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Use the dimensionless quantum phase $S$, so that $\psi =R{e}^{iS}$ with $R\ge 0$. Work locally away from wavefunction nodes, where $R$ and $S$ are differentiable. Differentiating the wavefunction for substitution in the Time-dependent Schrodinger equation gives:Cancel $e^{iS}$ in the time-dependent equation and equate real and imaginary parts. The resulting Madelung equations areThe second equation is a Hamilton-Jacobi equation for the action $\hslash S$, with an additional quantum potential. To see the meaning of the first, multiply it by $2R$ and set $\rho =R^2$. It becomes the probability continuity equationThus the probability density is transported by the velocity field $\hslash \nabla S/m$. The Madelung equations are a local rewriting of the linear wave equation; their apparent nonlinearity comes from expressing a complex wavefunction in modulus and phase variables.

For the separated stationary state, restore its time factor $e^{-iEt/\hslash }$. Away from radial nodes, the quantum phase is $S=k\varphi -Et/\hslash$, up to a constant $0$ or $\pi$ where the real radial function has fixed sign. The guidance equation in plane polar coordinates therefore gives the Bohmian circulation of an angular-momentum eigenstateThe direction is $\hat{\mathbf{\varphi }}$ for $k>0$ and $-\hat{\mathbf{\varphi }}$ for $k<0$; the velocity is zero for $k=0$. Each admissible trajectory is a circle:The orbital angular momentum along the trajectory is $m{r}_0{v}_{\varphi }=\hslash k$. The origin or any zero-amplitude circle is excluded from this local formula. For the real degenerate superposition constructed above, the spatial quantum phase is constant on each nodal sector, so its Bohmian mechanics velocity is instead zero. Thus the circular motion is a conclusion about the separated angular-momentum eigenstate, not an arbitrary energy eigenstate.

The Born rule depends only on transition probabilitiesbetween rays. A physical transformation must preserve these probabilities. By Wigner theorem, such a ray transformation is induced by a unitary or antiunitary operator. For ordinary continuously connected transformations one chooses the unitary branch, so

The states $U(g_1)U(g_2)|\psi ⟩$ and $U(g_1{g}_2)|\psi ⟩$ represent the same ray. They may therefore differ by a state-independent quantum phase:Thus quantum transformations form a projective unitary representation.

If $g$ is a symmetry of the Hamiltonian operator $H$, it also obeysequivalently $[U(g),H]=0$. For a differentiable one-parameter subgroup, writewith a self-adjoint generator $Q$. Differentiating the symmetry equation at $s=0$ yieldsFor a time-independent $Q$, the Heisenberg picture equation isHence the generator is a conserved observable. Conversely, a self-adjoint conserved $Q$ commutes with $H$ and its unitary group generates symmetries. This is the conserved generator of a continuous quantum symmetry.

Forthe phase of the $n$th stationary state isReflection in the midpoint givessoThe minus sign is a global quantum phase and therefore does not change the physical state. The relative even--odd phases first acquire this reflection pattern at $T$. Applying it twice givesIf $f$ vanishes on the right half of the well, then $\psi (x,T)$ vanishes on the left half. The Born rule consequently gives zero probability of finding the particle in $0\le x\le a/2$ at time $T$.

For two nonorthogonal input states $|\alpha ⟩,|\beta ⟩$, preservation of inner products by $G$ givesHence the two output-program states agree up to phase. Any two qubit states can be joined through a state nonorthogonal to both, so after absorbing phases
$|{\tilde{P}}_U⟩$ is independent of the input.

Using the conventionpart (ii) applies the extra phase ${\omega }^{j\beta \eta }$ to term $j$. The resulting first register is the quantum Fourier transform of $|\alpha +\beta \eta \mathrm{mod}N⟩$. Applying its inverse therefore produces

For a neutron in the Earth's Newtonian gravitational potential ${\Phi }_E$, the Schrodinger equation isNear the surface, ${\Phi }_E(z)=gz$ up to an additive constant.

The Colella–Overhauser–Werner experiment uses crystal slabs as coherent beam splitters and mirrors in a neutron interferometer. The two paths contain horizontal segments of length $s$ separated vertically by $r$, then recombine. The gravitational potential energy difference is $m_{n}gr$. A neutron of speed $v$ spends time $s/v$ on a horizontal segment, so the gravitationally induced relative quantum phase isUsing the de Broglie wavelength $\lambda =h/(m_{n}v)$ and $\hslash =h/(2\pi )$ givesRotating the apparatus changes the vertical projection of its enclosed area from zero to $rs$. The number of full interference oscillations is thereforeHere $rs=(\sqrt{10}\mathrm{cm}{)}^2=10^{-3}{\mathrm{m}}^2$, soThe expected change is therefore $\boxed{9}$ oscillations to the nearest integer.

Because state vectors differing by a quantum phase represent the same ray, a unitary realization of a group action need only satisfy