Action of the time reversal operator on the QM wave equation

[qtm912]

Homework Statement

Apply the time reversal operator T to the following plane wave equation: Ψ = exp [i (kx - Et)]

Relevant Equations

T[f(t)] = T[f(-t)]

Applying the time reversal operator to the plane wave equation: Ψ = exp [i (kx - Et)]

T[Ψ ] = T{exp [i (kx - Et)]} = exp [i (kx + Et)]

This looks straightforward as I have simply applied the 'relevant equation' however my doubt is in relation to the possible action of operator T on the i in the exponent with which I have assumed the T operator commutes as in this case the i seems to be just a number. But elsewhere in my QM course it was proven that T acting on i switches its sign so it anti commutes. Applying this to the equation above it would lead instead to the result :

T[Ψ ] = T{exp [- i (kx + Et)]}

So in the alternative formulation the T operator acts by setting both : t -> -t and i -> -i

Which of these is correct and if both are wrong what is the right answer?

On a related point, if say 'i ' and 't ' appear together as a product in some quantum mechanical expression, following the latter prescription by switching the signs of the product 'i* t' would leave that expression invariant which would imply time reversal invariance wherever they appeared together in such a form. This was another source of doubt in my mind though I suppose some physical quantities are time reversal invariant and it may be a perfectly ok result.

Related question : it was actually shown that T and i anticommute - but this requires us to treat i as a matrix iI and write TiI T^-1

I am therefore quite confused about how to think about all of this.

 

[BigJDubs]

Any clarification would be greatly appreciated.The correct answer is that T[Ψ ] = exp [i (kx + Et)]. The time reversal operator only changes the sign of the time-dependent part of the wave function, not the imaginary unit i. The reason it is possible to change the sign of the time-dependent part without changing the sign of the imaginary unit is because the time-dependent part is multiplied by i, which does not change when the sign of the time-dependent part is changed. As such, the time reversal operator only affects the time-dependent part of the wave function, not the imaginary unit i.