Parity
Classically, if one takes a trajectory \(\textbf x(t)\) and reflects it about the origin to obtain the reflected trajectory \(\textbf x'(t)=-\textbf x(t)\), then the momentum of the particle \(\textbf p=m\dot{\textbf x}\) is correspondingly reflected \(\textbf p’=m\dot{\textbf x’}=-m\dot{\textbf x}=-\textbf p\). Thus, a parity transformation \(\textbf x\mapsto -\textbf x\) maps the entire state \((\textbf x,\textbf p)\) of the particle in phase space to \((-\textbf x,-\textbf p)\).
When working with quantum mechanics on \(\textbf R^3\) so that \(\mathcal H\cong L^2(\textbf R^3\to\textbf C,d^3\textbf x)\), one defines the discrete parity operator \(\Pi\in U(\mathcal H)\cap i\frak u\)\((\mathcal H)\) to be a unitary, Hermitian, and involutory operator \(\Pi=\Pi^{\dagger}=\Pi^{-1}\) that acts on states \(|\psi\rangle\in\mathcal H\) via a spatial inversion of their wavefunction:
\[\langle\textbf x|\Pi|\psi\rangle:=\langle-\textbf x|\psi\rangle\]
For instance, for an \(\textbf X\)-eigenstate \(|\psi\rangle=|\textbf x\rangle\), the above definition implies that \(\Pi|\textbf x\rangle=|-\textbf x\rangle\) whereas for a \(\textbf P\)-eigenstate \(|\psi\rangle =|\textbf p\rangle\) one has \(\Pi|\textbf p\rangle =|-\textbf p\rangle\) thanks to \(\langle -\textbf x|\textbf p\rangle=e^{i(-\textbf x)\cdot\textbf p/\hbar}/(2\pi\hbar)^{3/2}=e^{i\textbf x\cdot(-\textbf p)/\hbar}/(2\pi\hbar)^{3/2}=\langle\textbf x|-\textbf p\rangle\) so it completely mirrors the classical parity transformation.
Noting for arbitrary \(|\psi\rangle\in\mathcal H\) that \(\langle\psi|\Pi^{\dagger}\textbf X\Pi|\psi\rangle=-\langle\psi|\textbf X|\psi\rangle\), this yields \(\Pi^{\dagger}\textbf X\Pi=-\textbf X\). One can compute that the momentum operator \(\textbf P\) conjugates via the parity operator \(\Pi\) in exactly the same manner as the position operator \(\textbf X\), namely \(\Pi^{\dagger}\textbf P\Pi=-\textbf P\), so \(\textbf P\) is said to be a vector operator (the position operator \(\textbf X\) is also trivially a vector operator as the whole definition of vector operator is based on how \(\textbf X\) itself conjugates via \(\Pi\)). It follows that the orbital angular momentum \(\textbf L:=\textbf X\times\textbf P\) does not conjugate via \(\Pi\) in the same manner, but instead \(\Pi^{\dagger}\textbf L\Pi=\Pi^{\dagger}\textbf X\times\textbf P\Pi=(-\textbf X)\times(-\textbf P)=\textbf X\times\textbf P=\textbf L\) which can be taken to define a pseudovector operator (this is nothing new; in classical mechanics, if one views the dynamics of a system in a “mirror” (not the kind hanging on one’s wall, but really a sort of “inverting mirror”), the angular momentum vector \(\textbf L=\textbf x\times\textbf p\) is also a pseudovector precisely because both \(\textbf x\) and \(\textbf p\) are vectors and two wrongs make a right). The spin angular momentum \(\textbf S\) and the total angular momentum \(\textbf J\) are also pseudovector operators. There is of course also the notion of a scalar operator (e.g. the non-relativistic kinetic energy operator \(T:=\textbf P^2/2m\)) and a pseudoscalar operator (e.g. the dot product of a vector and a pseudovector operator \(\textbf P\cdot\textbf L\)) (and more generally tensor operators (especially spherical tensor operators as in the Wigner-Eckart theorem) and pseudotensor operators). The following table summarizes these ideas pretty well:
where always keep in mind that the reference is based on the position operator \(\textbf X\) transforming under spatial rotations as \(e^{i\Delta\boldsymbol{\phi}\cdot\textbf L/\hbar}\textbf Xe^{-i\Delta\boldsymbol{\phi}\cdot\textbf L/\hbar}\mapsto e^{\Delta\boldsymbol{\phi}\cdot\boldsymbol{\mathcal{L}}}\textbf X\) and under parity as \(\Pi^{\dagger}\textbf X\Pi=-\textbf X\).
Given that \(\Pi\in U(\mathcal H)\cap i\frak u\)\((\mathcal H)\), it follows that \(\Lambda_{\Pi}\subseteq U(1)\cap \textbf R=\{-1,1\}\) and indeed \(\Pi\)-eigenstates with definite parity \(+1\) are called even while those with \(-1\) are called odd.
We know that a Hamiltonian \(H\) is rotationally symmetric iff it fulfills the first criteria in the table above for being a scalar operator (and consequently conserves angular momentum). One can also consider situations in which the Hamiltonian \(H\) fulfills the second criteria (independent of whether or not it also fulfills the first), namely whether or not \([H,\Pi]=0\) (in which case parity is conserved in all processes governed by that Hamiltonian \(H\)). For instance, it is known that \([H_{\text{electromagnetism}},\Pi]=[H_{\text{strong interaction}},\Pi]=[H_{\text{gravity}},\Pi]=[H_{\text{Higgs}},\Pi]=0\) but a famous experiment of Wu et al. showed that \([H_{\text{weak interaction}},\Pi]\neq 0\) so parity is conserved in all electromagnetic or strong interactions (giving rise to spectroscopic selection rules, etc.) but not in weak interactions (e.g. parity conservation is violated in radioactive \(\beta\) decays).
