"Do you know that thought experiment with the cat in the box with the poison? Theory requires the cat be both alive and dead until observed. Well, I actually performed the experiment. Dozens of times. The bad news is that reality doesn't exist. The good news is we have a new cat graveyard."
The Schrödinger's Cat thought experiment is increasing in popularity as a Motif in fiction. To elaborate, Erwin Schrödinger's thought experiment involved a cat sealed inside a box with a Geiger counter attached to it. Inside that Geiger counter is a small amount of radioactive substance. The box is set up so that when the radioactive substance decays, a device smashes a vial of toxic gas, killing the cat inside. The thing is, after an hour, until the box is open, there is no way to determine whether the cat inside is alive or dead, thus allowing the cat to be both alive and dead.
The thought experiment was originally a critique of the Copenhagen interpretation of Quantum Physics, where the cat can be living and dead at the same time as a superposition of "realities". The paradox should be obvious. Note that the paradox only exists in a flawed understanding of the Copenhagen interpretation, the classic interpretation of quantum mechanics. According to the many-worlds interpretation, what actually happens is that the entire world splits into a superposition of worlds where the cat is alive, and those where it is dead.
The reason the experiment is interesting is that in quantum mechanics, things like subatomic particles routinely exist in superpositions of states, and correct predictions of the outcomes of certain experiments can only be made by taking these effects into account*
A classical example is the double-slit experiment for electrons where the resulting interference pattern is only expected if you assume the electron is in a superposition of states
. The experiment posits that if we can accept that small particles (like the radioactive trigger) can be in these kinds of states, then we have to accept that large objects (like the cat) can also be in these kinds of states, raising the question of why we don't routinely observe that.
This thought experiment immediately became popular as a metaphor for uncertainty of the truth, fate, Quantum Physics, how quantum physics can do anything, and whatever the hell is inside that box. Ironically, the original experiment was used to show how stupid the idea of superposition was.
To elaborate further, we're dealing with radioactive nuclei; a nucleus is at the centre of an atom, and some nuclei are radioactive, so they do something called "radioactive decay". Heisenberg's Uncertainty Principle shows that it's impossible to directly observe this decay because it involves particles so small that any equipment we have to monitor them—even the smallest, like photons and electrons—would interfere with what they're doing, so we can't get reliable results about an electron's both momentum and position—one or the other, but not both. Therefore, any particular radioactive nucleus may, at any moment, be either decayed, or undecayed. That much isn't controversial.
The Copenhagen interpretation says that, because of this, each particle is 50% decayed and 50% undecayed. Schrödinger thought that this idea was Insane Troll Logic, so he extended it to show: if the decay of a radioactive nucleus will set off a device which will release toxic gas into a box that contains a cat, and if (as per the Copenhagen interpretation) the nucleus is 50% decayed and 50% undecayed, the device is 50% triggered and 50% untriggered, so the cat is simultaneously 50% killed and 50% still alive.
As mentioned earlier, the supposed paradox relies on a flawed understanding of Quantum Physics. The flaw lies in not knowing what it means to observe something; observation is not the passive thing like in the world of the everyday, but an active process. In the theoretical box (no physicist would actually do this; it's highly unethical), the observer isn't the scientist, it's not even the cat; it's the Geiger counter, and it keeps the wave function collapsed into either 100% alive or 100% dead, and not 50% both. To learn more, see the Quantum Physics page. In short, the Copenhagen Interpretation doesn't predict a cat simultaneously alive and dead, but rather a cat that is fully alive until it is fully dead. To put it another way, even famous and intelligent physicists like Schrödinger can be completely wrong.
Sometimes though, it's just used to show off the writer'scleverness. Or not.
Here's a video that explains the principle in question.
As you might know, even This Very Wiki is fond of using this. It's the Trope Namer for:
Schrödinger's Cat may or may not exist in these works, you'll have to check to know for sure:
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Anime & Manga
To Aru Majutsu no Index uses Schrödinger's Cat in order to explain the Esper abilities: the state of the "cat" outside of their mind is whatever the user's mind deems it to be, allowing the manifestation of the abilities.
Misaka Imouto tries to name her cat Schrödinger, but Touma shoots that idea down. That name is taboo for cats.
Umineko no Naku Koro ni has this as a recurring motif, and is used to explain the fantasy scenes as something that could have happened but not necessarily did happen.
The events on Rokkenjima is also refered to as this as no one except Eva knows and who chose not to reveal the truth so that Ange wouldn't find out the truth about her parents.
Similarly, Higurashi no Naku Koro ni uses the cat; most notably, Frederica Bernkastel uses it in one of her poems. In that poem, the cat dies, referring to the end of Minagoroshi-hen, as well as the fate Rika encounters a few nights after the Watanagashi.
Done a couple of times in Yu-Gi-Oh! GX. Bastion Misawa has it as a formula on his wall (dub only) and a character uses a card which uses Schrödinger's Cat as a basis.
