Unitary equivalence of QM (not for energy?)

[center o bass]

Since all the observables in QM is on the form

$$\langle \alpha |A| \beta \rangle$$

where A is an observable, one can transform the observables and states like

$$A \to A' = UAU^{-1} \ \ \ |\beta \rangle \to |\beta '\rangle = U |\beta \rangle$$
where U is a unitary transformatioin. These descriptions of the theory is equivalent because

$$\langle \alpha' |A'| \beta' \rangle = \langle \alpha |U^{-1} U A U U^{-1}| \beta \rangle = \langle \alpha |A| \beta \rangle.$$

However by using the Schrödinger equation one can show that the Hamiltonian transforms like

$$H = H' = UHU^{-1} + i\hbar \frac{dU}{dt} U^{-1}$$

which means that the expectation value of H in the transformed representation is

$$\langle \psi'| H'|\psi '\rangle = \langle \psi| H \psi \rangle + i\hbar \langle \psi |U^{-1}\frac{dU}{dt} |\psi \rangle \neq \langle \psi| H \psi \rangle.$$

What is the meaning of this inequivalence? Is not the expectation value of H supposed to be equal in two descriptions which differ by a unitary transformation?

 

[IsometricPion]

I think the assumption is usually that U≠U(t). This seems intuitively reasonable since time-dependent rotations (a subset of U(t)), such as the one required to transform classically from an inertial frame to a rotating frame, do not generally preserve the (classical) energy (they don't even preserve Newton's laws).

 

[Dickfore]

However by using the Schrödinger equation one can show that the Hamiltonian transforms like

$$H = H' = UHU^{-1} + i\hbar \frac{dU}{dt} U^{-1}$$

Please show this.

 

[center o bass]

Please show this.

$$i\hbar \dot{|\psi '(t) \rangle } = i\hbar (\dot{U}(t) |\psi(t) \rangle + U \dot{|\psi(t)\rangle} = (UH + i\hbar \frac{dU}{dt} ) |\psi(t)\rangle = (UHU^{-1} ih \frac{dU}{dt}U^{-1})|\psi'(t)\rangle = H' |\psi'(t) \rangle$$

where I've used the Schrödinger equation in the second equality and in the third equality I've expressed the non-transformed state in terms of the transformed one.

 

[center o bass]

I think the assumption is usually that U≠U(t). This seems intuitively reasonable since time-dependent rotations (a subset of U(t)), such as the one required to transform classically from an inertial frame to a rotating frame, do not generally preserve the (classical) energy (they don't even preserve Newton's laws).

But then what about the Heisenberg picture? There certainly U=U(t), in fact it's the inverse of the time evolution operator. Is the energy of the system different in the Heisenberg picture?

 

[IsometricPion]

But then what about the Heisenberg picture? There certainly U=U(t), in fact it's the inverse of the time evolution operator. Is the energy of the system different in the Heisenberg picture?

I was thinking of the usual simple cases, such as rotational transformations. You are correct that some time-dependent unitary transformations do not change the expectation of H.

where I've used the Schrödinger equation in the second equality

This is not quite correct see Wikipedia-Heisenberg picture.

 

[Dickfore]

Ok, what you are doing reminds me of the derivation of the Berry phase. Basically, because you make a time-dependent similarity transformation, if the system described by the ket $\vert \Psi(t) \rangle$ is isolated (and its time-evolution is given by Schroedinger's equation through the Hamiltonian H), then the system described by $\vert \psi'(t) \rangle \equiv U(t) \, \vert \psi(t) \rangle$ is, in general, open, and its energy is not conserved (and it's evolution is described by another time-dependent Hamiltonian $H' = U \, H \, U^{\dagger} + i \, \hbar \, \dot{U} \, U^{\dagger}$).

 

[jfy4]

wait, I think you might be mixing up some time dependence and the derivatives... maybe
consider the following:
$$|\psi(t)\rangle = U(t,t_0)|\psi(t_0)\rangle$$
Now if we re-do this derivation watching the $t$s carefully, we get a different result
$$\begin{align} i\hbar \frac{d}{dt}|\psi (t)\rangle &= i\hbar \frac{d}{dt}U(t,t_0)|\psi (t_0)\rangle =i\hbar \dot{U}(t,t_0)|\psi (t_0)\rangle = i\hbar \dot{U}(t,t_0)U^{-1}U|\psi (t_0)\rangle \\ &= i\hbar \dot{U}U^{-1}|\psi (t)\rangle = H|\psi (t)\rangle \end{align}$$
because the time derivative doesn't hit both the $U$ and the state. then
$$i\hbar \frac{d U}{dt} = HU$$
which is normal and I imagine $UHU^{-1}$ still holds... but I might be mixed up.

 

[center o bass]

wait, I think you might be mixing up some time dependence and the derivatives... maybe
consider the following:
$$|\psi(t)\rangle = U(t,t_0)|\psi(t_0)\rangle$$
Now if we re-do this derivation watching the $t$s carefully, we get a different result
$$\begin{align} i\hbar \frac{d}{dt}|\psi (t)\rangle &= i\hbar \frac{d}{dt}U(t,t_0)|\psi (t_0)\rangle =i\hbar \dot{U}(t,t_0)|\psi (t_0)\rangle = i\hbar \dot{U}(t,t_0)U^{-1}U|\psi (t_0)\rangle \\ &= i\hbar \dot{U}U^{-1}|\psi (t)\rangle = H|\psi (t)\rangle \end{align}$$
because the time derivative doesn't hit both the $U$ and the state. then
$$i\hbar \frac{d U}{dt} = HU$$
which is normal and I imagine $UHU^{-1}$ still holds... but I might be mixed up.

If I understand you correctly it seems like you've assumed that U is the time evolution operator, but I meant a general time dependent, unitary transformation.

 

[center o bass]

I was thinking of the usual simple cases, such as rotational transformations. You are correct that some time-dependent unitary transformations do not change the expectation of H.This is not quite correct see Wikipedia-Heisenberg picture.

I think it is, I did use the fact that for a state in the Schrödinger picture
$$i\hbar \dot{|\psi(t)\rangle }= H|\psi(t)\rangle.$$

I have to agree with your explanation though. Thanks :)

 

[IsometricPion]

Another way to derive what you did is:$$[U,H]=i\hbar{}\frac{dU}{dt}\Rightarrow{}[U,H]U^{-1}=UHU^{-1}-HUU^{-1}=UHU^{-1}-H=i\hbar{}\frac{dU}{dt}U^{-1}\Rightarrow{}UHU^{-1}=i\hbar{}\frac{dU}{dt}U^{-1}+H$$. However, this derivation is only correct for U that do not explicitly depend on time. Per the wikipedia page, in general $[H,A]=-i\hbar\frac{dA}{dt}+i\hbar\frac{\partial{A}}{ \partial{t}}$. The correspoding result being: $$UHU^{-1}=i\hbar{}(\frac{dU}{dt}-\frac{\partial{U}}{ \partial{t}})U^{-1}+H$$. When U is the time-evolution operator it so happens that $\frac{dU}{dt}=\frac{\partial{U}}{ \partial{t}}$, so the equation reduces to UHU^-1=H. This is simply a statement that H commutes with the time-evolution operator, something already known to be true.