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Schrodinger's Crouch aka Quantum Crouch is a glitch that allows the player to be in a state that is between crouching and standing.
Although Centaur1um was the first to document it in 2011, while exploring CEOOB, Gocnak found how to put it to tactical use. The map d2_prison_03 requires a button press to open a gate, but the button is positioned such that it could be pressed through the ceiling. Half-Life 2 Done Quick was able to press the button through the ceiling due to an older engine, but Gocnak found that performing a Schrodinger's Crouch and Gravity Jumping to get enough height was enough to hit the button.
Z1mb0bw4y later found that the glitch also worked in Portal, as the player could perform a Schrodinger's Crouch and fling out of a portal to hit a lower ceiling, effectively poking their head through and become fully capable of shooting a portal Out of Bounds.
When the player crouches, then uncrouches, and crouches again before the animation of standing up completes, the game glitches their stance into two states. The camera becomes seperated from the player's actual body, allowing the player to peek their view through ceilings.
The name is derived from the famous paradox "Schrödinger's Cat", where the player is in two states, both crouching and standing, until the player releases the crouch button.
The player is unable to jump while in a Schrodinger's Crouch, and it can prove to be difficult to reach a high ceiling. An orthodox way of gaining height without jumping would be a Gravity Jump in Half-Life 2 or manipulating portals in Portal.
Simply crouch, then uncrouch, and while the standup animation occurs, crouch again. Your camera will freeze in place, and how late you crouch in the animation determines the height at which the camera will freeze.
This trick is useful for activating buttons through ceilings. You can also use your weapons on the other side of the ceiling, for example, punting an Orb in d3_c17_10b start.
This trick also lets you to trick the AI so it'll shoot your head, which is actually below, making them miss their shot.
Another use is in Portal, where the player can peek their head out of a ceiling to shoot a portal Out of Bounds, for example, in Chambers 06-07. It can also be used to enter a portal at the edge to clip your view above where you could normally. This is done by performing it at a platform that you would need to crouch to get under. If you have a portal on the underside, then your view should go to the other portal (this may allow for post-patch save glitch; investigation needed).