Suppose a Hamiltonian H is time reversal symmetric: $ΘHΘ^{-1}=H$. And H can be decomposed under a complete basis $|k,α⟩$, i.e. $H=∑_k∑_{α,β}|k,α⟩⟨k,β|$, and I am trying to deduce the corresponding form for the time reversal symmetry of $H(k)=∑_{α,β}|k,α⟩⟨k,β|$. The answer should be $θ·H(k)·θ^{-1}=H(-k)$. Yeah, this is a general result of condensed matter physics when treating Hamiltonian in momentum space.

Here I encountered the problem of how should I treat $⟨k,β|θ^{-1}$. If I just make it equal $⟨-k,β|$ the right result can be deduced. But I didn't feel safe when I did this, especially I know it conflicts with Sakurai's treatment of time reversal operator:

In fact, we do not even attempt to define ⟨β|Θ. This is one place where the Dirac bra-ket notation is a little confusing. After all, that notation was invented to handle linear operators, not antilinear operators.

-Modern Quantum Mechanics(2nd ed.) by Sakurai, p292

So is it safe to treat $⟨ϕ|θ^{-1}$ as the bra of $θ|ϕ⟩$? If not, how can I complete the deduction above in a safer way?

  • $\begingroup$ Leslie E. Ballentine mentions in his book Quantum Mechanics - A Modern Development that "Messiah (1966) allows antilinear operators to act either to the left or to the right, but as a consequence he must caution his readers that $(\langle\xi|A)|\phi\rangle\ne\langle\xi|(A|\phi\rangle)$, and hence the common expression $\langle\xi|A|\phi\rangle$ becomes undefined.". So Quantum Mechanics by Albert Messiah may be of interest to you. $\endgroup$ – ummg Oct 24 '20 at 1:18

The complication is the following. Normally, the bra $\langle\Theta\psi|$ (corresponding to the ket $\Theta|\psi\rangle$) would be $\langle\psi|\Theta^\dagger$. The adjoint $\Theta^\dagger$ of an operator $\Theta$ is usually defined by $$\langle\psi|\Theta^\dagger|\phi\rangle=\langle\Theta\psi|\phi\rangle=(\langle\phi|\Theta\psi\rangle)^*=(\langle\phi|\Theta|\psi\rangle)^*.$$ (I included a bunch of equivalent definitions for convenience). However, this definition of the adjoint does not work for anti-linear operators. This is because the left-hand side of this equation would be anti-linear in $\psi$ while the right-hand expressions would be linear in $\psi$. Instead, for anti-linear operators, we define the adjoint, which I will denote as $\Theta^T$, using $$\langle\psi|\Theta^T|\phi\rangle=\langle\phi|\Theta|\psi\rangle=\langle\phi|\Theta\psi\rangle=(\langle\Theta\psi|\phi\rangle)^*.$$ With this definition of the adjoint, the anti-unitary condition becomes $\Theta^T=\Theta^{-1}$. However, it's not completely clear what $\langle\Theta\psi|$ is. There is a relationship, but it's not as simple to see in the bra-ket notation. We have to use the completeness relation $\int d\phi|\phi\rangle\langle\phi|=1$. $$\langle\Theta\psi|=\int d\phi \langle\Theta\psi|\phi\rangle\langle\phi|=\int d\phi \Big(\langle\psi|\Theta^T|\phi\rangle\Big)^*\langle\phi|.$$ I'm not sure if this can be simplified further.


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