Exponential of Unbounded Operator
Martin Mucha
quantum mechanics. It is often written in various textbooks that
Hamiltonian is a generator of evolution operator:
U(t) = exp(iHt/hbar)
and if we have a system is state |psi(0)> than at later time t the system
will be in
|psi(t)> = U(t)|psi(0)>
Exponential of operator is said to be an infinite sum - Taylor
series, but in most cases (Hydrogen atom let say) the Hamiltonian
is unbounded, so how is then defined the infinite sum?
It is possible to solve this problem through the theory of semigroups
thinking of the set of operators {U(t)} where t is any positive parameter
as a semigroup generated by (unbounded) operator iH/hbar ? It would be
fine if there is some kind of Hille-Yosida theorem, that makes sure that
such a semigroup exists and is unique, but I think that in this case the
assumption of the theorem are not met.
Can anyone shed a light on the subject?
Thanks,
Martin Mucha
John Baez
Martin Mucha <mmuc...@mbox.troja.mff.cuni.cz> wrote:
>Exponential of operator is said to be an infinite sum - Taylor
>series, but in most cases (Hydrogen atom let say) the Hamiltonian
>is unbounded, so how is then defined the infinite sum?
It's not so easy to define exp(itH) via an infinite sum when H
is unbounded --- you can do it, but only after you've built up
some mathematical machinery.
>It is possible to solve this problem through the theory of semigroups
>thinking of the set of operators {U(t)} where t is any positive parameter
>as a semigroup generated by (unbounded) operator iH/hbar ? It would be
>fine if there is some kind of Hille-Yosida theorem, that makes sure that
>such a semigroup exists and is unique, but I think that in this case the
>assumption of the theorem are not met.
You want to learn Stone's theorem! Try section VIII.4 in Reed and
Simon's "Functional Analysis", which is the first volume of their
series "Methods of Modern Mathematical Physics". By the way, this
series is great if you're trying to understand the math of quantum
mechanics in a rigorous way. If you're really interested in the
Taylor series approach to exp(itH), try section X.6 of the second
volume. But don't try to read that until you know Stone's theorem!
Daniel Grubb
>I am a bit confused about the way unbounded operators are handled in
>quantum mechanics. It is often written in various textbooks that
>Hamiltonian is a generator of evolution operator:
> U(t) = exp(iHt/hbar)
>and if we have a system is state |psi(0)> than at later time t the system
>will be in
> |psi(t)> = U(t)|psi(0)>
>Exponential of operator is said to be an infinite sum - Taylor
>series, but in most cases (Hydrogen atom let say) the Hamiltonian
>is unbounded, so how is then defined the infinite sum?
Essentially, you have to do a spectral decomposition of H and
exponentiate on every 'eigenvalue'. More specifically, write
H=\int x dE(x)
where E is a spectral measure. Then
exp(iHt/hbar)=\int exp(ixt/hbar) dE(x).
This formula is a generalization of what happens in finite
dimensions, but relies on the original spectral decomposition,
which in turn requires some form of normality of H. Since H
is self-adjoint, this is fine.
Rudin's book 'Functional Analysis' gives a nice treatment of
this from a mathematicians point of view. I'm not sure how a
physicist would take it.
--Dan Grubb