Blog - quantum superposition

John Baez

Hi! I'm going through this post now, adding explanations of terms that are undefined, and fixing grammar and formatting:

• Blog - quantum superposition

A question: is it really wise to use $\psi_+$ as the name for the state whose energy is $E_0 - \Delta$, and $\psi_-$ as the name for the state whose energy is $E_0 + \Delta$? I guess I know why Piotr is doing this, but it could be a bit confusing. Maybe I'll add an explanation.

By the way, the term 'superposition' is never defined. I'll also add a definition of that, since that's the topic of the post!

Piotr Migdal

Thanks John!

When it comes to the sign convention, each choice has its drawbacks (there need to be a minus sign somewhere). I haven't figured out which choice is the least confusing (it may be also a matter of preference).

As I think right now, the following convention should be the clearest:

• positive terms $\langle 0 | H |1\rangle$,
• the ground state with minus (i.e. $\propto |0\rangle - |1\rangle$).

Pros:

• same sign for the superposition and energy,
• positive off-diagonal terms in $H$,
• arguably, more natural for discrete systems (vide: singlet vs triplet).

Cons:

• different convention from the continuous variable variant (but should in matter in this post?).

What you think about this convention?

Superposition - I will add it today. (As always, a problem with being to accustomed to a word/concept, which is non-trivial for the newcomers.)

BTW: Should I use this thread or Potential blog post series on quantum community detection?

John Baez

Hi, Piotr. If you're working on this particular blog post, please put comments in this thread. The idea is that each blog post has a thread on the Forum with the same title, but with the word "Blog - " in front.

I'll see if you defined "superposition". When I write posts, or edit other people's posts, I go through and look at each word and decide whether or not we're assuming the readers will already know that word. If not, I define it or figure out a way to avoid it. Whenever it's possible to avoid a technical term, it's good to do so.

In a post called "Quantum superpositions", where we are explaining quantum superpositions, we clearly need to define that concept - at least in some rough way.

You also used the word "delocalization"... and I either defined it or avoided it, or promised myself that I would do so.

I'll try to finish this up soon, like tomorrow.

John Baez

What you think about this convention?

Sounds good. But the main thing is to use words to dispel any confusion that might arise.

Piotr Migdal

Great! I will try to do my changes on Sat.

John Baez

Please let me know when you're ready. I made a lot of changes myself.

Piotr Migdal

Thanks - I saw changes and I like them. You also added 'superposition' as a synonym of 'linear combination' - is it enough or do you think that more is needed?

I made a few minor things, including changing the sign convention for $|\psi_\pm \rangle$.

So, I am ready!

John Baez

Okay, I'll try to publish this soon.

Regarding "superposition", my main change was to say:

Note that

$$ |\psi \rangle = \alpha \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \beta \begin{bmatrix} 0 \\ 1 \end{bmatrix}. $$ So, we say the electron is in a 'linear combination' or 'superposition' of the two states

$$ |1\rangle = \begin{bmatrix} 1 \ 0 \end{bmatrix}, $$ (where it's near the first proton) and the state $$ |2\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}. $$ (where it’s near the second proton).

${}$

Before, it had said

The state of the electron can be described as a complex, two-dimensional vector:

$$ |\psi\rangle = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}. $$ That is, the electron is in a superposition of being in the state $$ |1\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, $$ (which is around the first proton) and the state $$ |2\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}. $$ (which is around the second proton).

In that old version the reader had to

1) make up the formula $|\psi \rangle = \alpha \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \beta \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ for themselves and

2) guess that a state $|\psi\rangle$ obeying this kind of formula is called a superposition.

That would be quite a mental feat for anyone who didn't already know what a superposition was.

In the new version the reader merely has to guess that a thing like $\alpha \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \beta \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ is called a 'superposition'. I should probably add more explanation to make this guessing process easier. If they already know what a linear combination is, it should be fairly easy. Otherwise, it may take work.

Piotr Migdal

I saw the diff.

Just now I got some feedback from my girlfriend. It seems that there are still issues with things being to implicit for anyone by a physicist. So, if you don't mind, would like to clarify a few things (mostly around complex numbers and introducing 'coherence').

Piotr Migdal

I've just made my changes.

In particular I:

• added some comments on complex numbers and bra-ket,
• made it explicit that state can be measured,
• explained/removed 'coherence' depending on its place in the text,
• made it more explicit that $p_1$ in the density matrix sum is NOT $|\alpha|^2$.

Unless I introduced more confusion (or grams, typos, ...) than cleared, this post should be ready to go.

Graham Jones

I'm no physicist, but this sounds wrong to me:

In quantum mechanics each possible configuration is described by a complex number called ‘probability amplitude’.

I'd say

In quantum mechanics each possible configuration is weighted by a complex number called an ‘amplitude’.

Piotr Migdal

@GrahamJones 'Probability amplitude' - I wanted to keep it (as it is the full name). With 'weighted' I am undecided - on one hand it is less vague than 'described', whereas on the other it implies some statistical mixing/weighting, with probabilities.

John Baez

I think 'weighted' is more accurate than 'described', since the complex number doesn't 'describe' the configuration.

I got distracted today, so I'll do a bit of work on it now and try to publish it tomorrow.

Piotr Migdal

I saw you have some high-stake offers :).

OK, I changed it to "weighted".

Tomorrow - sounds cool!

John Baez

I published the article:

• Quantum superposition, Azimuth Blog, 13 March 2013.

As usual I made a lot of small changes while reformatting the TeX for the blog. If there's something you don't like, let me know.

Congratulations and thanks!

Piotr Migdal

Thank you a lot! (Both for editorial help and a spot!)

When it comes to reformatting - in the line defining superposition there is $|$ missing in $|\psi\rangle$ (as I now see, it stems from a typo in the draft).

John Baez

I'll fix it, if I haven't yet.

JimStuttard

I found this an excellent easy to understand explanation :) and I could actually do the exercies I;ve tried.

A typo:

But if we get infinite velocities we see that there is something wrong (if one ins).

John Baez

On the blog version I deleted that mysterious parenthetical remark and made other last-minute optimizations.

JimStuttard

+1