Wednesday, 22 May 2019

Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?


For an observable A and a Hamiltonian H, Wikipedia gives the time evolution equation for A(t)=eiHt/AeiHt/ in the Heisenberg picture as


ddtA(t)=i[H,A]+At.


From their derivation it sure looks like At is supposed to be the derivative of the original operator A with respect to t and ddtA(t) is the derivative of the transformed operator. However, the Wikipedia derivation then goes on to say that At is the derivative with respect to time of the transformed operator. But if that's true, then what does ddtA(t) mean? Or is that just a mistake?


(I need to know which term to get rid of if A is time-independent in the Schrodinger picture. I think it's At but you can never be too sure of these things.)



Answer



There is no mistake on the Wikipedia page and all the equations and statements are consistent with each other. In AHeis.(t)=eiHt/AeiHt/

the letter A in the middle of the product represents the Schrödinger picture operator A=ASchr. that is not evolving with time because in the Schrödinger picture, the dynamical evolution is guaranteed by the evolution of the state vector |ψ.



However, this doesn't mean that the time derivative dASchr./dt=0. Instead, we have dASchr.dt=ASchr.t

Here, ASchr. is meant to be a function of xi,pj, and t. In most cases, there is no dependence of the Schrödinger picture operators on t - which we call an "explicit dependence" - but it is possible to consider a more general case in which this explicit dependence does exist (some terms in the energy, e.g. the electrostatic energy in an external field, may be naturally time-dependent).


In Schrödinger's picture, dxi,Schr./dt=0 and dpj,Schr./dt=0 which is why the total derivative of ASchr. with respect to time is given just by the partial derivative with respect to time. Imagine, for example, ASchr.(t)=c1x2+c2p2+c3(t)(xp+px)

We would have dASchr.(t)dt=c3(t)t(xp+px).
These Schrödinger's picture operators are called "untransformed" on that Wikipedia page. The transformed ones are the Heisenberg picture operators given by AHeis.(t)=eiHt/ASchr.(t)eiHt/
Their time derivative, dAHeis.(t)/dt, is more complicated. An easy differentiation gives exactly the formula involving [H,AHeis.] that you quoted as well. ddtAHeis.(t)=i[H,AHeis.(t)]+AHeis.(t)t.
The two terms in the commutator arise from the t-derivatives of the two exponentials in the formula for the Heisenberg AHeis.(t) while the partial derivative arises from dASchr./dt we have always had. (These simple equations remain this simple even for a time-dependent ASchr.; however, we have to assume that the total H is time-independent, otherwise all the equations would get more complicated.) The two exponentials on both sides never disappear by any kind of derivative, so obviously, all the appearances of A in the differential equation above are AHeis.. The displayed equation above is the (only) dynamical equation for the Heisenberg picture so it is self-contained and doesn't include any objects from other pictures.


In the Heisenberg picture, it is no longer the case that dxHeis.(t)/dt=0 (not!) and the similar identity fails for pHeis.(t) as well. AHeis.(t) is a general function of all the basic operators xi,Heis.(t) and pj,Heis.(t), as well as time t.


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