
The Mathematics of Quantum Mechanics [pdf] - sajid
https://uwaterloo.ca/institute-for-quantum-computing/sites/ca.institute-for-quantum-computing/files/uploads/files/mathematics_qm_v21.pdf
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monktastic1
Most intro courses on QM that I've seen cover exclusively problems of
operators with continuous spectra (like position and momentum). For those, the
mathematics of L2 space is critical (and, in my experience, woefully
neglected). Because multiple observables are rarely treated, tensor products
and entanglement are less important.

For intro to QIT, thoroughly understanding the C^2 space and tensor products
is critical. This book appears to be useful for that. Unfortunately, it can be
hard to make the jump to the former way of looking at things, without someone
explicitly pointing out how to think of L2 as a Hilbert space. Maybe this is
just intuitive to some, but it wasn't for me.

And the converse seems to be true, too: you can apparently master the former
without having a clue about the latter. As Scott Aaronson points out:

> Today, in the quantum information age, the fact that all the physicists had
> to learn quantum this way seems increasingly humorous. For example, I've had
> experts in quantum field theory -- people who've spent years calculating
> path integrals of mind-boggling complexity -- ask me to explain the Bell
> inequality to them. That's like Andrew Wiles asking me to explain the
> Pythagorean Theorem.

[https://www.scottaaronson.com/democritus/lec9.html](https://www.scottaaronson.com/democritus/lec9.html)

~~~
kgwgk
This reminds me of Peter Woit, a theoretical physicist who has written a book
on "Quantum Theory, Groups and Representations", asking if QM is a
probabilistic theory.

[https://www.math.columbia.edu/~woit/wordpress/?p=10533](https://www.math.columbia.edu/~woit/wordpress/?p=10533)

By the way, this example illustrates the point I made in another comment: you
can have a long career in physics and even become professor of theoretical
physics without ever caring about the interpretation of QM.

Edit: I can't resist copying the opening of a recent comment in that thread.

\----

John Baez says:

September 18, 2018 at 4:05 pm

Peter Woit wrote:

"The state of the world is described at a fixed time by a state vector, which
evolves unitarily by the Schrodinger equation. No probability here."

And perhaps no physics here, either, unless we say how the state of the world
is described by that vector: that is, how we can use the vector to make
predictions of experimental results.

(A long and insightful comment by Baez follows)

------
audunw
This looks really nice. But can someone recommend something more like "The
Practical Experiments of Quantum Mechanics"?

I've read some articles and seen some video courses on Quantum Mechanics, and
it seems that there's always way more focus on the mathematics and
abstract/simplified physics, and not much description of the actual
experiments. I get that the main challenge of teaching it is understanding the
mathematics, but it's hard to stay focused when - even though I understand the
equations - I'm not sure what the mathematics describes physically. Or when I
do understand it, that I have no idea how one would go about preparing the
state.

I think I've gotten a half decent understanding of the Stern–Gerlach
experiment, but that's about it.

~~~
SiempreViernes
Famously there is no consensus of how to interpret the equations of quantum
mechanics, which naturally restricts introductory discussion to the
mathmatical aspects.

~~~
kgwgk
I'm not sure the lack of consensus is the reason. One can learn QM and use it
for a whole career without ever worrying about these issues.

I'm not sure why do you mean by "introductory discussion" but I think "popular
science" books focus mostly on the controversial parts. At the university
level, at least in my experience, you study a lot of classical physics, early
quantum physics (1900-1925), and mathematical physics before taking a first
look at Quantum Mechanics (Schroedinger/Heisenberg). And you may never be
exposed to the relativistic extensions.

~~~
SiempreViernes
I meant the discussion in the intro courses; Griffiths just plonks down the
Schrodinger equation on page one and you just have to eat it.

In contrast a textbook about thermodynamics will be all steam engines at the
start, or maybe talk about a thermometer. Solid, sensible things and no talk
about Gibbs free energy or any of the deep theory that exists.

