- #211
- 24,488
- 15,033
I don't understand, where the problem might be. I was obviously wrong in trying to write down the operator for local thermal equilibrium. I'm even not sure anymore, whether it is possible to introduce such an idea, but first we should clarify, whether we agree on the general scheme. For simplicity let's work in the most simple example of a non-relativistic single particle in one dimension.
Let's write everything down in terms of a general picture. The time-dependence of the fundamental operators ##\hat{x}## and ##\hat{p}## are by some self-adjoint operator ##\hat{H}_0## that of the states by some self-adjoint operator ##\hat{H}_1##. Both can be also explictly time dependent, i.e., ##\hat{H}_j=\hat{H}_j(\hat{x},\hat{p},t)##, ##j \in\{0,1 \}## and the Hamiltonian is given by ##\hat{H}=\hat{H}_0+\hat{H}_1##.
The equations of motion for operators representing observables ##\hat{O}(\hat{x},\hat{p},t)## has a time dependence due to the time dependence of ##\hat{x}## and ##\hat{p}## and an explicit time dependence. ##\hat{x}## and ##\hat{p}## by definition have no explicit time-dependence. Thus we have
$$\mathrm{d}_t \hat{O}=\frac{1}{\mathrm{i} \hbar} [\hat{O},\hat{H}_0] + \partial_t \hat{O},$$
where ##\partial_t## refers only to the explicit time dependence. For the fundamental operators we have
$$\mathrm{d}_t \hat{x}=\frac{1}{\mathrm{i} \hbar} [\hat{x},\hat{H}_0], \\
\mathrm{d}_t \hat{p}=\frac{1}{\mathrm{i} \hbar} [\hat{p},\hat{H}_0].$$
The statistical operator, representing the state of the system is somewhat special. It's not special in fullfilling the equation of motion as any other operator,
$$\mathrm{d}_t \hat{\rho}=\frac{1}{\mathrm{i} \hbar} [\hat{\rho},\hat{H}_0]+\partial_t \hat{\rho}. \qquad (1)$$
On the other hand it must fulfill the von Neumann-Liouville equation,
$$\mathring{\hat{\rho}}=\frac{1}{\mathrm{i} \hbar} [\hat{\rho},\hat{H}] +\partial_t \hat{\rho}=0,$$
and this makes it special in some sense.
Combining the last two equations by eliminating ##\partial_t \hat{\rho}## in (1), we find
$$\mathrm{d}_t \hat{\rho} = \frac{1}{\mathrm{i} \hbar} [\hat{\rho},\hat{H}_0-\hat{H}] = -\frac{1}{\mathrm{i} \hbar} [\hat{\rho},\hat{H}_1].$$
This is the formalism in and arbitrary picture, usually called a Dirac picture. A often used special case is the interaction picture in QFT, where ##\hat{H}_0## is the quadratic part of the hamilton functional of field operators and ##\hat{H}_1=\hat{H}_I## is the part with higher-order monomials in the hamilton density, the "interaction part".
The special cases we discussed so far are
The Heisenberg picture:
$$\hat{H}_0=\hat{H}_{\text{H}} \; \Rightarrow \; \hat{H}_1=0.$$
Thus the statistical operator is constant in time (and thus explicitly time dependent except for equilibrium states!):
$$\mathrm{d}_t \hat{\rho}_{\text{H}}=0.$$
The time dependence of observables' operators is given by
$$\mathrm{d}_t \hat{O}_{\text{H}}=\frac{1}{\mathrm{i} \hbar} [\hat{O}_{\text{H}},\hat{H}_{\text{H}}]+\partial_t \hat{O}_{\text{H}}.$$
The Schrödinger picture:
$$\hat{H}_0=0 \; \Rightarrow \; \hat{H}_1=\hat{H}_{\text{S}}.$$
The time dependence on the statistical operator (state) is given by
$$\mathrm{d}_t \hat{\rho}_{\text{S}}=-\frac{1}{\mathrm{i} \hbar} [\hat{\rho}_{\text{S}},\hat{H}_{\text{S}}],$$
and that of observables' operators by
$$\mathrm{d}_t \hat{O}_{\text{S}}=\partial_t \hat{O}_{\text{S}}.$$
The equivalence of all pictures is proven by the fact that the transformation from one picture to another is given by a (time-dependent) unitary transformation.
