1-D simple harmonic oscillator

[indigojoker]

I was just wondering what the difference was in the 1-D simple harmonic oscillator in the Heisenberg picture versus the Schrödinger picture?

 

[dextercioby]

What is generally the difference b/w the Heisenberg and the Schrödinger pictures ?

 

[indigojoker]

dextercioby, please correct me if I'm wrong but this is my understanding of the two pictures (involving time):

In the Schrödinger picture, the time evolution operator operates on the state vector and the dynamic variabs x and p do not change. So:

x->x
p->p
|a> -> U|a>

However, in the Heisenberg picture, the dynamic variables x and p changes, while the state vector stays the same:

let *=dagger

x(t)->U*xU
p(t)->U*pU
|a> -> |a>

i guess I'm confused as to how this related to the SHO?

 

[CompuChip]

The difference is if we tag the time dependence onto the states or onto the operators.
In one picture (Schrödinger) the operators are not time-dependent, and the states generally are. In the Heisenberg picture it's the other way around.
Of course, they're completely equivalent (when you calculate the physically relevant quantities like expectation values etc. you get the same answer in both), it's just that sometimes one is more efficient than the other.