How can the relations between states be determined using commutators?

By using commutation relations, such as the commutator of a and a dagger to the n, the relationships between creation and annihilation operators and the states can be determined.

Q: Why is there a factor of square root of n in the relation of a lowering operator?

The factor of square root of n is present in the relation of a lowering operator because it ensures the overall normalization of the states.

Q: Can momentum be determined for the states in the harmonic oscillator?

No, momentum cannot be determined for the stationary states in the harmonic oscillator since they are eigenstates and do not have momentum.

Summary & Key Takeaways

• Creation and annihilation operators change the state by adding or removing a factor and can be represented as a combination of a and a dagger operators.

• The commutator of the operators and its factorial representation help determine the precise relations between states.

• Creation operators act as lowering operators, reducing the energy of the state by one, while annihilation operators act as raising operators, increasing the energy level.