What is the inner product in state (ket/Hilbert) space.

In summary, in Quantum mechanics, books typically introduce a vector space known as the ket-space and use the Riesz representation theorem to associate each ket with an element in the linear dual space. The inner product of two kets, such as |x\rangle and |y\rangle, is written as \langle x | y\rangle and there appears to be a flaw in the logic, as the Riesz rep theorem requires knowledge of the inner product on the state space. The inner product in QM is often defined as a Hilbert space, but concrete realizations can also be useful. A state space is described as H/C*, where H is a Hilbert space and C* is the complex numbers without 0. It is
  • #1
jdstokes
523
1
In Quantum mechanics books they usually first introduce a vector space called the ket-space and then associate using (Riesz representation theorem I believe) to each ket a corresponding element of the linear dual space.

Then they write the inner product of [itex]|x\rangle[/itex] and [itex]|y\rangle[/itex] (say) by calling on the dual to [itex]|x\rangle[/itex]:

[itex]\langle x | y\rangle[/itex].

There appears to be a flaw in the logic here. To employ the Riesz rep theorem we must already have knowledge of the inner product on the state space.

How is this inner product explicitly defined in QM?? I can't see it written anywhere in Sakurai.
 
Physics news on Phys.org
  • #2
jdstokes said:
In Quantum mechanics books they usually first introduce a vector space called the ket-space and then associate using (Riesz representation theorem I believe) to each ket a corresponding element of the linear dual space.

Then they write the inner product of [itex]|x\rangle[/itex] and [itex]|y\rangle[/itex] (say) by calling on the dual to [itex]|x\rangle[/itex]:

[itex]\langle x | y\rangle[/itex].

There appears to be a flaw in the logic here. To employ the Riesz rep theorem we must already have knowledge of the inner product on the state space.

How is this inner product explicitly defined in QM?? I can't see it written anywhere in Sakurai.

Often it's enough to say state space is a Hilbert space, and to use formally the properties of Hilbert spaces, inner products, operators, etc. Sometime concrete realizations are useful, like, for example, the completion of the inner product space of square-integrable functions.
 
  • #3
jdstokes said:
In Quantum mechanics books they usually first introduce a vector space called the ket-space and then associate using (Riesz representation theorem I believe) to each ket a corresponding element of the linear dual space.

Then they write the inner product of [itex]|x\rangle[/itex] and [itex]|y\rangle[/itex] (say) by calling on the dual to [itex]|x\rangle[/itex]:

[itex]\langle x | y\rangle[/itex].

There appears to be a flaw in the logic here. To employ the Riesz rep theorem we must already have knowledge of the inner product on the state space.

How is this inner product explicitly defined in QM?? I can't see it written anywhere in Sakurai.

more formally a system is described by the state space which is: H/C*.

H= Hilbert space
C*= C-(0)

IN other word is a projective space... sometimes called tha rays space.
a state vector is always normalized and what is important in QM is the relative phase between state vectors.

regards
marco
 

Related to What is the inner product in state (ket/Hilbert) space.

What is the inner product in state (ket/Hilbert) space?

The inner product in state space, also known as the scalar product, is a mathematical operation that takes two vectors and returns a scalar value. In quantum mechanics, state space is represented by either kets or Hilbert space, and the inner product is used to calculate the probability of measuring one state when the system is in another state.

How is the inner product calculated in state space?

The inner product is calculated by multiplying the complex conjugate of one vector by another vector. In state space, the vectors are represented by kets or Hilbert space, and the inner product is denoted by \langle \psi \mid \phi \rangle, where \psi and \phi are the two vectors.

What is the significance of the inner product in quantum mechanics?

In quantum mechanics, the inner product is used to calculate the probability of observing a particular state from a given set of states. It is also used to calculate the expectation value of an observable, which is the average value of the measurement outcome for a given system.

How does the inner product relate to the concept of orthogonality in state space?

If the inner product of two vectors in state space is zero, then the two vectors are said to be orthogonal. This means that the two vectors are perpendicular to each other, and have no overlap. In quantum mechanics, orthogonal states have a probability of zero of transitioning into one another.

Are there any other applications of the inner product in physics?

Yes, the inner product is a fundamental concept in linear algebra, and is used in various branches of physics, such as electromagnetism, quantum field theory, and general relativity. It is also used in signal processing and image analysis to measure the similarity between two signals or images.

Similar threads

  • Quantum Physics
Replies
7
Views
950
  • Quantum Physics
Replies
8
Views
2K
  • Quantum Physics
2
Replies
61
Views
2K
Replies
11
Views
1K
Replies
8
Views
1K
  • Quantum Physics
Replies
2
Views
1K
  • Calculus
Replies
4
Views
966
  • Quantum Physics
Replies
2
Views
1K
Replies
4
Views
1K
Replies
21
Views
2K
Back
Top