Anyone who has studied quantum mechanics know the following relation: $ 2 \otimes 2 = 3 \oplus 1 $  But how did a man woke up and said "Hell yeah, I'll use tensor product of two spin $1/2$ to simulate the interaction of two particles with spin $1/2$" ? Why didn't he start with the direct sum ?  (And then, group theory made the magic leading to the relation above)  In fact, i'm wondering this because i don't fully understand why we use the tensor product to unite the two Hilbert's space.
My text introduces multi-quibt quantum states with the example of a state that can be &quot;factored&quot; into two (non-entangled) substates. It then goes on to suggest that it should be obvious1 that the joint state of two (non-entangled) substates should be the tensor product of the substates: that is, for example, that given a first qubit $$\left|a\right\rangle = \alpha_1\left|0\right\rangle+\alpha_2\left|1\right\rangle$$ and a second qubit $$\left|b\right\rangle = \beta_1\left|0\right\rangle+\beta_2\left|1\right\rangle$$ any non-entangled joint two-qubit state of $\left|a\right\rangle$ and $\left|b\right\rangle$ will be $$\left|a\right\rangle\otimes\left|b\right\rangle = \alpha_1 \beta_1\left|00\right\rangle+\alpha_1\beta_2\left|01\right\rangle+\alpha_2\beta_1\left|10\right\rangle+\alpha_2\beta_2\left|11\right\rangle$$ but it isn't clear to me why this should be the case. It seems to me there is some implicit understanding or interpretation of the coefficients $\alpha_i$ and $\beta_i$ that is used to arrive at this conclusion. It's clear enough why this should be true in a classical case, where the coefficients represent (where normalized, relative) abundance, so that the result follows from simple combinatorics. But what accounts for the assertion that this is true for a quantum system, in which (at least in my text, up to this point) coefficients only have this correspondence by analogy (and a perplexing analogy at that, since they can be complex and negative)? Should it be obvious that independent quantum states are composed by taking the tensor product, or is some additional observation or definition (e.g. of the nature of the coefficients of quantum states) required?  1:  (bottom of p. 18) &quot;so the state of the two qubits must be the product&quot; (emphasis added).
GPG Error while downloading software: "Clearsigned file isn't valid, got 'NODATA'" Correcting for multiple kruskal wallis tests I am running a kruskal wallis test a number of times (37) and wanted to know if I have to adjust the p-values of the kruskal tests before I look any further into the post hocs? While I was investigating a problem I had with lexical closures in Javascript code, I came along this problem in Python:  flist = []  for i in xrange(3):     def func(x): return x * i     flist.append(func)  for f in flist:     print f(2)   Note that this example mindfully avoids lambda. It prints "4 4 4", which is surprising. I'd expect "0 2 4".   This equivalent Perl code does it right:  my @flist = ();  foreach my $i (0 .. 2) {     push(@flist, sub {$i * $_[0]}); }  foreach my $f (@flist) {     print $f-&gt;(2), "\n"; }   "0 2 4" is printed.  Can you please explain the difference ?    Update:   The problem is not with i being global. This displays the same behavior:  flist = []  def outer():     for i in xrange(3):         def inner(x): return x * i         flist.append(inner)  outer() #~ print i   # commented because it causes an error  for f in flist:     print f(2)   As the commented line shows, i is unknown at that point. Still, it prints "4 4 4". Permutations with repeating digits My question is this : how many distinct two digit numbers can be produced from numbers $4, 3, 3, 1$? When applying the formula $$\frac{4!}{(4-2)!2!}$$ you come up with $6$, yet when doing the problem manually, I come up with $7$ numbers, namely $41, 43, 14, 13, 33, 31, 34$. Can anybody explain this discrepancy?  Thanks. In programming terminology, what do you call the difference between these instructions: In Windows 7, How to query times, when the computer was locked? Is there a way to get a list of the following times:  whenever windows is locked (using windows+l combo)  whenever windows is unlocked in the past week?   I've tried poking in the event log, to no avail. Many thanks! Received W-2 but was never paid I just now received a W-2 from a company that I worked for 3 days as a contractor. I had to leave country for 2 months due to a family emergency. Since the employment duration was very short, I was not paid. I was told that they don't want to charge the client because 3 days is not enough contribution.  Since I had to leave country the same day, I did not argue with them and told it's okay if they feel right.  Now I have a W-2 from them saying they have paid me about $1500. I contacted them but I have not received any reply yet.  I just want to be prepared. What can I do here? Why is sample standard deviation a biased estimator of $\sigma$? According to the Wikipedia article on  the sample SD   $$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2}$$  is a biased estimator of the SD of the population. It states that $E(\sqrt{s^2}) \neq \sqrt{E(s^2)}$.   NB. Random variables are independent and each $x_{i} \sim N(\mu,\sigma^{2})$   My question is two-fold:   What is the proof of the biasedness? How does one compute the expectation of the sample standard deviation    My knowledge of maths/stats is only intermediate. Is there a way to reach a `protected` member of another object from a derived type? class MyBase {     protected object PropertyOfBase { get; set; } }  class MyType : MyBase {     void MyMethod(MyBase parameter)     {         // I am looking for:         object p = parameter.PropertyOfBase;  // error CS1540: Cannot access protected member 'MyBase.PropertyOfBase' via a qualifier of type 'MyBase'; the qualifier must be of type 'MyType' (or derived from it)     } }   Is there a way to get a protected property of a parameter of a type from an extending type without reflection? Since the extending class knows of the property through its base type, it would make sense if possible. Is there a term for the moment when a lot of users visit/log onto your website at the same time? Is there a term for the moment when a lot of users visit/log onto your website at the same time? In my class design, I use abstract classes and virtual functions extensively. I had a  feeling that virtual functions affects the performance. Is this true? But I think this performance difference is not noticeable and looks like I am doing premature optimization. Right? Etiquette, rules, or advice on which site to pose a question? (Or, why are there so many StackExchange sites?) Is it possible to fail an ability check on purpose? Here are the general guidelines regarding ability checks:     The GM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results.   