You could start a QM book talking about the double slit experiment, but would
have to write a lot of "one interpretation" and "some views say this". Or you
could just brazenly pretend there is no controversy and make every reader that
knows of it very suspicious indeed.

~~~
andrepd
>You could start a QM book talking about the double slit experiment

That is indeed how my favorite QM book starts: Cohen-Tannoudji. See also
Landau for one that starts from experiment and not theory.

~~~
kgwgk
Yes, but it quickly cuts to the meat :-) (I mean C-T/D/L, that’s the book I
was talking about in a sibling comment)

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donpdonp
Fry: Hey, professor, what are you teaching this semester? Prof. Farnsworth:
Same thing I teach every semester, the mathematics of quantum neutrino fields.
I made up the title so that no student would dare take it. Fry: [writing]
Mathematics of wonton burrito meals... I'll be there! Prof. Farnsworth: Please
Fry; I don't know how to teach. I'm a professor.

------
jabl
Semi off-topic, for a more in depth treatment I recommend "Mathematics of
Classical and Quantum Physics" by Byron and Fuller.

It's a graduate level text, so probably somewhat dense if you haven't been
exposed to the material in some way before, but as a mostly refresher I found
it very good.

And, it's a Dover book, so it's very cheap.

~~~
westoncb
I just spent about half an hour reading from the preview on Amazon, and I've
got to say I was super impressed with the style—very clear and focused on the
most interesting things.

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forgotpwd16
Since it isn't said in the title, this is based on the abstract formulation ie
Dirac (material usually covered extensively in QMII courses) and the book is
suitable to high school students interested in a QM primer for use in quantum
computation yet don't want to delve in physics or mathematics. Also it is
mostly self-contained. About 60 pages (13-70) is math background (complex
numbers and linear algebra). The QM part is exactly 30 pages (71-101). The
final pages are some extra stuff (proofs, trigonometry, etc).

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oytis
Looks like "Mathematics of Quantum Computing" to me. All the analysis fun is
missing.

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unnouinceput
" Unfortunately, most high school mathematics around the world do not teach
linear algebra" Yeah, lol no. Most high-schools do actually around the world.
Only in US most don't. Funny, stupid but funny

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lisper
This is a really terrific and accessible introduction to QM math.

If you get to the end, it is worth noting the similarity of the tensor product
and the join operation in a relational database.

~~~
hamad
"the tensor product and the join operation in a relational database." can you
elaborate more. very interesting!

~~~
lisper
> can you elaborate more

Hope, you'll have to figure it out for yourself. It's not hard. If you just
look at the details of how these two operations are defined it should be
obvious.

~~~
gunnihinn
As a PhD in mathematics who converted to development, please, go into the
details of what you mean.

I know tensor products. I know SQL joins. I see superficial similarities
between the two, but no unifying underlying principle.

The world has had quite enough of nonsense thrown around because it's
"obvious". Put your money where your runaway mouth is.

~~~
lisper
Um, can we tone this down a notch please? I didn't intend for this to be a
deep insight, just the observation that both the tensor product and the join
operation involve taking all possible pairs of the components of the input
data. The only formal difference between the two is that the components of the
tensor product are ordered while the components of a join are not (they are a
set).

~~~
kgwgk
The tensor product is not exactly like the Cartesian product because not all
the elements of the tensor product of the Hilbert spaces representing each
qbit can be written as the tensor product of the individual qbits.

In the cross join case, all the elements are pairs of elements from the
original tables. In the Hilbert space of a composite system only the separable
states can be written as the tensor product of states of the subsystems.

