Why is there a * in the 2nd integral for self-adjoint ODEs?

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Discussion Overview

The discussion revolves around the notation and properties of inner products in the context of self-adjoint ordinary differential equations (ODEs), particularly focusing on the use of the conjugate symbol (*) in complex inner product spaces and the conventions of bra-ket notation in quantum mechanics.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the necessity of the conjugate symbol (*) in the inner product notation, particularly in relation to their textbook's presentation.
  • Another participant explains that the conjugate is essential in complex inner product spaces to maintain certain properties, suggesting that the absence of the conjugate in the second integral may indicate a real inner product space.
  • There is a discussion about whether it is always the bra that is conjugated, with some participants noting that physicists consistently conjugate the bra while mathematicians may vary in their approach.
  • One participant mentions the concept of the Hilbert adjoint, clarifying that the conjugate transpose of a bra is a ket and vice versa, emphasizing the independence of which vector is conjugated in an inner product.
  • Another participant expresses a desire to adhere to the convention of conjugating the bra and seeks clarification on the implications of this preference in terms of mathematical understanding.
  • The intuitive nature of Dirac notation is discussed, with one participant highlighting how it facilitates understanding of inner products and operations involving bras and kets.
  • There is an inquiry about whether the preference for conjugating the bra implies any deeper mathematical significance, to which a participant responds that it does not, but rather simplifies computations.
  • A participant reflects on their learning journey, expressing a commitment to improving their mathematical understanding and addressing gaps in their knowledge.

Areas of Agreement / Disagreement

Participants express varying views on the conventions of conjugation in inner products and the implications of bra-ket notation. There is no consensus on whether the preference for conjugating the bra implies deeper mathematical meaning, indicating ongoing debate and exploration of these concepts.

Contextual Notes

The discussion touches on the differences between complex and real inner product spaces, as well as the varying conventions in mathematical and physical contexts, which may lead to confusion or differing interpretations.

ognik
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Hi, my textbook claims $ <u|\mathcal{L}v> =\int_{a}^{b}u^*\mathcal{L}v \,dx = \int_{a}^{b} u(p_0u''+p_1u'+p2u) \,dx$, u,v matrices or functions

My only query is why $u^*$, and where did the * get to in the 2nd integral? I am used to $ <f|g>=\int_{a}^{b}f(x)g(x) \,dx $ ...
 
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The $^*$ is necessary in complex inner product spaces, such as you encounter in quantum mechanics. It's necessary to preserve some of the properties we expect inner products to have. As for it not appearing in the second integral, I suspect you're dealing with a real inner product space there, in which case the conjugate is unnecessary.
 
thanks ... and is it always the Bra part that is conjugated? And it's nothing to do with a conjugated Ket vector becoming a Bra?
 
ognik said:
thanks ... and is it always the Bra part that is conjugated? And it's nothing to do with a conjugated Ket vector becoming a Bra?

Here you have to be careful. Physicists are quite consistent here - the bra is conjugated and the ket is not. Mathematicians are inconsistent; some authors are linear in the first term, others in the second. I think the physics notation is superior because of how it suggests what's actually going on.

As for a conjugated ket becoming a bra, that's called taking the Hilbert adjoint. Technically, the conjugate transpose of a bra is a ket, and vice versa. This process takes a vector from one inner product space, and sends it into the dual space. This process is independent of which vector - the bra or the ket - you conjugate in an inner product.
 
Useful info thanks - I'm trying to become a physicist, so I'll stick with conjugating the Bra. I had seen that they were conjugate transposes of each other which was a tad confusing, so now I know they CAN, but will by default conjugate the Bra with inner products - hopefully I haven't misunderstood anything?

How does using the Bra suggest what's going on please? I thought it had something to do with needing one vector to be a col, the other a row - which of course doesn't work with other than 1 x n matrices...
 
Well, if you stick a bra $\langle y|$ together with a ket $|x\rangle$ you get a bra-ket, or bracket: $\langle y|x\rangle$, which is the inner product. Or if you have an expression like $|x\rangle\langle y|$, and you "multiply" it with a ket $|z\rangle$, you get $|x\rangle\langle y|z\rangle$. So I like the Dirac notation - I think it's intuitive, and helpful.
 
Just confirming, when you said it suggests what's going on, were you referring to the inner product? My query was more about physicists preferring the bra being conjugated instead of the ket, does conjugating the bra imply something deeper?
 
ognik said:
Just confirming, when you said it suggests what's going on, were you referring to the inner product?

Yep!

My query was more about physicists preferring the bra being conjugated instead of the ket, does conjugating the bra imply something deeper?

It does not imply anything mathematically deeper. It makes computations with Dirac notation a bit more straight-forward, that's all.
 
Gotcha, sorry if I sometimes appear pedantic, I have discovered too many small (but often significant) gaps and inaccuracies in my maths and am determined to increase my overall 'mathematical maturity' as well as do the course :-)
 

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