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\)
Appendix C Symbols
| Symbol |
Description |
Location |
| \(z=x+iy\) |
the rectangular form of a complex number \(z\) with real part \(x\) and imaginary part \(y\)
|
Definition 2.1 |
| \(\mathrm{Re}\, z,\; \Re\, z\) |
the real part of a complex number \(z\) or the real axis in the complex plane |
Notation 2.1 |
| \(\mathrm{Im}\, z,\; \Im\, z\) |
the imaginary part of a complex number \(z\) or the imaginary axis in the complex plane |
Notation 2.1 |
| \(z^*,\;\bar{z}\) |
complex conjugate of a complex number \(z\)
|
Definition 2.3 |
| \(|z|\) |
norm of a complex number \(z\)
|
Definition 2.5 |
| \(z = re^{i\phi}\) |
the exponential form of a complex number \(z\)
|
Definition 2.9 |
| \(M^T\) |
transpose of a matrix \(M\)
|
Paragraph |
| \(A \left|v\right> = \lambda \left|v\right>\) |
the eigenvalue/eigenvector equation for a linear operator \(A\)
|
Definition 4.1 |
| \([M,N]\equiv MN-NM\) |
the commutator of two operators \(M\) and \(N\)
|
Definition 5.1 |
| \(N^\dagger=N^*{}^T\) |
the Hermitian conjugate (or adjoint) of a matrix \(N\) \(A\)
|
Definition 5.2 |
| \(P_v^2 = P_v\) |
the definition of a projection operator \(P_v\)
|
Definition 5.3 |
| \(P_v = |v \rangle \langle v|\) |
the bra-ket form of a projection operator \(P_v\)
|
Paragraphs |
| \(\left\{\sigma_x, \sigma_y, \sigma_z\right\}\) |
Pauli matrices |
Paragraph |
| \(\frac{dy}{dx}\) |
Leibniz notation for derivative |
Notation 6.1 |
| \(y^\prime\) |
Lagrange notation for derivative (w.r.t. \(x\)) |
Notation 6.1 |
| \(\dot{y}\) |
Newton notation for derivative (w.r.t. \(t\)) |
Notation 6.1 |
| \(\forall\) |
“for all” |
Paragraph |
| \(\left\langle \vec{u}\vert \vec{v}\right\rangle\) |
the inner product of the vectors \(\left\vert \vec{u}\right\rangle\) and \(\left\vert \vec{v}\right\rangle\)
|
Definition 14.2 |
| \(\LL\) |
linear differential operator |
Notation 15.1 |
| \(\delta_{ij}\) |
the Kronecker Delta |
Definition 16.1 |
| \(\Theta(x)\) |
the step function (alternatively, theta or Heaviside function) |
Definition 16.5 |
| \(\delta(x)\) |
the Dirac delta function (alternatively, distribution) |
Definition 16.9 |
