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## Main Question or Discussion Point

Since all the observables in QM is on the form

[tex]\langle \alpha |A| \beta \rangle[/tex]

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

[tex] A \to A' = UAU^{-1} \ \ \ |\beta \rangle \to |\beta '\rangle = U |\beta \rangle[/tex]

where U is a unitary transformatioin. These descriptions of the theory is equivalent because

[tex]\langle \alpha' |A'| \beta' \rangle = \langle \alpha |U^{-1} U A U U^{-1}| \beta \rangle = \langle \alpha |A| \beta \rangle.[/tex]

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

[tex] H = H' = UHU^{-1} + i\hbar \frac{dU}{dt} U^{-1}[/tex]

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

[tex] \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.[/tex]

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?

[tex]\langle \alpha |A| \beta \rangle[/tex]

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

[tex] A \to A' = UAU^{-1} \ \ \ |\beta \rangle \to |\beta '\rangle = U |\beta \rangle[/tex]

where U is a unitary transformatioin. These descriptions of the theory is equivalent because

[tex]\langle \alpha' |A'| \beta' \rangle = \langle \alpha |U^{-1} U A U U^{-1}| \beta \rangle = \langle \alpha |A| \beta \rangle.[/tex]

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

[tex] H = H' = UHU^{-1} + i\hbar \frac{dU}{dt} U^{-1}[/tex]

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

[tex] \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.[/tex]

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?