- #1

tom.stoer

Science Advisor

- 5,766

- 160

## Main Question or Discussion Point

Can you please tell me were I am wrong?

I define the entropy (as usual) as

[tex]S[\rho] = -k_B \text{tr}(\rho\ln\rho)[/tex]

The time development of the density operator is

[tex]\rho(t) = U(t)\rho(0) U^\dagger(t)[/tex]

That means

[tex]S[\rho(t)] = S[U(t)\rho(0) U^\dagger(t)] = S[\rho(0)][/tex]

where I used the invariance of any trace of the density operator w.r.t. a general unitary transformation (here: time translation)

But that means that the entropy my system is always constant

[tex]\frac{dS}{dt} = 0[/tex]

Where is my mistake?

I define the entropy (as usual) as

[tex]S[\rho] = -k_B \text{tr}(\rho\ln\rho)[/tex]

The time development of the density operator is

[tex]\rho(t) = U(t)\rho(0) U^\dagger(t)[/tex]

That means

[tex]S[\rho(t)] = S[U(t)\rho(0) U^\dagger(t)] = S[\rho(0)][/tex]

where I used the invariance of any trace of the density operator w.r.t. a general unitary transformation (here: time translation)

But that means that the entropy my system is always constant

[tex]\frac{dS}{dt} = 0[/tex]

Where is my mistake?