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DanielC
DanielC
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Let's consider a quantum system in a state $\left| \psi \right>$ relative to an observer using a coordinate frame $F$. We now introduce a second observer, using a corrdinatecoordinate frame $F'$, related to $F$ by a rotation by an angle $\theta$ about some fixed axis (foras an example, the details of the transformation do not really matter). Now in quantum mechanics, we represent the transformation between those two reference frames by a unitary operator $\hat{U}$. This operator can be applied to the state $\left| \psi \right>$ to obtain the state of the system relative to $F'$. On the other hand, if $\hat{A}$ represents some observable relative to $F$, then $\hat{U}^{\dagger}\hat{A}\hat{U}$ represents the same observable relative to $F'$.

Now my question is:my question is: When changing frames, do we transform both the states and the observables, or do we only transofrmtransform one of those? Is it analogous to the SchrodingerSchrödinger and HeisnebergHeisenberg pictures of time evolution, where we choose whether it is the states or operators that evolve in time? If we transform only the states or only the operators, is the choice related to whether we view the transformation as active or passive?

Let's consider a quantum system in a state $\left| \psi \right>$ relative to an observer using a coordinate frame $F$. We now introduce a second observer, using a corrdinate frame $F'$, related to $F$ by a rotation by an angle $\theta$ about some fixed axis (for example, the details of the transformation do not really matter). Now in quantum mechanics, we represent the transformation between those two reference frames by a unitary operator $\hat{U}$. This operator can be applied to the state $\left| \psi \right>$ to obtain the state of the system relative to $F'$. On the other hand, if $\hat{A}$ represents some observable relative to $F$, then $\hat{U}^{\dagger}\hat{A}\hat{U}$ represents the same observable relative to $F'$.

Now my question is: When changing frames do we transform both the states and the observables, or do we only transofrm one of those? Is it analogous to the Schrodinger and Heisneberg pictures of time evolution, where we choose whether it is the states or operators that evolve in time? If we transform only the states or only the operators, is the choice related to whether we view the transformation as active or passive?

Let's consider a quantum system in a state $\left| \psi \right>$ relative to an observer using a coordinate frame $F$. We now introduce a second observer, using a coordinate frame $F'$, related to $F$ by a rotation by an angle $\theta$ about some fixed axis (as an example, the details of the transformation do not really matter). Now in quantum mechanics, we represent the transformation between those two reference frames by a unitary operator $\hat{U}$. This operator can be applied to the state $\left| \psi \right>$ to obtain the state of the system relative to $F'$. On the other hand, if $\hat{A}$ represents some observable relative to $F$, then $\hat{U}^{\dagger}\hat{A}\hat{U}$ represents the same observable relative to $F'$.

Now my question is: When changing frames, do we transform both the states and the observables, or do we only transform one of those? Is it analogous to the Schrödinger and Heisenberg pictures of time evolution, where we choose whether it is the states or operators that evolve in time? If we transform only the states or only the operators, is the choice related to whether we view the transformation as active or passive?

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Kuba
Kuba
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Symmetry transformations of states and operators

Let's consider a quantum system in a state $\left| \psi \right>$ relative to an observer using a coordinate frame $F$. We now introduce a second observer, using a corrdinate frame $F'$, related to $F$ by a rotation by an angle $\theta$ about some fixed axis (for example, the details of the transformation do not really matter). Now in quantum mechanics, we represent the transformation between those two reference frames by a unitary operator $\hat{U}$. This operator can be applied to the state $\left| \psi \right>$ to obtain the state of the system relative to $F'$. On the other hand, if $\hat{A}$ represents some observable relative to $F$, then $\hat{U}^{\dagger}\hat{A}\hat{U}$ represents the same observable relative to $F'$.

Now my question is: When changing frames do we transform both the states and the observables, or do we only transofrm one of those? Is it analogous to the Schrodinger and Heisneberg pictures of time evolution, where we choose whether it is the states or operators that evolve in time? If we transform only the states or only the operators, is the choice related to whether we view the transformation as active or passive?