Re: operator satisfying relations [H,T] = i hbar
- From: Igor Khavkine <igor.kh@xxxxxxxxx>
- Date: Thu, 20 Dec 2007 22:45:55 +0000 (UTC)
On 2007-12-19, p.kinsler@xxxxxxxx <p.kinsler@xxxxxxxx> wrote:
mingyuming@xxxxxxxxx wrote:
In below, I quote John Baez's original argument from his website:
But a physically realistic Hamiltonian must be bounded below!
Hmm. I once got that remark thrown at me about the Hamiltonian for
the parametric oscillator, which being cubic, isn't bounded below.
Nevertheless it _is_ bounded below in the region where the approximations
used in its construction are valid; i.e. for low excitations. I consider
that physically realistic enough for practical purposes.
Similar remarks, about the boundedness from below of physically
realistic Hamiltonians, are made often. It should be pointed out,
though, that their validity strongly depends on what is meant by
"physically realistic".
Let me give an example. Think back to projectile calculations in
introductory physics. The only force present is a uniform and constant
acceleration due to gravity. This is none other than a linear potential.
This potential has no minimum, and the total energy is unbounded from
below. However, this observation doesn't seem to bother any of the
introductory physics students or their instructors, and rightly so.
The reason is that they are not looking for the minimum of the potential
or for stable/periodic particle motion. Instead, they are looking for
the trajectory of the projectile particle between leaving and hitting
the ground. This trajectory is part of a larger trajectory where the
ground is absent and the projectile comes from and escapes to infinity.
While this trajectory itself is not physically realistic, parts of it
are and there is no harm in studying the entire trajectory before
deciding which part of it is relevant.
Coming back to quantum mechanics, it's simple to formulate a
Hamiltonian for the analogous problem. It is again not bounded from
below. This implies that there is no ground state. However, there is
still an entire L^2 space of states, just none of them can claim to be
of minimal energy. The Hamiltonian can be extended to be self-adjoint
and a unitary evolution operator U(t) = exp(-iHt) can be defined. So,
even if one is considering a linear potential together with infinite
potential walls (such as the surface of the Earth), this evolution
operator may give a good approximation to the short time dynamics of a
wave packet distant from any of the potential walls. While this may not
be as simple as in classical mechanics, this use of U(t) is still
possible and I think falls into a physically realistic scenario (if only
as an approximation).
A similar discussion can be made for the inverted harmonic oscillator.
Igor
.
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