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Blog - quantum superposition

edited March 2015 in Azimuth Blog

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

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!

Comments

  • 1.
    edited March 2015

    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?

    Comment Source: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](https://forum.azimuthproject.org/discussion/1479/potential-blog-post-series-on-quantum-community-detection#latest)?
  • 2.

    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.

    Comment Source: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.
  • 3.

    What you think about this convention?

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

    Comment Source:> What you think about this convention? Sounds good. But the main thing is to use words to dispel any confusion that might arise.
  • 4.
    edited March 2015

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

    Comment Source:Great! I will try to do my changes on Sat.
  • 5.

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

    Comment Source:Please let me know when you're ready. I made a lot of changes myself.
  • 6.
    edited March 2015

    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!

    Comment Source: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!
  • 7.
    edited March 2015

    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.

    Comment Source: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.
  • 8.

    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').

    Comment Source: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').
  • 9.
    edited March 2015

    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.

    Comment Source: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.
  • 10.

    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’.

    Comment Source: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’.
  • 11.
    edited March 2015

    @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.

    Comment Source:@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.
  • 12.
    edited March 2015

    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.

    Comment Source: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.
  • 13.

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

    OK, I changed it to "weighted".

    Tomorrow - sounds cool!

    Comment Source:I saw you have some high-stake offers :). OK, I changed it to "weighted". Tomorrow - sounds cool!
  • 14.
    edited March 2015

    I published the article:

    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!

    Comment Source:I published the article: * [Quantum superposition](https://johncarlosbaez.wordpress.com/2015/03/13/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!
  • 15.

    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).

    Comment Source: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).
  • 16.

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

    Comment Source:I'll fix it, if I haven't yet.
  • 17.

    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).

    Comment Source: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).
  • 18.

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

    Comment Source:On the blog version I deleted that mysterious parenthetical remark and made other last-minute optimizations.
  • 19.

    +1

    Comment Source:+1
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