Why explain with both vectors and functions

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Discussion Overview

The discussion revolves around the use of different mathematical representations in quantum mechanics (QM), specifically the transition between bra-ket notation (complex vectors) and wave functions (psi). Participants explore the reasons for this shift and express confusion regarding the switching between these representations during explanations.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions why QM explanations switch between bra-ket notation and functions, suggesting that both are fundamentally the same but expressing confusion about the pedagogical approach.
  • Another participant suggests that understanding the level of mathematics is crucial for providing an appropriate explanation, hinting that functions in a Hilbert space can be viewed as vectors.
  • A later reply proposes that the introduction to QM often follows a formalized structure that includes specific systems, which may necessitate the use of both representations to prepare students for more complex concepts.

Areas of Agreement / Disagreement

Participants express confusion about the switching of mathematical representations, and while some insights are shared, there is no consensus on the best approach or the reasons behind the pedagogical choices.

Contextual Notes

Participants note that the introduction of quantum mechanics often involves specific examples that may influence the choice of mathematical representation, but the discussion does not resolve the underlying confusion about the pedagogical strategy.

mraptor
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Why when explaining/introducing QM things get explained with the <bra|ket> complex vectors... then somewhere in between they start using functions (psi)..
It gets confusing when epxlanation start switching between them back and forth...

My understanding is they are the same thing...
Is there some purpose of this exercise...
Similar thing with the Operators... when the explanation goes around the the braket vectors.. they use matrices... then the moment find operators to be differential operators we switch to functions.

I understand it is logical to do this, but why we don't they stick with one of the mathematical abstraction and go with it the whole time..

sorry if it is a stupid question.. I'm neither mathematician nor physicist.
May be that is why they are confusing to me ;)
 
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mraptor said:
I'm neither mathematician nor physicist.
Perhaps you should say what level of maths you do understand, else it's hard to know at what level to pitch an answer. I could say that functions in a Hilbert space are vectors in that space, but I suspect this wouldn't be very enlightening...
 
mraptor said:
Why when explaining/introducing QM things get explained with the <bra|ket> complex vectors... then somewhere in between they start using functions (psi)..
It gets confusing when epxlanation start switching between them back and forth...

My understanding is they are the same thing...
Is there some purpose of this exercise...
Similar thing with the Operators... when the explanation goes around the the braket vectors.. they use matrices... then the moment find operators to be differential operators we switch to functions.

I understand it is logical to do this, but why we don't they stick with one of the mathematical abstraction and go with it the whole time..

sorry if it is a stupid question.. I'm neither mathematician nor physicist.
May be that is why they are confusing to me ;)

Because a first introduction to quantum mechanics has been somewhat formalized as the introduction to a few specific systems: The particle in the box, the harmonic potential and propagation through a potential barrier (followed by the hydrogen atom in a later course). These are sort of three examples that are considered the "take-away" from a first course in QM. That puts teaching in a place where they want to introduce the more robust and generalized bra-ket notation which you'll need for continued learning in quantum and the fact that these problems specifically are best worked out with integrals and calculus. They want to show you a few classic pictures to give you a sense of the key features of quantum while prepping you for later generalizations.
 
I see, thank you
 

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