So let's consider two pictures, labeled A and B. The time evolution for the operators of not explicitly time-dependent operators and that of the statistical operator are unitary transformations. We set withoug loss of generality the initial time (where the system is prepared) to ##t_0=0## and assume that both pictures coincide at this time.
Any not explicitly time-dependent observable operator fulfills the equation of motion as the fundamental operators. Let's take ##x##. For each picture ##j \in \{A,B \}## we have
$$\mathrm{d}_t \hat{x}^{(j)}=\frac{1}{\mathrm{i} \hbar} [\hat{x}^{(j)},\hat{H}_0^{(j)}].$$
This can be (at least formally) solved by a unitary time-evolution operator ##\hat{C}^{(j)}##, fulfilling the EoM
$$\mathrm{d}_t \hat{A}^{(j)}=\frac{\mathrm{i}}{\hbar} \hat{A}^{(j)}, \quad \hat{A}^{(j)}(0)=\hat{1}.$$
Then
$$\hat{x}^{(j)}(t)=\hat{A}^{(j)\dagger}(t) \hat{x}^{(j)}(0) \hat{A}^{(j)}(t).$$
The same holds true for the EoM of the statistical operator
$$\mathrm{d}_t \hat{\rho}^{(j)}=-\frac{1}{\mathrm{i} \hbar} [\hat{\rho}^{(j)},\hat{H}_1^{(j)}].$$
The corresponding time-evolution operator is defined by the EoM
$$\mathrm{d}_t \hat{C}^{(j)}=-\frac{\mathrm{i}}{\hbar} \hat{H}_1^{(j)} \hat{C}^{(j)}, \quad \hat{C}^{(j)}(0)=\hat{1}$$
and
$$\hat{\rho}^{(j)}(t)=\hat{C}^{(j)\dagger}(t) \hat{\rho}^{(j)}(0) \hat{C}^{(j)}(t).$$
Now the transformation from one picture to the other is found by
$$\hat{\rho}^{(B)}(t)=\hat{C}^{(B)\dagger}(t) \hat{\rho}^{(B)}(0) \hat{C}^{(B)}(t) = \hat{C}^{(B)\dagger}(t) \hat{\rho}^{(A)}(0) \hat{C}^{(B)}(t) = \hat{C}^{(B)\dagger}(t) \hat{C}^{(A) \dagger}(t) \hat{\rho}^{(A)}(t) \hat{C}^{(A) \dagger}(t) \hat{C}^{(B)}(t).$$
Thus the unitary transformation from picture A to picture B is given by
$$\hat{B}^{(AB)}(t)=\hat{C}^{(A) \dagger}(t) \hat{C}^{(B)}(t).$$
In the same way one derives from the EoM of the observable operators that this transformation should be given by the operator
$$\hat{D}^{(AB)}(t)=\hat{A}^{(A) \dagger}(t) \hat{A}^{(B)}(t).$$
To have both pictures equivlent we should have ##\hat{B}^{(AB)}=\hat{D}^{(AB)}##. This can be proven as follows: Define
$$\hat{G}^{(j)}=\hat{A}^{(j) \dagger} \hat{C}^{(j)}, \quad j \in \{A,B \}.$$
Then from the EoM of the time-evolution operators we get
$$\mathrm{d}_t \hat{G}^{(j)}=-\frac{\mathrm{i}}{\hbar} \hat{A}^{(j) \dagger} [\hat{H}_0^{(j)}(t)+\hat{H}_1^{(j)}(t)] \hat{C}^{(j)}(t) \\
=-\frac{\mathrm{i}}{\hbar} \hat{A}^{(j) \dagger} \hat{H}^{(j)}(t) \hat{C}^{(j)}(t) \\
=-\frac{\mathrm{i}}{\hbar} \hat{A}^{(j) \dagger} \hat{H}^{(j)}(t) \hat{A}^{(j)}(t) \hat{G}^{(j)}(t) \\
=-\hat{H}(0) \hat{G}^{(j)}(t).$$
Since also ##\hat{G}^{(j)}(0)=\hat{1}##, ##\hat{G}^{(j)}## fulfills the same EoM and initial condition for both ##j \in \{A,B\}## and thus
$$\hat{G}^{(A)}=\hat{G}^{(B)}$$
This implies
$$\hat{A}^{(A)\dagger} \hat{C}^{(A)} = \hat{A}^{(B)\dagger} \hat{C}^{(B)},$$
