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The principle states that it is impossible to simultaneously measure the exact position and momentum of a particle. It's mathematically expressed as $\Delta x \Delta p \geq \frac{\hbar}{2}$. To reduce uncertainty in one variable, increase uncertainty in the other.

Quantum entanglement occurs when pairs or groups of particles are generated in such a way that the quantum state of each particle cannot be described independently. Actions performed on one entangled particle affect the other instantaneously, no matter the distance, which was referred to as 'spooky action at a distance' by Einstein.

Solution involves solving the Schrödinger equation for the system, which gives quantized energy levels. Solutions include sine functions that represent the stationary states of the particle. Energy levels are given by $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$, where $n$ is a quantum number, $L$ the length of the box, $m$ the particle mass, and $\hbar$ the reduced Planck constant.

Spin is an intrinsic form of angular momentum carried by elementary particles. It is calculated by the operator $\hat{S} = \sqrt{s(s+1)}\hbar$, where $s$ is the spin quantum number. Electrons have a spin of $1/2$, which leads to two possible orientations, +1/2 (up) and -1/2 (down).

The Pauli exclusion principle states that no two fermions can occupy the same quantum state simultaneously. For a system of two electrons in an atom, this means they must have opposite spins if they are in the same orbital, leading to a singlet state with spins cancelling out. In different orbitals, they can be in a triplet state with parallel spins.

Quantum tunneling allows particles to pass through a barrier that would be insurmountable in classical physics. This phenomenon can be explained by wave-particle duality and is described mathematically by the transmission coefficient through a potential barrier. Applications include the scanning tunneling microscope and semiconductor devices.

The double-slit experiment demonstrates the wave-particle duality of quantum objects. When not observed, particles exhibit an interference pattern, indicating wave-like behavior. Once observed, they behave like particles. This reveals the role of the observer in affecting the outcome of quantum events.

Quantum decoherence describes the loss of quantum coherence, whereby a system ceases to exhibit purely quantum effects like superposition and becomes classically distinguishable. This occurs due to the interaction with the environment, which acts as a measurement and disrupts the superposition.

The Stern-Gerlach experiment passes a beam of silver atoms through a non-uniform magnetic field. This splits the beam into two, corresponding to the two possible spin states. It demonstrates superposition by showing that the silver atom's spin is in a superposition of up and down states until measured.