References: Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 4.3.
The fourth postulate of non-relativistic quantum mechanics concerns how states evolve with time. The postulate simply states that in non-relativistic quantum mechanics, a state satisfies the Schrödinger equation:
where is the Hamiltonian, which is obtained from the classical Hamiltonian by means of the other postulates of quantum mechanics, namely that we replace all references to the position by the quantum position operator with matrix elements (in the basis) of
and all references to classical momentum by the momentum operator with matrix elements
In our earlier examination of the Schrödinger equation, we assumed that the Hamiltonian is independent of time, which allowed us to obtain an explicit expression for the propagator
The propagator is applied to the initial state to obtain the state at any future time :
What happens if , that is, there is an explicit time dependence in the Hamiltonian? The approach taken by Shankar is a bit hand-wavy, but goes as follows. We divide the time interval into small increments . To first order in , we can integrate 1 by taking the first order term in a Taylor expansion:
So far, we’ve been fairly precise, but now the hand-waving starts. We note that the term multiplying consists of the first two terms in the expansion of , so we state that to evolve from to , we multiply the initial state by . That is, we propose that
[The reason this is hand-waving is that there are many functions whose first order Taylor expansion matches , so it seems arbitrary to choose the exponential. I imagine the motivation is that in the time-independent case, the result reduces to 4.]
In any case, if we accept this, then we can iterate the process to evolve to later times. To get to , we have
The snag here is that we can’t, in general, combine the two exponentials into a single exponential by adding the exponents. This is because and will not, in general, commute, as the Baker-Campbell-Hausdorff formula tells us. For example, the time dependence of might be such that at , is a function of the position operator only, while at , becomes a function of the momentum operator only. Since and don’t commute, , so .
This means that the best we can usually do is to write
The propagator then becomes, in the limit
This limit is known as a time-ordered integral and is written as
One final note about the propagators. Since each term in the product is the exponential of times a Hermitian operator, each term is a unitary operator. Further, since the product of two unitary operators is still unitary, the propagator in the time-dependent case is a unitary operator.
We’ve defined a propagator as a unitary operator that carries a state from to some later time , but we can generalize the notation so that is a propagator that carries a state from to , that is
We can chain propagators together to get
Since the Hermitian conjugate of a unitary operator is its inverse, we have
We can combine this with 20 to get
That is, the Hermitian conjugate (or inverse) of a propagator carries a state ‘backwards in time’ to its starting point.