Subject: Re: CCL:time-dependent and time-independent QM

From: hinsen/at/cnrs-orleans.fr
Date: Mon, 12 Jul 2004 20:17:43 +0200

On Jul 12, 2004, at 16:57, Shobe, David wrote:

> A possibly related question: there are two versions of the 
> Hamiltonian, the time-dependent and the time-independent.  The former 
> represents the evolution of a dynamic system; the latter represents 
> the steady states of the system, and is the Hamiltonian used in most 
> quantum-chemistry problems.  I've never really understood the 
> relationship between the time-dependent and time-independent 
> equations, especially since this is where all the "weirdness" of QM 
> comes into play.

The Hamiltonian is the same in the time-dependent and the 
time-independent Schroedinger equations. The time-dependent one is the 
starting point, it describes the time evolution of an arbitrary quantum 
state of a system. The time-independent equation results if one then 
asks which quantum states have no time evolution (other than a variable 
phase factor). The interest in these states is twofold:

1) They describe the stable states, which one is often interested in.
2) They contain all the time-dependent information as well, because the
    stable states (eigenstates of the Hamiltonian) are a basis of state 
space.
    For an arbitrary initial quantum state, calculate the projections on 
the
    stable states. Then multiply each projection with the (trivial) 
time-dependent
    phase factor and the corresponding stable state. Sum up everything, 
and you
    have the time evolution of the initial state.

Because of 2), the time-independent equation can be considered a 
particular approach to solve the time-dependent equation.

> Wouldn't the isomers simply be different solutions to the same non-B-O 
> Hamiltonian, just as the various states of a molecule are different 
> solutions?  (By "the various states" I mean the ones that within the 
> B-O approx. are labeled vibrational, rotational, and electronic).

Assuming that by "solution to a Hamiltonian" you mean its eigenstates, 
no, not in general, though often to a good approximation.

This leads back to the original question by Eric Scerri. I am not 
familiar with the articles he refers to, but I don't see why the B-O 
approximation would be crucial. With or without B-O, quantum mechanics 
gives the probability distribution of the atomic configurations. One 
can then search for well-defined maxima and call them "molecular 
structures", but that is an additional interpretation step made for 
convenience of discussion only. I suppose that all chemists are aware 
that molecules are not static, the atoms move around all the time. So 
"molecular structure" is a concept introduced as a convenience, it 
cannot be deduced strictly from quantum mechanics.

---------------------------------------------------------------------
Konrad Hinsen
Laboratoire Lion Brillouin, CEA Saclay,
91191 Gif-sur-Yvette Cedex, France
Tel.: +33-1 69 08 79 25
Fax: +33-1 69 08 82 61
E-Mail: hinsen/at/llb.saclay.cea.fr
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From: "Shobe, David" <dshobe^at^sud-chemieinc.com>
Date: Mon, 12 Jul 2004 16:14:28 -0400

Where does the part where measuring a property of the system "resets" the quantum state of the system come in?   That's always been mysterious: how does the system "know" it's being measured?  Does the uncertainty principle rest on an assumption that one cannot make a measurement w/o temporarily adding one or more other particles to the system?  And can you really do that temporarily, or are you permanently enlarging the system when you add particles?  

Another problem I have with understanding the time dependent Schroedinger equation is that it's a differential equation, so you need initial/boundary conditions to solve it, right?  But the postulates imply that you can't know all the initial conditions...or can the initial "psi-0" function incorporate both what's known about the initial state and what's unknown?

--David Shobe, Ph.D., M.L.S.
S|d-Chemie, Inc.
phone (502) 634-7409
fax (502) 634-7724

Don't bother flaming me: I'm behind a firewall.

-----Original Message-----
From: hinsen^at^cnrs-orleans.fr [mailto:hinsen^at^cnrs-orleans.fr]
Sent: Monday, July 12, 2004 2:18 PM
To: Shobe, David
Cc: CCL computational chemistry list (E-mail); scerri^at^chem.ucla.edu
Subject: Re: CCL:time-dependent and time-independent QM

---------------------------------------------------------------------
Konrad Hinsen
Laboratoire Lion Brillouin, CEA Saclay,
91191 Gif-sur-Yvette Cedex, France
Tel.: +33-1 69 08 79 25
Fax: +33-1 69 08 82 61
E-Mail: hinsen^at^llb.saclay.cea.fr
---------------------------------------------------------------------

---

From: hinsen!at!cnrs-orleans.fr
Date: Mon, 12 Jul 2004 22:52:04 +0200

On 12.07.2004, at 22:14, Shobe, David wrote:

> Where does the part where measuring a property of the system "resets"  
> the quantum state of the system come in?   That's always been  
> mysterious:

That depends on who you ask, I think that discussion is still  
open-ended. The "collapse of the wave function" is part of the original  
Copenhagen interpretation of the meaning of the wave function. However,  
there are other interpretations. As the term "interpretation" implies,  
the choice of one does not change the maths - the equations and wave  
functions remain the same.

The real problem in that discussion is that "measurement" needs to be  
defined. In a pure quantum world, there are particles interacting, all  
subject to the same laws, and the whole universe is governed by a  
single huge wave function. No measurements, no problems. Various  
definitions of measurements have been proposed, but none has yet been  
found to be without problems. Some say that measuring implies human  
consciousness - hence the paradox known as "Schrvdinger's cat". Others  
say that measurement is an interaction with a classical object, but  
that would imply that quantum mechanics needs to be defined in terms of  
its own limiting case, classical mechanics.

>  how does the system "know" it's being measured?  Does the uncertainty  
> principle rest on an assumption that one cannot make a measurement w/o  
> temporarily adding one or more other particles to the system?  And can  
> you

The strict mathematical formulation of the uncertainty principle is in  
terms of wave functions, thus independent of any one interpretation.  
Many formulations in plain language depend on a choice of  
interpretation, however.

> Another problem I have with understanding the time dependent  
> Schroedinger equation is that it's a differential equation, so you  
> need initial/boundary conditions to solve it, right?  But the  
> postulates imply that you

Yes, you need to know the initial quantum state.

>  can't know all the initial conditions...or can the initial "psi-0"  
> function incorporate both what's known about the initial state and  
> what's unknown?

Right. That's again part of the interpretation problem. But not the  
worst in practice, as the question of time propagation of a given  
quantum state rarely occurs in real-life questions. That is perhaps  
another reason why the time-independent equation is so much more used  
in practice.

------------------------------------------------------------------------ 
-------
Konrad Hinsen
Laboratoire Leon Brillouin, CEA Saclay,
91191 Gif-sur-Yvette Cedex, France
Tel.: +33-1 69 08 79 25
Fax: +33-1 69 08 82 61
E-Mail: hinsen!at!llb.saclay.cea.fr
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