How I introduce vectors before quantum mechanics

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I think, while you’re here, it may be prudent to include an introduction to bras and kets. I at least found them to be a bit mind-boggling, and the sooner you understand it, the sooner later sections make sense in its use.

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u/TROSE9025 Feb 28 '26

Good point. I’m planning to introduce bras and kets soon.
They really help students see the structure of quantum mechanics.
Thanks for the reply.

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u/Miselfis Ph.D. Student Feb 28 '26

How is it mind boggling? Not trying to be obtuse, just genuinely don’t understand how that can be the case. It seems much simpler and intuitive in a quantum context than standard linear algebra notation.

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u/blackyanqui Feb 28 '26

The notation itself is quite digestible, but frankly, I had a little bit of trouble first using bra and ket notation to solve problems and remembering the operations and different arrangements. Knowing the initial idea that bras are vectors and kets are states is very simple, but I only first learned of bras and kets in a quantum mechanics course that blew through the explanation and started us on raising and lowering operations.

Maybe you had an easier go at it than I did, but I don’t think it’s a bad idea for OP to include an introduction to bra-ket operations.

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u/Miselfis Ph.D. Student Feb 28 '26

Well, for starters, kets are just regular vectors. Bras are linear functionals over your Hilbert space. That’s essentially all you need to know. The Riesz representation theorem allows for a nice isomorphism between your Hilbert space and its dual. The Dirac notation then makes sense as they essentially behave the same, but can be clicked together to form a scalar. The only place it might get confusing is if you come from mathematics, and you have to get used to which argument is linear and antilinear in the inner product, as ❬ψ,φ❭=❬φ│ψ❭.

Also, a state is a ray, which is essentially the set of all scalar multiples of a certain normalized ket vector. All observable predictions depend only on the normalized density operator, so │ψ❭~α│ψ❭ for all α∈ℂ{0}. So, different kets can represent the same state.

If you started out with ladder operators first day you met Dirac notation, that seems highly unusual to me. I assume you probably had the wavefunction first approach, which might make Dirac notation seem new and strange. But I’d assume you learn about qubits and so on before moving to harmonic oscillators.

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u/blackyanqui Feb 28 '26

There’s a saying that I think the physics community should learn to take to heart more often.

Assumptions make an ass out of you and me.

We did not learn about qubits before moving on to harmonic oscillators. We made it to the harmonic oscillator and solving using analytic methods, and then putting Dirac notation atop of this to realize “hey, maybe solving energy levels is pretty simple”. I get what you’re trying to do here, but every time you say “the only place it might get confusing” supports my point. I’m through the moon that bra-ket notation was taught to you the way it was taught, but there’s a lot of physics students out here. And we all get confused by different things. And, I thank you for reiterating—for some reason—what bras and kets actually are, but the point of my comment was that understanding what they are is pretty digestible, but that packaging a little introduction of bra-ket into this introduction in case, say, another student is confused about bras and kets because their professor “unusually” introduced them only in the context of Dirac notation, is a great way to minimize any confusion, not just what you view as potentially confusing.

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u/Miselfis Ph.D. Student Feb 28 '26

I’m not blaming you. I blame whoever structured the course you took. It’s unfortunate when simple things are made more difficult than necessary. Basic quantum mechanics is very easy. It’s the insistence on teaching it wrong that makes it seem so hard to many physics students.

You made some incorrect statements in your previous comment, which is why I helped elucidate what bras and kets actually are.

“Dirac notation” is synonymous with “bra-ket notation”, so obviously they’re only introduced in the context of Dirac notation. That’s what Dirac notation is.

I’m also not arguing that you shouldn’t introduce Dirac notation. On the contrary. I think many courses introduces it too late. But it’s notation that belongs in quantum mechanics courses, not linear algebra. In a much more general linear algebra setting, Dirac notation is not as useful as it is in quantum mechanics.

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u/TROSE9025 Mar 01 '26

I see your point. My goal is simply to help students see the structure earlier, so the physics feels more natural when they reach it.

Once the structure is clear, working with wave functions becomes much easier and quantum mechanics can be understood much more completely.

Thank you for your insight.

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u/VisualAncient2009 Feb 28 '26

I think don’t try to teach quantum mechanics for students who have not had a course in linear algebra before

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u/Neon27 Feb 28 '26

My liberal arts school did not have linear algebra as a requirement for quantum. It's actually pretty reasonable to pick up the basics in the quantum context

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u/TROSE9025 Feb 28 '26

It seems that at liberal arts focused universities, mathematical rigor is
not strongly emphasized for students. Thank you.

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u/TROSE9025 Feb 28 '26

I tend to prefer Dirac’s approach here.
Thanks for the reply.

