## Does this equation mean anything to you?

• Imagine you are sipping tea or coffee while discussing various issues with a broad and diverse network of students, colleagues, and friends brought together by the common bond of physics, graduate school, and the physics GRE.

Izaac
Posts: 63
Joined: Tue Mar 20, 2012 2:24 am

### Does this equation mean anything to you?

$A(t)=e^{+iHt/\hbar}Ae^{-iHt/\hbar}$

Srednicki begins his textbook on QFT by a handful of equations ("In order to prepare for QFT, you should recognize these:"). All of them I grasp but that one.
Obviously it smells like QM, but I've never seen that exact equation, whatever the A and H mean. Anyway it seems to reduce to just $A(t)=A$, which is even more confusing.
So, anyone recognize something familiar in that equation? Thanks in advance.

PS: I'm pretty sure the answer is so simple as to mark me "noobie" up to the thousand and first generation...

Posts: 19
Joined: Tue Feb 12, 2013 7:52 pm

### Re: Does this equation mean anything to you?

http://en.wikipedia.org/wiki/Heisenberg_picture

You should probably brush up on QM and Classical Mechanics before starting in on QFT. A is just a general operator, H is the Hamiltonian. It does not necessarily reduce to A(t) = A if A does not commute with H.

Izaac
Posts: 63
Joined: Tue Mar 20, 2012 2:24 am

### Re: Does this equation mean anything to you?

Oh! Thanks, now I see. Amusingly, it was discretely mentioned in the book I'm using (Griffiths) on a smallish note bottom of page, without much development.

bfollinprm
Posts: 1203
Joined: Sat Nov 07, 2009 11:44 am

### Re: Does this equation mean anything to you?

$A = A(0)$

Then

$A(t)|{\phi}> = e^{iHt}Ae^{-iHt}|\phi>$

means (from right to left), in the Schrodinger picture you probably learned,

1. Devolve $|\phi>$ to its state at time $t = 0$

2. Evaluate $A$ acting on $|\phi>$ there.

3. Evolve $|\phi>$ back to $t = t$. This should be the same as $A(t)|\phi>$

This is only trivial if $A$ is time-translation invariant (e.g. is the same for all time).