Hellsing actually has a character named Schrödinger, and he has a power to be 'everywhere and nowhere' — allowing him to always be in all plot relevant spots at once, as well as to Retcon any damage he suffers. And he's a Cat Boy.
In All-Star Superman, the Ultrasphinx traps Lois Lane in a state of quantum uncertainty, between life and death, unless someone can answer its riddle. Superman answers it and saves her.
In one The Mighty Thor story, Donald Blake references Schrödinger's cat in describing the Odinsleep, saying the sleeper is in a state between life and death.
Played with in The Incredible Hercules, where Amadeus Cho finds himself playing a deadly Role-Playing Game (itself a Schrödinger's Butterfly of a sort), in which his character, Mastermind Excello, encounters a box containing a cat that is literally both dead and alive in the lair of a Mad Scientist.
Done wonderfully in the glee fanfic "Magical Thinking", which can be read here 
In Academy Blues, Laura develops a spell called "Schrodinger", which places her in a state between life a death, and allows her to literally be anywhere within a large radius. The spell is somewhat unstable, and she tends to fade in and out in different locations at random times. The spell encloses her body at the location she initiated the spell in a box of light.
A Serious Man by The Coen Brothers: References to Schrodinger's Cat occurs multiple times as a general theme, both on a literal and metaphorical level. It can be seen as a metaphor for general uncertainty in the world. For instance:
It's discussed literally in the film since Larry is a physics teacher
Larry's marriage is in completely the opposite state to what he thought it was,
When Larry both is and isn't being bribed.
Larry Gopnik: It doesn't make sense. Either [your son] left the money or he didn't.
Mr. Park: Please. Accept the mystery.
Reb Groshkover in the opening scene is one. He is suspected to be an undead dybbuk, but no one is sure whether he is either alive or dead until he's stabbed to make sure.
The end: Larry has been called to discuss his MRI results and Danny may be hit by a tornado. In our minds Larry and his son are both dead and not dead.
Dirk Gently's Holistic Detective Agency saw Dirk begin to explain Schrödinger's cat normally, before talking about how he intervened in the experiment because the cat had gone missing. Dirk turns out to be making it up to see if Richard, who has been acting very oddly, is capable of spotting complete nonsense (like someone actually performing an experiment where the outcome you're testing is what happens before you look) when he hears it.
The Salmon of Doubt features half of a cat. Not a cat chopped in half and lying still, the very lively and normal-acting front part of a cat, which just kind of ends midway along. The cat's name is Gusty Winds, and near the end of what Douglas Adams had written before he died, Dirk comes across a road sign that says "Gusty winds may exist."
In the Discworld novels, Death does not understand how this would work. A pretty thing would come to pass if he couldn't tell if a cat was dead or not.
Pratchett makes multiple references to this. In The Unadulturated Cat, the theory is used to explain the ability of Offscreen Teleportation that all cats seem to have (apparently, the cat was so distressed by being locked in a box with radioactive materials, it became an instant expert in teleportation)
In Lords and Ladies the motif is spoofed once again, when the cat locked in the box turns out to be Greebo.
Technically, a cat locked in a box may be alive or it may be dead. You never know until you look. In fact, the mere act of opening the box will determine the state of the cat, although in this case there were three determinate states the cat could be in: these being Alive, Dead, and Bloody Furious.
An incarnation of Schrödinger's Cat appears in Wyrm. It looks and acts a lot like the Cheshire Cat from Alice in Wonderland. But it's actually an avatar of Wyrm.
In Crushed by Laura and Tom McNeal, this becomes a critical plot point after Wickham reveals his past to Audrey.
Live Action TV
In Stargate SG-1, Carter explains why her cat is called Schrödinger to a scientifically advanced alien. (Season 1, episode 17, "Enigma".)
Amusingly, the alien agrees with Schrödinger when she finishes: anyone who thinks that might be true has misunderstood the laws of physics.
Flash Forward lets us know that Simon is a scientist because he explains it when we first meet him.
On The Big Bang Theory, Sheldon uses it as a metaphor about whether or not Penny and Leonard's date will be successful or not.
He also gets its origin wrong, claiming it was an explanation for the Copenhagen interpretation rather than a critique of it.
The third season premiere of Six Feet Under opens with Nate trapped in a Mental World with recurring references to Schrödinger's Cat, highlighting the fact that he's in an in-between state while undergoing risky brain surgery and won't know where he is until he actually takes a risk and "opens the box."
Though it makes a lot more sense when you realise it's an aspect of Seraph that achieved enlightenment.
In Nethack a monster known as the Quantum Mechanic will sometimes drop a box when it dies. When you open the box either a living cat will jump out or a cat corpse will be found inside. What's more, examining the code shows that which will be found is undetermined until you check — as opposed to every other container in the game, the contents of which are determined when the box is spawned.
Ghost Trick has a rather... literal interpretation of the trope. To be exact, by the end of the game, Sissel is de-facto a dead-immortal cat.