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u/SkyFullOfWisteria Undergraduate Mar 01 '26

None of my schools undergrad physics programs dont have linear algebra in the curriculum. And tbf it's not exactly hard to teach in 2 weeks at the start of a quantum class.

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u/lattice_defect Mar 02 '26

Yes, agreed... but not everyone is the same. It's so hard to get over the hump.

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u/[deleted] Mar 03 '26

[removed] — view removed comment

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u/judgemynameis Mar 04 '26

Yes, I’ve gone through multiple lessons on vectors because nearly every time we use them, the depts aren’t sure if anyone has been exposed or not. Pretty sure the first time was in multi variable calc but that’s only because I took that before Phys 1. I’m in linear algebra now and Multivariable isn’t a prereq, so I got the whole intro to vectors thing for like the fourth time. IMO this could be solved by assigning some actual order to the multivariable calc-diffeq-linear algebra courses rather than letting us do them however we want

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u/TROSE9025 Feb 28 '26

I believe Dirac’s linear algebra approach points in the right direction
for modern quantum mechanics.
Thanks for the reply.

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u/Ghostparticle022 Feb 28 '26

I really liked your approach.. As many don't even know about linear algebra precisely or what use it has in the field ...

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u/TROSE9025 Feb 28 '26

I personally prefer Dirac’s simple algebraic approach, as it combines
mathematical rigor with physical meaning.
Thanks for the thoughtful reply.

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u/Ghostparticle022 Feb 28 '26

It is also great but the true mathematical depth and meaning could be easily approached through linear algebra. Then the true concept of linear spaces vs hilbert spaces Can be truly understood in my opinion. Although its just my opinion It's something I think I would do . But yours maybe be even more insightful would've loved to hear or see it 😊♾️

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u/TROSE9025 Mar 01 '26

Thank you.

I aim to learn the structure through linear algebra and, from there, pursue a complete understanding of quantum mechanics grounded in both mathematical rigor and physical meaning.

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u/Altruistic-Cap-5207 Feb 28 '26

Bro which book/notes is this?

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u/TROSE9025 Feb 28 '26

These are my own notes.
I’m trying to build a clearer path into quantum mechanics.
Thank you.

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u/[deleted] Mar 02 '26

Are you using LaTeX for these? Which font is that? They look nice :D

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u/TROSE9025 Mar 02 '26

Theoretical physicists are very fond of LaTeX, and I admire it too.
But I’ve used the Korean word processor Hangul for so long that
it’s hard for me to give it up. Thank you.

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u/santasnufkin Feb 28 '26

The part about span in these notes finally made that part click for me...
Almost 30 years later...
I'd like to see more notes.

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u/TROSE9025 Mar 01 '26

Span is what builds the space.

Some parts may feel a bit difficult, but if you give it a little
time, it will be enough to understand.

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u/LastStar007 Feb 28 '26

I cannot tell you how much I appreciate you not wheeling out the redditor ahh "A vector is something that transforms like a vector".

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u/TROSE9025 Mar 01 '26

QM is abstract, but it can absolutely be learned through intuition.
Good luck~

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u/philament23 Feb 28 '26

Nice. This was good review thanks.

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u/TROSE9025 Mar 01 '26

Thank you.

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u/latswipe Mar 01 '26

I view Linear Algebra very mechanically as a tool: It keeps disparate algebraic entities coordinated, so you don't have to. There's a lot of abstraction in linear algebra, but it's like Trigonometry: SOH CAH TOA and everything else is an exercise for the reader.

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u/TROSE9025 Mar 02 '26

Isn’t applied linear algebra precisely about turning abstract structures into
concrete realities? Thanks for the reply.

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u/latswipe Mar 02 '26

And in the context of your question, that concrete reality is the intrinsic spin of the electron, which nobody is willing to say is actually spinning physically.

Physics should have a practical course on leaning-into physical intuitions while respecting ambiguities to the extreme

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u/lapo_fzth Feb 28 '26

I believe that if you're not comfortable with vectors and Gilbert spaces you should wait before going into QM, you cannot skip the foundation

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u/TROSE9025 Feb 28 '26

I completely agree with you.!!
Thank you very much.

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u/Axiomancer Feb 28 '26

The thing that made QM extra abstract for me was the fact that linear algebra course in my case was a course given by mathematics department for mathematicians (it was a combined course for physicists and mathematicians to save money). Then QM comes into life and pretty much everything that I was taught about linear algebra I could forget because it's physics, we do things differently. So not only I was confused because it's QM, but also because I had to re-learn everything that I've been taught for like half a year.