and thus after some algebra
$$\hat{B}^{(AB)}=\hat{C}^{(A) \dagger} \hat{C}^{(B)} = \hat{A}^{(A)} \hat{A}^{(B) \dagger} = \hat{D}.$$
In addition we have shown that the physically, i.e., measurable quantities, of QT are picture independent, because the time evolution probability to find the value ##o## when measuring the observable ##O## is governed by the picture-independent operator ##\hat{G}=\hat{G}^{(A)}=\hat{G}^{(B)}##. To see this let ##|o,\alpha \rangle## be a complete orthonormal set of eigenvectors of ##\hat{O}## with eigenvalue ##o##. Then the said probability is
$$P(o,t)={\sum_{\alpha}} {_{(j)}\langle o,\alpha,t|}\hat{\rho}^{(j)}(t)|\langle o,\alpha,t \rangle_{(j)} \\
=\sum_{\alpha} \langle o,\alpha,0|\hat{A}^{(j) \dagger}(t) \hat{C}^{(j)}(t) \hat{\rho}(0) \hat{C}^{(j) \dagger}(t) \hat{A}^{(j)}(t)|o,\alpha,0 \rangle \\
=\sum_{\alpha} \langle o,\alpha,0|\hat{G}(t) \hat{\rho}(0) \hat{G}^{\dagger}(t)|o,\alpha,0 \rangle.$$
From this one reads off that the picture-independent unitary time-evolution operator ##\hat{G}## can be interpreted as the time-evolution operator for the Schrödinger-picture ##\hat{\rho}^{(S)}## or ##\hat{G}^{\dagger}## as the time-evolution operator for the eigenstates of observable operators (and thus the observable operators themselves) in the Heisenberg picture.
Let's write everything down in terms of a general picture. The time-dependence of the fundamental operators ##\hat{x}## and ##\hat{p}## are by some self-adjoint operator ##\hat{H}_0## that of the states by some self-adjoint operator ##\hat{H}_1##. Both can be also explictly time dependent, i.e., ##\hat{H}_j=\hat{H}_j(\hat{x},\hat{p},t)##, ##j \in\{0,1 \}## and the Hamiltonian is given by ##\hat{H}=\hat{H}_0+\hat{H}_1##.
The equations of motion for operators representing observables ##\hat{O}(\hat{x},\hat{p},t)## has a time dependence due to the time dependence of ##\hat{x}## and ##\hat{p}## and an explicit time dependence. ##\hat{x}## and ##\hat{p}## by definition have no explicit time-dependence. Thus we have
$$\mathrm{d}_t \hat{O}=\frac{1}{\mathrm{i} \hbar} [\hat{O},\hat{H}_0] + \partial_t \hat{O},$$
where ##\partial_t## refers only to the explicit time dependence. For the fundamental operators we have
$$\mathrm{d}_t \hat{x}=\frac{1}{\mathrm{i} \hbar} [\hat{x},\hat{H}_0], \\
\mathrm{d}_t \hat{p}=\frac{1}{\mathrm{i} \hbar} [\hat{p},\hat{H}_0].$$
The statistical operator, representing the state of the system is somewhat special. It's not special in fullfilling the equation of motion as any other operator,
$$\mathrm{d}_t \hat{\rho}=\frac{1}{\mathrm{i} \hbar} [\hat{\rho},\hat{H}_0]+\partial_t \hat{\rho}. \qquad (1)$$
On the other hand it must fulfill the von Neumann-Liouville equation,
$$\mathring{\hat{\rho}}=\frac{1}{\mathrm{i} \hbar} [\hat{\rho},\hat{H}] +\partial_t \hat{\rho}=0,$$
and this makes it special in some sense.
