Propagator Operator: Clarifying H Acting & Function of (t-t')

In summary, the conversation is discussing the formula for the propagator, U(t,t'), and how the Hamiltonian acts on it. There is some confusion about whether the Hamiltonian should be a function of (t-t') or not, but ultimately it is determined that the formula is correct as written and there is no need to be careful with the ordering of the operator and time-dependence. The derivation only applies to time-independent Hamiltonians.
  • #1
jewbinson
127
0
So in the attachment, in fact (6), the formula for the propagator rectangled in red...

is the Hamiltonian ACTING on (t-t')?

is the Hamiltonian a function of (t-t')?

or should it be (this is what I think), to be more clear

U(t,t') = exp((-i/h)(t-t')H), so that when acting on a state |psi>, we have

U(t,t')|psi> = exp((-i/h)(t-t')H|psi>)

where H operates on whatever comes next. The last one makes most sense to me because U is overall an operator, so the RHS should also be an operator.

Unless it means like this:

U(t,t')|psi> = exp((-i/h)H[(t-t')|psi>]) ?

I'm really not sure which one...
 

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  • #2
It is just the Hamiltonian multiplied by (t-t'), so it is what you think it is. Here, you don't have to be careful with the ordering of the operator and the time-dependence, because the Hamiltonian never operates on time. Time and the Hamiltonian always commute.

Also keep in mind that this derivation applies only to time-independent Hamiltonians.
 

1. What is a propagator operator?

A propagator operator is a mathematical tool used in quantum mechanics to calculate the probability of a particle transitioning from one state to another over a given period of time. It takes into account the time evolution of a quantum system.

2. How does the propagator operator work?

The propagator operator, also known as the Green's function, acts on the wave function of a quantum system at two different times (t and t') and calculates the probability amplitude of the system transitioning from the initial state at t' to the final state at t. It is represented by the symbol H(t-t').

3. What is the function of the propagator operator?

The propagator operator is used to solve the Schrödinger equation, which describes the time evolution of a quantum system. It allows us to calculate the probability of a particle transitioning from one state to another over a specific period of time.

4. How is the propagator operator related to the Hamiltonian operator?

The Hamiltonian operator, represented by the symbol H, is the total energy of a quantum system. The propagator operator is derived from the Hamiltonian operator and is used to calculate the time evolution of the system.

5. Can the propagator operator be used for any quantum system?

Yes, the propagator operator can be used for any quantum system as long as the Hamiltonian operator is known. It is a fundamental tool in quantum mechanics and is used to study a wide range of physical phenomena, from the behavior of elementary particles to the properties of atoms and molecules.

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