Operator Conventions

Operators represent transformations, observables, projectors, time-evolution maps, density matrices, and symmetry actions. The default convention is to state the operator first as an abstract object and then specify a representation when needed.

For example, momentum may be discussed abstractly as $P$, as a differential operator in the position representation, or as multiplication by $p$ in the momentum representation. These are related descriptions, not different physical momenta.

Notation Policy

• Use uppercase letters such as $A$, $H$, $P$, and $U$ for abstract operators when the meaning is clear.
• Use hats, as in $\hat x$ or $\hat p$, when they prevent confusion between operators and classical variables or eigenvalues.
• Use $I$ or $\mathbf 1$ for the identity operator, with local pages declaring which is clearer.
• Use $A^\dagger$ for the adjoint.
• Use $A^{-1}$ only when an inverse exists on the relevant space or domain.
• State domains when unbounded operators or self-adjointness issues matter.

Observables and Self-Adjointness

In finite-dimensional examples, observables are represented by Hermitian matrices. In infinite-dimensional quantum mechanics, the precise condition is self-adjointness, including domain data. A formally Hermitian differential expression is not automatically a self-adjoint operator.

Expectation values in a normalized pure state use

$$\langle A\rangle_\psi =\langle\psi\vert A\psi\rangle.$$

For density operators,

$$\langle A\rangle_\rho =\operatorname{Tr}(\rho A).$$

Matrix Elements

In an orthonormal basis $\{\lvert n\rangle\}$, matrix elements are written

$$A_{mn}=\langle m\vert A\vert n\rangle.$$

The column vector representing $A\lvert n\rangle$ has entries $A_{mn}$ indexed by the output basis label $m$. This convention matches ordinary matrix multiplication on column vectors.

Products, Functions, and Order

Operator order matters. In general,

$$AB\ne BA.$$

The adjoint of a product reverses order:

$$(AB)^\dagger=B^\dagger A^\dagger.$$

Functions of self-adjoint operators are defined by spectral methods in the mathematically precise treatment. In finite dimensions, diagonalizing the matrix gives the same practical rule: apply the function to eigenvalues and keep the eigenvectors.

Common Mistakes

• Using hats inconsistently so that the same symbol means both an operator and an eigenvalue.
• Calling every Hermitian-looking differential expression self-adjoint without checking domains.
• Reversing matrix-element indices.
• Forgetting that operator multiplication is not commutative.
• Treating $A^\dagger A=AA^\dagger$ as automatic. It holds for normal operators, not all operators.
• Using $\operatorname{Tr}(\rho A)$ before checking that the product is trace-class in infinite-dimensional settings.

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1980.

Exercises

1. Show that $(AB)^\dagger=B^\dagger A^\dagger$ is consistent with matrix multiplication.

Solution

In matrix notation, the adjoint is transpose plus complex conjugation. Transposing a product reverses order, and complex conjugation distributes over products, so

$$(AB)^\dagger =(AB)^{*T} =B^{*T}A^{*T} =B^\dagger A^\dagger.$$

2. For

$$\sigma_z= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}, \qquad \lvert\psi\rangle =\frac{1}{\sqrt2} \begin{pmatrix} 1\\ 1 \end{pmatrix},$$

compute $\langle\sigma_z\rangle_\psi$.

Solution

First,

$$\sigma_z\lvert\psi\rangle =\frac{1}{\sqrt2} \begin{pmatrix} 1\\ -1 \end{pmatrix}.$$

Then

$$\langle\psi\vert\sigma_z\vert\psi\rangle =\frac12(1-1) =0.$$