Combining the last two equations by eliminating ##\partial_t \hat{\rho}## in (1), we find
$$\mathrm{d}_t \hat{\rho} = \frac{1}{\mathrm{i} \hbar} [\hat{\rho},\hat{H}_0-\hat{H}] = -\frac{1}{\mathrm{i} \hbar} [\hat{\rho},\hat{H}_1].$$
This is the formalism in and arbitrary picture, usually called a Dirac picture. A often used special case is the interaction picture in QFT, where ##\hat{H}_0## is the quadratic part of the hamilton functional of field operators and ##\hat{H}_1=\hat{H}_I## is the part with higher-order monomials in the hamilton density, the "interaction part".
The special cases we discussed so far are
The Heisenberg picture:
$$\hat{H}_0=\hat{H}_{\text{H}} \; \Rightarrow \; \hat{H}_1=0.$$
Thus the statistical operator is constant in time (and thus explicitly time dependent except for equilibrium states!):
$$\mathrm{d}_t \hat{\rho}_{\text{H}}=0.$$
The time dependence of observables' operators is given by
$$\mathrm{d}_t \hat{O}_{\text{H}}=\frac{1}{\mathrm{i} \hbar} [\hat{O}_{\text{H}},\hat{H}_{\text{H}}]+\partial_t \hat{O}_{\text{H}}.$$
The Schrödinger picture:
$$\hat{H}_0=0 \; \Rightarrow \; \hat{H}_1=\hat{H}_{\text{S}}.$$
The time dependence on the statistical operator (state) is given by
$$\mathrm{d}_t \hat{\rho}_{\text{S}}=-\frac{1}{\mathrm{i} \hbar} [\hat{\rho}_{\text{S}},\hat{H}_{\text{S}}],$$
and that of observables' operators by
$$\mathrm{d}_t \hat{O}_{\text{S}}=\partial_t \hat{O}_{\text{S}}.$$
The equivalence of all pictures is proven by the fact that the transformation from one picture to another is given by a (time-dependent) unitary transformation.
So let's consider two pictures, labeled A and B. The time evolution for the operators of not explicitly time-dependent operators and that of the statistical operator are unitary transformations. We set withoug loss of generality the initial time (where the system is prepared) to ##t_0=0## and assume that both pictures coincide at this time.
Any not explicitly time-dependent observable operator fulfills the equation of motion as the fundamental operators. Let's take ##x##. For each picture ##j \in \{A,B \}## we have
$$\mathrm{d}_t \hat{x}^{(j)}=\frac{1}{\mathrm{i} \hbar} [\hat{x}^{(j)},\hat{H}_0^{(j)}].$$
This can be (at least formally) solved by a unitary time-evolution operator ##\hat{C}^{(j)}##, fulfilling the EoM
$$\mathrm{d}_t \hat{A}^{(j)}=\frac{\mathrm{i}}{\hbar} \hat{A}^{(j)}, \quad \hat{A}^{(j)}(0)=\hat{1}.$$
Then
$$\hat{x}^{(j)}(t)=\hat{A}^{(j)\dagger}(t) \hat{x}^{(j)}(0) \hat{A}^{(j)}(t).$$
The same holds true for the EoM of the statistical operator
$$\mathrm{d}_t \hat{\rho}^{(j)}=-\frac{1}{\mathrm{i} \hbar} [\hat{\rho}^{(j)},\hat{H}_1^{(j)}].$$
The corresponding time-evolution operator is defined by the EoM
$$\mathrm{d}_t \hat{C}^{(j)}=-\frac{\mathrm{i}}{\hbar} \hat{H}_1^{(j)} \hat{C}^{(j)}, \quad \hat{C}^{(j)}(0)=\hat{1}$$
and
