Relation for Inner Product with States from a Complete Set

In summary, the conversation discusses a relation found in a book about qm 3-body scattering theory, where it is shown that the inner product is equal to zero. This leads to the conclusion that another term is also equal to zero because the p-states form a complete set. The conversation then delves into how a ket can be expressed with respect to the complete set of states and what the only possibility for the ket is if the inner product is zero for every state in the set.
  • #1
tommy01
40
0
Hi.

I've found the following relation (in a book about the qm 3-body scattering theory):

[tex]<\Omega^{\pm}^{\dagger} \Psi_n|p>= ... = 0[/tex]

where [tex]|p>[/tex] is a momentum eigenstate.
So it is shown, that the inner Product is zero. Then they conclude that [tex]\Omega^{\pm}^{\dagger}|\Psi_n> = 0[/tex] because the p-states form a complete set.

How can this formally be shown?

thanky you.
 
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  • #2
First, something a little more general.

How is a ket [itex]\left| \psi \right>[/itex] expressed with respect to the complete set of states [itex]\left| p \right>[/itex]?
 
  • #3
This isn't mentioned in the book. But i assume [tex] |\Psi>=\int |\mathbf{p}><\mathbf{p}|\Psi> d\mathbf{p}[/tex] as usual. Or what you mean?
thanks for the quick reply.
 
  • #4
tommy01 said:
This isn't mentioned in the book. But i assume [tex] |\Psi>=\int |\mathbf{p}><\mathbf{p}|\Psi> d\mathbf{p}[/tex] as usual. Or what you mean?
thanks for the quick reply.

Yes.

Now, what is the only possibility for [itex]|\Psi>[/itex] if [itex]<\mathbf{p}|\Psi>[/itex] is zero for every [itex]\mathbf{p}[/itex]?
 
  • #5
thanks a lot.
 

1. What is the inner product in relation to states from a complete set?

The inner product is a mathematical operation that takes two vectors and returns a scalar value. In the context of states from a complete set, the inner product is used to calculate the probability amplitude between two quantum states.

2. How is the inner product related to quantum mechanics?

In quantum mechanics, the inner product is used to calculate the probability amplitudes between quantum states. It is a fundamental concept in understanding the behavior of quantum systems and plays a crucial role in the formulation of quantum mechanics.

3. What is a complete set of states in quantum mechanics?

A complete set of states in quantum mechanics is a set of states that can be used to describe any other state in the system. These states are usually orthogonal, meaning that their inner product is equal to zero, and they form a basis for the system's vector space.

4. How is the inner product related to the concept of superposition in quantum mechanics?

In quantum mechanics, superposition refers to the ability of quantum states to exist in multiple states at the same time. The inner product is used to calculate the probability amplitudes between these superposed states, providing a mathematical framework for understanding superposition.

5. Can the inner product be used to measure the similarity between two quantum states?

No, the inner product should not be interpreted as a measure of similarity between two quantum states. It is a mathematical operation that calculates the probability amplitude between two states, but it does not provide information on their overall similarity or differences.

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