For instance, the Hamiltonian of both the quantum harmonic oscillator and any central potential are automatically parity-conservative, so energy eigenspaces have definite parity; in the case of the former one can use ladder operators to show that the energy eigenstates \(|0\rangle,|1\rangle,…\) (as labelled by the eigenvalue of the number operator) have alternating parity \(\Pi|n\rangle=(-1)^n|n\rangle\) whereas in any central potential one has \(\Pi|n,\ell,m_{\ell}\rangle =(-1)^{\ell}|n,\ell,m_{\ell}\rangle\) as one can show by just looking at how the highest weight spherical harmonics \(Y_{\ell}^{\ell}(\theta,\phi)=e^{i\ell\phi}\sin^{\ell}\theta\) transform under \((\theta,\phi)\mapsto (\pi-\theta,\phi+\pi)\) and using the ladder operators \(L_{\pm}\) to prove the \(m_{\ell}\)-independence of the parity.
As an application of the above idea, if one has a system of two quantum particles interacting via a potential energy \(V=V(|\textbf X_1-\textbf X_2|)\), then by boosting into the center-of-mass frame it becomes clear that such a Hamiltonian will be rotationally symmetric with respect to their relative position \(\textbf X:=\textbf X_1-\textbf X_2\) and so the \(H\)-eigenstates of such a system will look like a radial part times a spherical harmonic. The parity operator on \(\mathcal H=\mathcal H_1\otimes\mathcal H_2\) is \(\Pi:=\Pi_1\otimes\Pi_2\), so if the composite state is in a \(\Pi\)-eigenstate \(|\pi\rangle=|\pi_1\rangle\otimes|\pi_2\rangle\), then \(\Pi|\pi\rangle=\Pi_1\otimes\Pi_2|\pi_1\rangle\otimes|\pi_2\rangle\) so in particular \(\pi=\pi_1\pi_2(-1)^{\ell}\) (not sure how fully airtight that argument is but the result is correct for the intrinsic parities which derive from quantum field theory).
Example: if a \(\pi^-\) pion scatters inelastically off a deuteron \(d^+=(p^+,n^0)\) to create two neutrons \(n^0\) so that \(\pi^-+d^+\to n^0+n^0\), then because the weak interaction is not involved, one must have conservation of parity \(\pi’=\pi\). But \(\pi=\pi_{\pi^-}\pi_{d^+}(-1)^{\ell_{(\pi^-,d^+)}}\) and \(\pi’=\pi^2_{n^0}(-1)^{\ell_{(n^0,n^0)}}\). Now if one accepts that \(\ell_{(\pi^-,d^+)}=0\), that \(\pi_{d^+}=\pi_{p^+}\pi_{n^0}(-1)^{\ell_{(p^+,n^0)}}\), that \(\pi_{p^+}=\pi_{n^0}\) due to an approximate, internal \(SU(2)\) isospin symmetry of the Standard Model, that \(\ell_{(p^+,n^0)}=0\) is in an “\(s\)-wave bound state”, and that \(\ell_{(n^0,n^0)}=1\) in order to conserve total angular momentum and respect the fermionic nature of the identical neutrons \(n^0\) created, then one deduces \(\pi_{\pi^-}=-1\) is a pseudoscalar particle (since it is given that \(s_{\pi^-}=0\)). Thus, this example demonstrates that parity conservation (together with angular momentum conservation in this case) can be a useful tool for deducing the intrinsic parities of hadrons in experiments.
Time Reversal
Again, begin with a discussion of classical mechanics first before moving onto the quantum mechanical case. Here, if one takes a trajectory \(\textbf x(t)\) and, rather than negating the output as was done for a parity transformation, negate the input instead to obtain the time-reversed trajectory \(\textbf x'(t)=\textbf x(-t)\). This correspondingly means that \(\textbf p'(t)=m\dot{\textbf x}'(t)=-\textbf p(-t)\) and \(\textbf L'(t)=-\textbf L(-t)\), etc. Note that electromagnetic fields also transform under time reversal \(\rho\mapsto\rho,\textbf J\mapsto -\textbf J,\textbf E\mapsto\textbf E,\textbf B\mapsto -\textbf B\).
Since the phrase “time reversal” and the Greek letter “theta” \(\Theta\) both start with the letter “t”, it seems customary to denote the quantum mechanical time reversal operator by \(\Theta\). One thing that’s immediately strange is that because the Schrodinger equation is first-order in time, it is basically like a heat/diffusion equation whose solution space is certainly not closed under “naive” time reversal \(|\psi(t)\rangle\mapsto |\psi(-t)\rangle\). Instead, thanks to the factor of \(i\) it turns out that the correct definition of the time reversal operator \(\Theta\) is:
\[\Theta|\psi(t)\rangle:=|\psi(-t)\rangle^*\]
Where one can now check that \(i\hbar\frac{\partial}{\partial t}\Theta\psi=H\Theta\psi\) is indeed a valid solution of the Schrodinger equation. One can check that this definition ensures that it behaves in accordance with classical time reversal, for instance:
\[\]
For all the training we’ve done with linear algebra, here it will be necessary to work with antilinear algebra (the two coincide over real vector spaces, but unfortunately Hilbert spaces are complex in QM).