Question

In the topology online course by TU Delft, the time-reversal operator acting on a system of spin-1/2 particles is introduced as

τ=iσ_yK.
I understand this acts on the matrix H in
H=c Hc, and σ_y acts on the spin label.

My question is, does it have any significance that iσ_y is chosen? Why not σ_x or σ_y only? Does it matter?

Answer 1

The time-reversal operator τ=iσ_yK in quantum mechanics is specifically chosen for spin-1/2 particles to achieve the required properties of time reversal, specifically reversing the spin states and ensuring a return to the original state upon two successive applications.

Step-by-step explanation:

The significance of the time-reversal operator τ=iσ_yK when acting on a system of spin-1/2 particles lies in its ability to reverse the direction of time without affecting the spatial coordinates of the system. The choice of iσ_y (where σ_y is the Pauli spin matrix corresponding to the y-axis and i is the imaginary unit) is essential because it ensures that upon two successive applications of this operator, we get back the original state, as expected from a time-reversal operation (since time reversal squared should do nothing). This is due to the fact that the square of iσ_y is -1 (from the properties of Pauli matrices and the square of imaginary unit i), and hence when applied twice, it gives a minus sign that needs to be canceled by the K operator (complex conjugation operator) twice, which restores the original state.

The operator K alone would not reverse the spins, and using σ_x or σ_y without the imaginary unit would not yield the required minus sign upon two applications. Hence, the time-reversal operator τ=iσ_yK uniquely achieves the reversal of spins as well as the proper mathematical properties associated with time reversal in quantum mechanics.