Simple quantum mech notation question

[Syrus]

Homework Statement

I see in chapters on perturbation theory and time-dependent variation, the Hamiltonian (usually expressed as H) is now printed as H(t). This is still the same Hamiltonian, correct? I assume this notation simply helps to signify that the total energy varies with time? If so, wouldn't the formal (and seemingly more revealing) expression be H(r,t)?

Similarly, in some instances, the hamiltonian is expressed as H(r). I assume this is to explicitly indicate that the total energy is a function of position.

Homework Equations

The Attempt at a Solution

 

[Oxvillian]

Remember what the Hamiltonian is - it's an operator living in a Hilbert space. As such it maps state vectors onto other state vectors. If it does this in a time-dependent way, then it's time dependent.

For example, an electron in a time-dependent electric field has a time-dependent Hamiltonian.

Spatial dependence makes less sense, however. The electron just has one Hamiltonian - it doesn't have a whole field of Hamiltonians depending on where it is. Maybe what you're thinking of is the matrix element of this one Hamiltonian in the position representation, <r'|H|r>.