Reconciling the quantum mechanical behavior of subatomic particles with the ordinary behavior of macroscopic systems is still difficult, and this is the point of the thought experiment known as Schrodinger’s cat.
In a purely Newtonian world, particles would have a location and a velocity. If you send particles through a pair of slits, each particle either goes through one or the other. A standard college physics experiment shows that electrons do not behave like this. The electrons act as though each electron passes through both slits, illustrating the wave-like nature of electrons. At the moment the electron passes through, you can’t specify the location as a point. The location is a superposition of states.
While it is much more difficult to imagine macroscopic systems which behave as superpositions of quantum states, quantum mechanics predicts that this happens. Schrodinger’s cat is a hypothetical cat in a black box whose fate depends on the state of a subatomic particle. The state of the cat is a superposition of the living and dead states until you look at the particle or cat.
In a Newtonian world, probabilities would be real densities. In a quantum mechanical world, probabilities are described by complex distributions, the squares of whose magnitudes total 1. This allows destructive interference, as seen in the double-slit experiment. That’s why an even mixture of ψalive and ψdead is (ψalive + ψdead) / √2, not (ψalive + ψdead) / 2.
We mixed a depiction of Schrodinger’s cat with the internet LOLcat meme, which combines adorable images of cats with captions in broken English, usually as spoken by the cat. In the LOLcat vernacular, the wavefunction of the cat is: ψIM IN UR BOX = (ψHAI + ψOH NOES!) / √2. We have a non-LOLed Schrodinger’s cat shirt as well.
Modern transistors are based on quantum mechanical effects. This allows the basic switches to be extremely fast and small, but our computers are still Turing machines. They still follow the same model of computation as the old vacuum-tube giants, or pencil and paper computations. Future computers called quantum computers may rely on manipulation of particles in superpositions of quantum states. These are fundamentally more powerful than Turing machines, and they may allow polynomial-time solutions to problems which take exponentially long on Turing machines.