Annihilation and Creation operator problem

In summary, the conversation discussed a problem involving the matrix element of x^2 and how to obtain it using the annihilation and creation form for x. The individual was unsure of where to start, but was reminded of the formula x = \sqrt{\hbar / 2mw}(a^\dagger + a) and was instructed to square it and apply it to the states. The conversation also touched on the effect of the creation and annihilation operators on the states and the importance of orthonormal basis states. Finally, it was clarified that 1 and 2 correspond to different states, not just numbers. The conclusion was reached that the answer to the problem should be zero.
  • #1
Trogdor27
4
0
I've looked through all my course notes, but I just don't even know where to start with this problem.

The problem:

[tex] \langle 1 \mid x^2 \mid 2 \rangle [/tex]

Use the annihilation and creation form for x to obtain the above matrix element.

What I do know:

I know that [tex] x = \sqrt{\hbar / 2mw}(a^\dagger + a)[/tex]But where do I go from here? Any help or pointers are much appreciated!
 
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  • #2
So...square that x...and then apply it to your states. What is the effect of the creation operator on the state 2? The effect of the annihilation operator? Lastly, consider that your basis states should be orthonormal.
 
  • #3
Ah - I didn't realize that 1 & 2 correspond to different states, I thought they were just numbers. It all makes sense now, and if I'm right the answer should be zero.

Thanks!
 

Related to Annihilation and Creation operator problem

1. What is the "Annihilation and Creation operator problem" in physics?

The "Annihilation and Creation operator problem" is a theoretical issue in quantum mechanics that arises when attempting to apply the principles of quantum field theory to systems with infinite degrees of freedom. It refers to the mathematical inconsistency that occurs when trying to use the annihilation and creation operators, which are used to describe the creation and destruction of particles, in systems with infinite degrees of freedom.

2. Why is the "Annihilation and Creation operator problem" important?

The "Annihilation and Creation operator problem" is important because it highlights the limitations of our current understanding of quantum mechanics. It also poses a challenge for physicists trying to unify quantum mechanics and general relativity, as the problem becomes even more pronounced in systems with gravity.

3. How do physicists attempt to solve the "Annihilation and Creation operator problem"?

Physicists have proposed various approaches to solve the "Annihilation and Creation operator problem," including regularizing the operators, introducing a cutoff, and using different mathematical techniques such as path integrals. However, none of these solutions have been universally accepted, and the problem remains an active area of research.

4. Are there any real-world implications of the "Annihilation and Creation operator problem"?

While the "Annihilation and Creation operator problem" is primarily a theoretical issue, it does have potential implications for our understanding of the universe. If the problem remains unsolved, it could limit our ability to make accurate predictions about physical phenomena, particularly in systems with infinite degrees of freedom such as black holes.

5. What are some current developments in the study of the "Annihilation and Creation operator problem"?

Recent developments in the study of the "Annihilation and Creation operator problem" include attempts to apply machine learning techniques to find solutions, as well as exploring the problem in the context of string theory and other proposed theories of quantum gravity. Additionally, there is ongoing research into the potential implications of the problem for our understanding of the fundamental laws of physics.

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