GLaDOS: Do you know that thought experiment with the cat in the box with the poison? Theory requires the cat be both alive and dead until observed. Well, I actually performed the experiment. Dozens of times. The bad news is that reality doesn't exist. The good news is we have a new cat graveyard.
The same comic later compares Chell to the cat, surviving in stasis but with no guarantee she'll ever be woken up.
There was an achievement in Portal 2 called Schrödinger's Catch, which involved catching a box before it hit the ground. Doing the catch the intended way as opposed to the easy way involves catching it with no idea of where it will land in a large room. Possibly also a reference to the comic.
Also mentioned by the Fact Core:
Fact Core: The Schrödinger's Cat paradox outlines a situation in which a cat in a box must be considered, for all intents and purposes, simultaneously alive and dead. Schrödinger created this paradox as a justification for killing cats.
One of Ratman's dens early in the game has a diagram of a cat apparently jumping out of a box, with quantum physics equations scribbled all over it.
Selected randomly as a gambling device in the Money-Making Game in Kingdom of Loathing, where the winner is decided by one player guessing whether the cat is alive or dead. Of course, you don't actually get to choose.
Nine Hours, Nine Persons, Nine Doors references the original paradox (albeit not by name) in the Safe End and True End. After The Reveal, Akane's discussion of how information flows and her existence are nods to the Copenhagen interpretation. The sub-tropes listed above are also used, although the player doesn't know that until afterward.
The sequel, Virtues Last Reward, actually discusses the concept in-game. K's identity is also an in-universe example. In most timelines, the person in the suit is Kyle Klim. In the cyan door pathways, however, it is Akane Kurashiki.
The post-credits sequence of Bio Shock Infinite. DeWitt thinks his daughter is in her room, but he's not sure. As he opens the door, the screen cuts to black. Given the infinite amount of universes that could exist, she is both in and not in the room, and it is left up to the player which version DeWitt finds.
Well, maybe. It would be in the strictest sense if absolutely nobody— including the author— knew what the last panel was before it was read. Then again, since humor is heavily subjective, until each individual person has read it they have no idea whether it'll be funny to them or not, so the quantum mechanics does work in that way.
In thisDM of the Rings strip, Gimli tries to abuse the concept (along with a poorly cast die) to become an UNCERTAINTY LICH.
This strip of Irregular Webcomic! references Schrödinger's thought experiment in the punchline, but you'll notice that the explanatory note below the comic starts off talking about Heisenberg. As DMM points out in the final paragraph, the punchline of the comic existed in a state of flux while he was writing The Rant.
Scenes From A Multiverse has fun with this in this strip, which briefly presents an entire civilization based on the uncertainty theory, including universal ownership of celebratory "cat boxes".
Captain SNES has a cat named Schrödinger, whose state in Videoland cannot be determined by the residents, as it is from the real world, and it generally proceeds to make fools of everyone it meets.
Homestuck had a sudden perspective switch after Nepeta, the resident cat troll, attacked Ax Crazy sober Gamzee. For the longest time, nobody knew whether she had survived the encounter. She didn't. Therefore, she was temporarily Schrodinger's Nepeta.
Linkara criticises the experiment in one of his reviews. In his top 11 mistakes, he admits he missed the point somewhat.
Used as a joke in a Futurama episode. The cops catch Erwin Schrödinger speeding with a box containing a cat, he's not clear whether it's alive or dead (until Fry opens the box and the cat claws his face), and there's also drugs in there.
There's also the Professor's cry of No fair! You changed the outcome by measuring it! at the racetrack.
Said by Velma in the series finale for Scooby Doo Mystery Incorporated while she is trying to comprehend the reveal of an alternate timeline where the Nibiru Entity never existed, and therefore never negatively influenced any of the character's lives, placing them in a happy town with no mysteries.
A famous quote by Stephen Hawking: "When I hear about Schrödinger's cat, I reach for my gun." *
A paraphrase of the, eh, more (un)famous quote of Nazi Hermann Göring, who did the same hearing the word "Kultur".
This represents the thoughts of many physicists, because there are several aspects the thought experiment that bring up issues. The biggest problem with the analogy is that quantum physics typically only operates on the microscopic scale of atoms and subatomic particles, not on the macroscopic scale of cats and poison vials.
Also, physicists such as John Gribbin (author of In Search of Schrödinger's Cat and Schrödinger's Kittens) wonder at what point particles go from being "quantum" to "ordinary". How big do you have to be to be normal? How small do you have to be to be in two places at once?
The rise of Quantum Mechanics Can Do Anything has led to references to the original thought experiment in unrelated fields. One publication in political science asks the reader to imagine international structure by invoking Schrödinger's cat and the Uncertainty Principle.
One internet hoax centered around a LiveJournal member soliciting donations to treat her injured cat. The scam fell apart quickly, with plenty of attention on the non-existent feline's name.
An entry in the Style Invitational (humor contest in the Washington Post) from 2010:
Curiosity only collapsed the quantum superposition of Schrödinger's cat into a definite state. What killed it was leaving it in a box for a week.