Kinetic momentum and magnetic pseudomomentum

For charge

e

π = p - e A, K = p + e A .

In the symmetric gauge,

[K_{i}, π_{j}] = 0, [K_{i}, K_{j}] = - i e ℏ ε_{ijk} B_{k} .

Table of contents
- Kinetic momentum Kinetic momentum and magnetic pseudomomentum
- Magnetic translation operator Kinetic momentum and magnetic pseudomomentum
  - Magnetic translation algebra Magnetic translation operator
    - Magnetic flux quantum Magnetic translation algebra

Kinetic momentum ( $Π = p - q A$ )

The kinetic momentum

Π = p - q A

equals

m v

for a nonrelativistic charged particle and is invariant under an electromagnetic gauge transformation.

Magnetic translation operator

For a lattice vector

r

,

T_{r} = exp (\frac{i}{ℏ} r \cdot K)

acts in the symmetric gauge as

(T_{r} ψ) (x) = exp (\frac{i e}{2ℏ} r \cdot (B \times x)) ψ (x + r) .

It commutes with

(p - e A)^{2} / (2 m) + V (x)

when

V (x + r) = V (x)

.

Magnetic translation algebra

For constant

B

,

T_{r} T_{r^{'}} = exp (\frac{i e}{ℏ} (r \times r^{'}) \cdot B) T_{r^{'}} T_{r} .

Magnetic flux quantum

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

\frac{e Φ}{ℏ} \in 2 π Z, equivalently Φ \in \frac{h}{e} Z .

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 34B / a / Solution

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