Symmetric gauge

For a constant magnetic field,is the symmetric gauge.

For charge $e$, defineIn the symmetric gauge,

For a charged particle in a uniform magnetic field, the symmetric gauge $\mathbf{A}=\mathbf{B}\times \mathbf{x}/2$ makes the displayed magnetic pseudomomentum a conserved quantity when no other position-dependent potential breaks translation symmetry. It differs from the kinetic momentum $\mathbf{\pi }=\mathbf{p}-e\mathbf{A}=m\dot{\mathbf{x}}$. With $\mathbf{B}=B{\mathbf{e}}_3$, Hamilton's equations for $H=|\mathbf{p}-e\mathbf{A}{|}^2/(2m)$ give ${\dot{p}}_1=(eB/2){\dot{x}}_2$ and ${\dot{p}}_2=-(eB/2){\dot{x}}_1$, proving conservation of $K_1=p_1-eB{x}_2/2$ and $K_2=p_2+eB{x}_1/2$. The charge factor is necessary unless it has already been absorbed into the field convention. In quantum mechanics the transverse components need not commute, even though each commutes with this Hamiltonian operator; magnetic translation algebra records that distinction.

For charges $+e,-e$ and relative coordinate $\mathbf{r}={\mathbf{r}}_2-{\mathbf{r}}_1$, total magnetic pseudomomentum is conserved in a uniform magnetic field. In an appropriate symmetric-gauge Lagrangian differing by a total derivative, it is the canonical centre-of-mass momentum. Mechanical centre-of-mass momentum instead equals $\mathbf{K}+e\mathbf{B}\times \mathbf{r}$ and need not be constant. The sector $\mathbf{K}=0$ is not generally a stationary-centre-of-mass frame.

In the zero-pseudomomentum sector, a planar relative circular orbit has two signed angular-frequency branches. Here $\mu =m_1{m}_2/(m_1+m_2)$, $e^{\ast }=e(m_1-m_2)/[2(m_1+m_2)]$, $R$ is separation, and the Coulomb constant is absorbed into $e^2$. The centre of mass also circles: ${\mathbf{R}}_{\mathrm{cm}}={\mathbf{R}}_0+eB\mathbf{r}/[(m_1+m_2)\omega ]$. This distinction is essential when reconstructing individual particle orbits.

The kinetic momentum $\mathbf{\Pi }=\mathbf{p}-q\mathbf{A}$ equals $m\mathbf{v}$ for a nonrelativistic charged particle and is invariant under an electromagnetic gauge transformation.

For a lattice vector $r$,acts in the symmetric gauge asIt commutes with $(p-eA{)}^2/(2m)+V(x)$ when $V(x+r)=V(x)$.

For constant $B$,

Magnetic translations along two lattice vectors commute when their cell flux obeys

For a nonvanishing superconducting order parameter $\psi =\sqrt{n_s}{e}^{i\theta }$, $j=(q{n}_s/m)(\hslash \nabla \theta -qA)$. Single-valuedness gives phase winding $2\pi N$ around a closed curve, hence the fluxoid and flux quantization in a superconductor identity above. Magnetic flux alone is $Nh/q$ when the current integral vanishes. If the entire spanning surface is in a region where $D\psi =0$, the field vanishes there and the winding is zero; nonzero quantized flux requires a hole or a vortex core where that condition does not hold.

For uniform superconducting carrier density, the displayed combination of enclosed magnetic flux and the circulating supercurrent is quantized on a closed contour where the order parameter does not vanish. It follows by integrating the gauge-invariant phase gradient. Magnetic flux alone is quantized only when the current contribution on that contour is negligible.

The magnitude of one magnetic flux unit for an electron-pair order parameter, whose electric charge has magnitude $2e$. The enclosing contour must satisfy the flux condition in fluxoid and flux quantization in a superconductor; it is not a statement that every finite area inside the material contains an integer number of these units.