Quantum Mechanics: Schrödinger vs Heisenberg picture

Pascal Szriftgiser¹ and Edgardo S. Cheb-Terrab² 

(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France

(2) Maplesoft

Within the Schrödinger picture of Quantum Mechanics, the time evolution of the state of a system, represented by a Ket , is determined by Schrödinger's equation:

where H, the Hamiltonian, as well as the quantum operators  representing observable quantities, are all time-independent.

Within the Heisenberg picture, a Ket  representing the state of the system does not evolve with time, but the operators representing observable quantities, and through them the Hamiltonian H, do.

Problem: Departing from Schrödinger's equation,

where in the right-hand-side we see the commutator of  with the Hamiltonian of the system.

Solution: Let  and  respectively be operators representing one and the same observable quantity in Schrödinger's and Heisenberg's pictures, and H be the operator representing the Hamiltonian of a physical system. All of these operators are Hermitian. So we start by setting up the framework for this problem accordingly, including that the time t and Planck's constant are real. To automatically combine powers of the same base (happening frequently in what follows) we also set combinepowersofsamebase = true. The following input/output was obtained using the latest Physics update (Aug/31/2016) distributed on the Maplesoft R&D Physics webpage.

Let's consider Schrödinger's equation

Now, H is time-independent, so (2) can be formally solved:  is obtained from the solution  at time , as follows:

To check that (4) is a solution of (2), substitute it in (2):

Next, to relate the Schrödinger and Heisenberg representations of an Hermitian operator O representing an observable physical quantity, recall that the value expected for this quantity at time t during a measurement is given by the mean value of the corresponding operator (i.e., bracketing it with the state of the system ).

So let  be an observable in the Schrödinger picture: its mean value is obtained by bracketing the operator with equation (4):

The composed operator within the bracket on the right-hand-side is the operator O in Heisenberg's picture,

Analogously, inverting this equation,

As an aside to the problem, we note from these two equations, and since the operator  is unitary (because H is Hermitian), that the switch between Schrödinger's and Heisenberg's pictures is accomplished through a unitary transformation.

Inserting now this value of  from (8) in the right-hand-side of (6), we get the answer to item a)

where, on the left-hand-side, the Ket representing the state of the system is evolving with time (Schrödinger's picture), while on the the right-hand-side the Ket is constant and it is , the operator representing an observable physical quantity, that evolves with time (Heisenberg picture). As expected, both pictures result in the same expected value for the physical quantity represented by .

To complete item b), the derivation of the evolution equation for , we take the time derivative of the equation (7):

To rewrite this equation in terms of the commutator  , it suffices to re-order the product  H   placing the exponential first:

Finally, to express the right-hand-side in terms of   instead of , we take the commutator of the equation (8) with the Hamiltonian

Combining these two expressions, we arrive at the expected result for b), the evolution equation of a given observable  in Heisenberg's picture