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Albfey
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In the Schrodinger picture, the operators don't change with the time, but the states do. So, what happen if my hamiltonian depend on time? Should I use the others pictures in these cases?
vanhees71 said:I think it's wrong in general since ##\hat{C}(t,t_0)## and ##\hat{H}(t)## do not commute in general.
vanhees71 said:In the Schrödinger picture all time dependence of operators that represent observables are by definition explict time dependence.
To be more specific, use as example a non-relativistic single particle. Then the fundamental set of operators are the position, momentum, and spin operators ##\hat{\vec{x}}##, ##\hat{\vec{p}}##, and ##\hat{\vec{\sigma}}##, all of which by definition do not explicitly depend on time.Albfey said:Yes, you are right, they don't comute in general. I confused it with another situation, I'm sorry.
I have already read it in a book, but I didn't understand what it means. The most books say that all operators are time-independent in this picture. So, how can they have an explicit dependence on time?I'm sorry for insisting on this question, I am really confused about pictures in quantum mechanics.
Maybe what the OP wants is to confirm that some version of the following claim:vanhees71 said:What should happen? You can (in principle) use any picture of time evolution you like for any Hamiltonian.
What do you mean by explicit? What does it mean to be implicit?vanhees71 said:In the Schrödinger picture all time dependence of operators that represent observables are by definition explict time dependence
Yes, the Schrodinger picture can be used for time-dependent Hamiltonians. In this picture, the state vector represents the time evolution of the quantum system, while the Hamiltonian remains constant. However, if the Hamiltonian is time-dependent, the state vector will also change with time.
The main difference between the Schrodinger picture and the Heisenberg picture is that in the Schrodinger picture, the state vector evolves with time, while the operators remain constant. In the Heisenberg picture, it is the operators that evolve with time, while the state vector remains constant.
In the Schrodinger picture, the expectation value of a time-dependent observable can be calculated by taking the inner product of the time-evolving state vector with the operator at a specific time. This allows for the calculation of time-dependent expectation values without explicitly considering the time-dependence of the operator.
Yes, the Schrodinger picture can be used for systems with time-dependent interactions. The time-dependence of the interactions can be incorporated into the Hamiltonian, and the state vector will evolve accordingly. However, the calculation of expectation values for time-dependent observables may become more complicated.
One advantage of using the Schrodinger picture for time-dependent systems is that it allows for a more intuitive understanding of the time evolution of the state vector. Additionally, it simplifies the calculation of expectation values for time-independent observables. However, for highly time-dependent systems, the Heisenberg picture may be more convenient to use.