QuantumPhysics.dvi

and may be viewed as a relativistic effect. For this reason, this operation is not normally

encountered in non-relativistic quantum mechanics, and we shall introduce it only later when

we discuss the Dirac equation.

Time-reversalreverses the direction of timet→−t. In classical mechanics, time reversal

leaves position and energy unchanged, but reverses the sign of momentum and angular mo-

mentum. In electrodynamics, the electric field and the electric charge density are unchanged,

but the sign of the magnetic field and the electric current are reversed. Under these transfor-

mations, the laws of mechanics and electrodynamics are then invariant under time reversal.

On the other hand, the fundamental laws of quantum mechanics are expressed via operators

acting on Hilbert space and the Schr ̈odinger equation. For example,

ih ̄

∂ψ

∂t

=−

̄h^2

2 m

∆ψ+V ψ (9.86)

This equation isnot invariantunderψ(t,x)→ψ(−t,x). What is needed in addition is the

operation of complex conjugation. The Schr ̈odinger equation is indeed invariant under

T : ψ(t,x)→ψ(−t,x)∗ (9.87)

Therefore, the operation of time reversal acts not by a linear transformation, but rather

by ananti-linear transformation, which involves an extra complex conjugation. We have

instead,

T|α〉=|α′〉 〈β′|α′〉=〈β|α〉∗

T|β〉=|β′〉 |〈β′|α′〉|^2 =|〈β|α〉|^2 (9.88)

Thus, probability is unchanged under time reversal, though the probability amplitude is

changed. As a result, we have

T(a|α〉) =a∗T|α〉 (9.89)

for any complex numbera.