$$\hat{\rho}^{(j)}(t)=\hat{C}^{(j)\dagger}(t) \hat{\rho}^{(j)}(0) \hat{C}^{(j)}(t).$$
Now the transformation from one picture to the other is found by
$$\hat{\rho}^{(B)}(t)=\hat{C}^{(B)\dagger}(t) \hat{\rho}^{(B)}(0) \hat{C}^{(B)}(t) = \hat{C}^{(B)\dagger}(t) \hat{\rho}^{(A)}(0) \hat{C}^{(B)}(t) = \hat{C}^{(B)\dagger}(t) \hat{C}^{(A) \dagger}(t) \hat{\rho}^{(A)}(t) \hat{C}^{(A) \dagger}(t) \hat{C}^{(B)}(t).$$
Thus the unitary transformation from picture A to picture B is given by
$$\hat{B}^{(AB)}(t)=\hat{C}^{(A) \dagger}(t) \hat{C}^{(B)}(t).$$
In the same way one derives from the EoM of the observable operators that this transformation should be given by the operator
$$\hat{D}^{(AB)}(t)=\hat{A}^{(A) \dagger}(t) \hat{A}^{(B)}(t).$$
To have both pictures equivlent we should have ##\hat{B}^{(AB)}=\hat{D}^{(AB)}##. This can be proven as follows: Define
$$\hat{G}^{(j)}=\hat{A}^{(j) \dagger} \hat{C}^{(j)}, \quad j \in \{A,B \}.$$
Then from the EoM of the time-evolution operators we get
$$\mathrm{d}_t \hat{G}^{(j)}=-\frac{\mathrm{i}}{\hbar} \hat{A}^{(j) \dagger} [\hat{H}_0^{(j)}(t)+\hat{H}_1^{(j)}(t)] \hat{C}^{(j)}(t) \\
=-\frac{\mathrm{i}}{\hbar} \hat{A}^{(j) \dagger} \hat{H}^{(j)}(t) \hat{C}^{(j)}(t) \\
=-\frac{\mathrm{i}}{\hbar} \hat{A}^{(j) \dagger} \hat{H}^{(j)}(t) \hat{A}^{(j)}(t) \hat{G}^{(j)}(t) \\
=-\hat{H}(0) \hat{G}^{(j)}(t).$$
Since also ##\hat{G}^{(j)}(0)=\hat{1}##, ##\hat{G}^{(j)}## fulfills the same EoM and initial condition for both ##j \in \{A,B\}## and thus
$$\hat{G}^{(A)}=\hat{G}^{(B)}$$
This implies
$$\hat{A}^{(A)\dagger} \hat{C}^{(A)} = \hat{A}^{(B)\dagger} \hat{C}^{(B)},$$
and thus after some algebra
$$\hat{B}^{(AB)}=\hat{C}^{(A) \dagger} \hat{C}^{(B)} = \hat{A}^{(A)} \hat{A}^{(B) \dagger} = \hat{D}.$$
In addition we have shown that the physically, i.e., measurable quantities, of QT are picture independent, because the time evolution probability to find the value ##o## when measuring the observable ##O## is governed by the picture-independent operator ##\hat{G}=\hat{G}^{(A)}=\hat{G}^{(B)}##. To see this let ##|o,\alpha \rangle## be a complete orthonormal set of eigenvectors of ##\hat{O}## with eigenvalue ##o##. Then the said probability is
$$P(o,t)={\sum_{\alpha}} {_{(j)}\langle o,\alpha,t|}\hat{\rho}^{(j)}(t)|\langle o,\alpha,t \rangle_{(j)} \\
=\sum_{\alpha} \langle o,\alpha,0|\hat{A}^{(j) \dagger}(t) \hat{C}^{(j)}(t) \hat{\rho}(0) \hat{C}^{(j) \dagger}(t) \hat{A}^{(j)}(t)|o,\alpha,0 \rangle \\
=\sum_{\alpha} \langle o,\alpha,0|\hat{G}(t) \hat{\rho}(0) \hat{G}^{\dagger}(t)|o,\alpha,0 \rangle.$$
From this one reads off that the picture-independent unitary time-evolution operator ##\hat{G}## can be interpreted as the time-evolution operator for the Schrödinger-picture ##\hat{\rho}^{(S)}## or ##\hat{G}^{\dagger}## as the time-evolution operator for the eigenstates of observable operators (and thus the observable operators themselves) in the Heisenberg picture.