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In a rotating shallow-water layer,is the horizontal scale on which gravity and Coriolis effects balance.

For a homogeneous layer of mean depth $H_0$ on an f-plane, linearization about rest gives

Two bound-state solutions of the same one-dimensional time-independent Schrodinger equation have constant Wronskian. Their decay at infinity makes this constant zero, so the two solutions are linearly dependent. Every bound-state energy is therefore nondegenerate.

The time-dependent Schrodinger equation evolves a quantum state according to

An energy eigenstate satisfiesFor a one-dimensional particle,

For a smooth circular hoop of radius $R$ rotating about its vertical diameter at angular speed $\omega$, a bead at angle $\theta$ from the downward vertical obeys

A pendulum of length $a$ whose pivot rotates at radius $L$ can remain at angle $\varphi$ from the downward vertical whenIf the connector is a spring of extension $x$, replace $a$ in the radius by $a+x$ and use vertical balance to determine $x$.

In a uniformly rotating frame, centrifugal acceleration points away from the rotation axis and equals $-\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{r})$.

In a frame rotating with angular velocity $\mathbf{\Omega }$, a moving particle has apparent Coriolis acceleration $-2\mathbf{\Omega }\times \mathbf{v}$.

A tensor network state contracts auxiliary indices of local tensors to encode a many-body wavefunction.

Quantum error correction encodes logical states into a larger Hilbert space so that a specified family of physical errors can be detected and reversed.

Topological quantum matter has robust low-energy properties controlled by global topology rather than local order parameters.

The qubit Hadamard basis consists of $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$ and $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$.

The partial trace over $B$ is the linear operation characterized byApplied to a density operator, it produces a reduced density matrix.

A pure quantum state is represented by a unit vector $|\psi ⟩$ up to global phase, or equivalently by the rank-one density operator $|\psi ⟩⟨\psi |$.

Quantum gravity seeks a theory in which gravity and quantum phenomena are described consistently.

The foundations of quantum mechanics study the physical and conceptual status of quantum states, measurement, nonlocal correlations, and possible extensions of quantum theory.

Quantum decoherence is the suppression of observable phase coherence when a system becomes entangled with unobserved environmental degrees of freedom. In a preferred basis, tracing out the environment damps off-diagonal entries of the system's reduced density matrix while leaving the corresponding populations unchanged.

Quantum information theory studies information processing with quantum states, measurements, channels, entanglement, and quantum entropy.

Shor's factoring algorithm uses quantum period finding to determine a modular order and then extracts factors classically with greatest common divisors.