"Observation 3.6.5: Interesting two-qubit states. Not every 4-dimensional
vector can be written as a Kronecker product of two 2-dimension vectors, e.g.,
you can have a two-qubit state |Ψ⟩ such that:

|Ψ⟩ = ̸= |ψ⟩|φ⟩, for any one-qubit state |ψ⟩ and |φ⟩

These types of states (called entangled states) are very intriguing and play a
fundamental role in quantum mechanics."

~~~
lisper
Interestingly, I was just about to edit my original comment to say exactly
that: there are tables which are exactly analogous to entangled states in that
they cannot be written as a cartesian product of two other tables.

But I have to stress that I do not intend this to be a deep insight, just an
interesting (IMHO) observation.

~~~
kgwgk
It is interesting that the basis of the tensor product of the spaces can be
constructed as the tensor product of each possible combination of the basis of
the spaces. Among other things, it makes clear that the dimension of Hilbert
space for the composite system is the product of the dimensions of the Hilbert
spaces. But this can distract us from the most interesting part, which is that
the space spanned by these basis vectors is much larger than the simple
cartesian product.

Focusing on superficial similarities is a two-edged sword: anchoring on a
familiar concept can help or make things more difficult. And it's guaranteed
to annoy people! (I'm mostly getting over it, but I still dislike the use of
the word "tensor" to refer to multi-dimensional arrays. Tensor has a meaning
in geometry and tensor algebra is not about doing linear algebra on 2-d slices
of a larger-dimensional object.)

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77pt77
Way too basic.

No infinite dimensional spaces. No discussion of self-adjoint operators

These are essential mathematical components of QM.

------
Escapado
Looks like a nice introduction with lots of concrete examples although while
reading through it I spotted a couple of minor things that are poorly
worded/wrong or other things are omitted which are rather interesting from a
quantum computing perspective.

~~~
whatshisface
Given that it's targeted at kids, the fact that they managed to word it at all
is a feat. The same goes for anything that was omitted.

~~~
Escapado
Not gonna argue with this, I just wanted to point out that it's not without
mistakes. For Students of the 12th grade it's super well done. For those more
concerned with correctness and completeness: For example saying a basis is a
_finite_ set of vectors is not generally correct. And I would like to have
seen that the Pauli matrices generate all single Qubit rotations. And saying
that a bra in itself is simply the conjugate transpose of a state or that
<bra|ket> would simply be a multiplication ommits all the cases in which the
state is not a simple numerical vector in some basis but perhaps a function of
many variables. So read it with caution, but read it nonetheless.

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codethief
This looks like a very good introduction to quantum mechanics, though I wish
the author had also added some physical motivation for _why_ the definitions
are the way they are, by referring to experimental results and such.

~~~
grigjd3
Not an intro to QM. Just reviews some of the math used in it.

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mcguire
Who is this for, exactly? It's introductory material for the "Quantum
Cryptography School for Young Students", but what is that?

One quote:

" _We’ll use special mathematics – complex numbers and linear algebra (vectors
and matrices). Unfortunately, most high school mathematics curricula around
the world do not teach linear algebra. It’s not very complicated. It’s really
just a different and clever way to add and multiply numbers together, but it’s
a very powerful tool._ "

~~~
jlgray
Looks like a summer school program at University of Waterloo:

"The Quantum Cryptography School for Young Students (QCSYS) is a unique,
eight-day enrichment program for students hosted by the Institute for Quantum
Computing (IQC) at the University of Waterloo. QCSYS will run August 10-17,
2018 with students arriving August 9 and departing August 18.

The school offers an interesting blend of lectures, hands-on experiments and
group work focused on quantum cryptography"

[https://uwaterloo.ca/institute-for-quantum-
computing/program...](https://uwaterloo.ca/institute-for-quantum-
computing/programs/quantum-cryptography-school-young-students)

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paulpauper
love the color and layout. really great for readability which you don't find
in many online textbooks

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diebir
Highly recommended:

[https://www.amazon.com/Lectures-Mechanics-Mathematics-
Studen...](https://www.amazon.com/Lectures-Mechanics-Mathematics-Students-
Mathematical/dp/082184699X)

Absolutely beautiful way to present quantum mechanics.

