I have a confusion regarding expressing operators as projectors in Schrodinger and Heisenberg pictures. Please help.(adsbygoogle = window.adsbygoogle || []).push({});

Consider a two-state system with |1> and |2>

We know that e.g. a raising operator can be expressed as: [itex]\hat{\sigma}_+=|2><1|[/itex]

But here's my line of thought:

In theSchrodinger picture:

[itex]\hat{\sigma}_+[/itex] is supposed to remain constant in time, while the two stationary states evolve as:

[itex]|1(t)>=e^{-\frac{iE_1 t}{\hbar}}|1(0)>[/itex] and [itex]|2(t)>=e^{-\frac{iE_2 t}{\hbar}}|2(0)>[/itex]

But this seems to suggest that [itex]\hat{\sigma}_+(t) = e^{-\frac{i(E_2-E_1)t}{\hbar}}\hat{\sigma}_+(0)[/itex], so the operator seems to be evolving, which it shouldn't be.

Similarly in theHeisenberg picture:

From the Heisenberg equation of motion we expect:

[itex]\hat{\sigma}_+(t) = e^{\frac{i(E_2-E_1)t}{\hbar}}\hat{\sigma}_+(0)[/itex]

And |1> and |2> are expected to be constant.

But if so, then the above equation states that:

[itex]|2><1| = e^{\frac{i(E_2-E_1)t}{\hbar}} |2><1| [/itex]

Which is paradoxical.

Where am I making a mistake?

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# Time evolution of operators as projectors - confusion

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