In some cases, you could want to fail an ability check. If you are climbing a wall an want to drop off, you fail simply because you stop attempting to climb, but in other cases, success and failure are both tied to the actual action and may both have desirable outcomes.  Example 1: Spell scrolls  An evil adversary wants you to cast a spell scroll for him, but you have him figured out and want to destroy his spell scroll instead, seemingly by accident. The rules on spell scrolls read:     If the spell is on your class’s spell list but of a higher level than you can normally cast, you must make an ability check using your spellcasting ability to determine whether you cast it successfully. The DC equals 10 + the spell’s level. On a failed check, the spell disappears from the scroll with no other effect.   A failure means that you successfully destroy the spell scroll.  Example 2: Amulet of the Planes  You are in possesion of the Amulet of the Planes. Your party is being overrun by terrible monsters, and the only way to save your party is to sacrifice yourself by sending yourself and the monsters to another plane of existence.     While wearing this Amulet, you can use an action to name a location that you are familiar with on another plane of existence. Then make a DC 15 Intelligence check. On a successful check, you cast the Plane Shift spell. On a failure, you and each creature and object within 15 feet of you travel to a random destination. Roll a d100. On a 1-60, you travel to a random location on the plane you named. On a 61-100, you travel to a randomly determined plane of existence.   By walking up to the monsters and failing the check, you and the monsters would be sent to another plane of existence.  The question  The rules seem to assume that you always want to succeed when you perform an ability check. Sometimes, however, you could want to fail on purpose. A sensible DM would rule that if you are skilled enough to succeed the check, then you would undoubtly be able to fail it on purpose as well. What does the rules as written say -- if anything -- about this circumstance?  According to RAW, is it possible to attempt an ability check and intentionally fail? Show that if the smallest prime factor $p$ of the positive integer $n$ exceeds $\sqrt[3]{n}$, then $\frac{n}{p}$ must be prime or 1.  I'm stuck trying to prove this. I tried this using contradiction, but I haven't been able to prove it.  My approach:  Let $p$ be the smallest prime factor of the positive integer $n$.  Let $p &gt; \sqrt[3]{n}$  Then we must show that $\frac{n}{p}$ is either a prime number or 1.  We know that $n = pe$ for some $e \in \mathbb{Z}$.  Thus, $\frac{n}{p} = e$  We also know that $p$ > $\sqrt[3]{n}$, so $p^3 &gt; n$, and $p^2 &gt; \frac{n}{p}$  So, assume $\frac{n}{p} = e$ is not prime nor 1, meaning it's a composite number. Let's try to get a contradiction.  If $\frac{n}{p}$ is composite, then $\exists a,b \neq 1: \frac{n}{p} = ab $  Now, here's where I get stuck. What can I do with this info? $\frac{n}{p} = ab $ and $p^2 &gt; \frac{n}{p}$. How to move and overwrite subdirectories (and files) to parent directory? Unsealed meat in freezer Below is my situation:  Is completely (vacuum) sealing leftover meat in the freezer absolutely necessary? I would always just use a plastic bag to cover the meat and fold it around couple of times so it's almost as good as sealed (usually grocery plastic bag is much bigger than the piece of meat) and have never had a problem with it last a decade or so I have been cooking.  But recently I got this new roommate who insists on me using special small plastic bags with sealers to use and he is being very paranoid, obsessive about it.  So my question is: Is he right? Is the sub-field of algebraic elements of a field extension of $K$ containing roots of polynomials over $K$ algebraically closed? Recommended books for advanced undergraduate electrodynamics What books are recommended for an advanced undergraduate course in electrodynamics? I'll be at my uncle's house just in case you need to reach me.      I'll be at my uncle's house should you need to reach me.   Could you tell me what is the difference between those? I am trying to design a circuit and was wondering how to design it that the 12V 800mAH NimH battery I am using will not overcharge. I have researched trickle charging and as far as I understand, a zener diode that regulates the voltage down to about 13.3-13.7V should do. I chose a 14V Zener, taking into account the .6-.7V drop.   I have heard that trickle charging is not the best for the battery, but am not sure how much more work designing a smart charger would take.  I was also thinking of using another diode to prevent flow back to the panels from the battery.         &ndash; Schematic created using  The three voltage sources on the left represent the solar cells, with the NiMH battery on the right.  I was wondering if I needed any additional components if I were to use three 5.5V solar cells in parallel outputting 40mA (16.5V total at 40mA, regulated to 13.3V by the zener) in order to prevent overcharging.  A similar question to mine was asked at  However, this dealt more with a larger lead-acid battery and used a voltage regulator with a schottky diode.   This link discusses almost my exact same question, but I was confused to the answer given which said that up to 1.45V per cell could be charged, taking my max charge voltage up to 14.5V.   Considering I am only outputting 40mA to an  800mAH battery, it seems that I may not need any more battery protection, but would love any input.