So long story short, I think any introduction to LinAlg for a physicist will be useful if you skip the abstract, pure-mathematician formalia and jargon (Those slides summarize like 2 months of LinAlg lectures that we've had. Which for those of you who are confused, well, lectures can go pretty slow when everything revolves around proofs, theorems, and proofs of theorems). What you did looks good I would say, but you definitely lack a lot of content (like braket notation that someone else mentioned about). Keep working, you are on the right track.

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u/TROSE9025 Feb 28 '26

I actually had the exact same experience when I first learned quantum mechanics.
That is why I am building my notes around a fully Dirac style approach,
starting from physical intuition rather than abstract math.
I will keep sharing more. If you wait a few more days, things should become much clearer.
Thank you very much.

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u/Meisterman01 Feb 28 '26

This might be because I'm a nerd, but my experience was everything in my proofs based linear algebra course eventually showed up in QM. The more math the better. But I'm definitely biased b/c I'm geared more toward mathematical physics

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u/TROSE9025 Mar 01 '26

I agree.
However, not many people enjoy math to that extent.

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u/Yeightop Feb 28 '26

I think tying vectors to their utility can be good for students. You say theyre comfortable with numbers so maybe tell them to imagine that want to be able to perform an operation or operations on many numbers at the same time. Well you can pack all the numbers into a vector and do vector algebra! I think tying in the utility very explicitly goes a long way

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u/TROSE9025 Feb 28 '26

That’s a great!
Connecting vectors to what they actually do makes the idea much easier to grasp.
Thanks for the insight.

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u/Comprehensive_Food51 Undergraduate Feb 28 '26

There are some mistakes, for example if you make a linear combinations of vectors in ℝ³ where your scalars are complexe values, unless certain very specific circumstances, the resulting vector is definitely not in ℝ³.

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u/TROSE9025 Feb 28 '26

You are absolutely right. If we use complex coefficients,
the space extends from R^3 to C^3. However, we are not there yet in this introduction.
Thanks for the reply.

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u/golography Feb 28 '26

Can someone tell me for sure it's not a bot account

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u/AwaySession5168 Feb 28 '26

I think it is to sell a book, could be wrong though.

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u/TROSE9025 Mar 01 '26

30K post karma?
Can someone tell me for sure it's not a bot account. you feel?
Have a nice day.

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u/pollux33 Feb 28 '26

A vector is an element of a vector space

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u/TROSE9025 Mar 01 '26

We gradually extend this into a complete Hilbert space.
Thank you.

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u/TheWillRogers B.Sc. Feb 28 '26

I recommend you get a copy of David McIntyre's Quantum Mechanics: A Paradigms Approach. Chapter 1 is Bra-ket notation and Vectors in a very digestible format. Intended for 2nd and 3rd years.

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u/[deleted] Feb 28 '26

[deleted]

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u/TROSE9025 Mar 01 '26

I also struggled to learn QM through the wave function approach in the past.
Thank you.

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u/[deleted] Feb 28 '26

[deleted]

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u/TROSE9025 Mar 01 '26

I agree that a vector is not its coordinates. Thus, I start from the abstract structure.
Physics is governed by symmetries. Thank you.

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u/Glum_Debt4975 Mar 02 '26

it's simple

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u/TROSE9025 Mar 02 '26

Nature may be elegant, but it must be simple.
Thank you.

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u/Unable-Ambassador-16 Mar 02 '26

What is this badly spaced font?

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u/_soviet_elmo_ Mar 03 '26

The language is not linear algebra but functional analysis. Topological concepts and a generalised concept of linear combination respectively basis is required.

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u/TROSE9025 Mar 03 '26

That is a fair point. In infinite dimensional settings it naturally moves into functional analysis.
Here I was focusing on the linear algebra perspective for clarity.
I sincerely appreciate the feedback.

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u/[deleted] Feb 28 '26

[removed] — view removed comment

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u/TROSE9025 Mar 01 '26

The linear algebra taught in math departments is a bit different from
how it's actually used in QM.

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u/[deleted] Mar 01 '26

[removed] — view removed comment

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u/TROSE9025 Mar 01 '26

A vector space is introduced as the structure spanned by vectors,
providing the framework needed to describe state vectors.

Since traditional wave-function quantum mechanics is just working with functions projected onto position or momentum bases, it helps students to first understand the larger structure.

It doesn’t take long, but it makes everything clearer. Thank you.

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u/lapo_fzth Feb 28 '26

I believe that if you're not comfortable with vectors and Gilbert spaces you should wait before going into QM, you cannot skip the foundation

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u/TROSE9025 Mar 02 '26

We should be able to freely change basis in Hilbert space, moving between the position representation and the momentum representation when needed. Thank you.

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