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In Heisenberg picture the operators evolve in time instead of states, which evolve in time in Schrödinger picture. But this also means that to solve the equations (at least numerically) one has to work with $N^2$ numbers instead of Schrödinger's $N$ (i.e. Hamiltonian matrix vs state vector).

So, why, despite being dramatically more inefficient for calculations, is Heisenberg picture still used?

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    $\begingroup$ Note that the hamiltonian is still Hermitean. So, before you even start to consider degrees of freedom, you should consider that you're not really manipulating $N^{2}$ numbers. $\endgroup$
    – Zo the Relativist
    Commented May 7, 2014 at 15:33
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    $\begingroup$ @JerrySchirmer well, $\frac12N^2$. Still $\mathcal O(N^2)$. $\endgroup$
    – Ruslan
    Commented May 7, 2014 at 19:01
  • $\begingroup$ well, consider that the hamiltonian always has a complete set of eigenvectors. How many independent components does it have when the hilbert space is expressed in this eigenbasis? $\endgroup$
    – Zo the Relativist
    Commented May 7, 2014 at 19:15
  • $\begingroup$ Well, if we always knew the eigenbasis, we'd have nothing to solve in many cases. $\endgroup$
    – Ruslan
    Commented May 7, 2014 at 19:27

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There are a couple of types of situation in which the Schrodinger picture can be problematic. If you are working in a relativistic context you might not want to be tied down to one particular reference frame in which the states are a function of time and extend across all of space.

The other situation in which the Schrodinger picture is problematic is any situation in which you don't understand what's going on. It is almost always easier to tell how a system has evolved in the Heisenberg picture than in the Schrodinger picture. In the Heisenberg picture you can pick a set of representative observables that span the vector space of operators that can act on the system under consideration, including the unitary operators. From how that set of observables changes you can usually reconstruct what unitary operator acted on them. So the Heisenberg picture observables contain more information than the Schrodinger picture state. So, for example, entangled quantum systems instantiate locally inaccessible information in that the observables can be dependent on a parameter but their expectation values are not. This helps explain correlations between the results of measurement on entangled quantum systems

http://xxx.lanl.gov/abs/quant-ph/9906007

http://arxiv.org/abs/1109.6223.

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