From Heisenberg to Superposition states

[skynelson]

Hi All,
I am trying to remember the logical argument that leads from Heisenberg's uncertainty principle to the existence of quantum superposition states.

Here's my sketchy version:

1) postulate of Quantization leads to non-commuting operators
2) This leads to Heisenberg Unc. Principle, and the concept of wave-like probability distribution (wave packet)
3) This leads to the evolution of the wave packet over time (Schrödinger Equation)
4) this leads to the concept of multiple simultaneous solutions to the Sch Equation (such as an infinite collection of solutions to the harmonic oscillator, each one representing a different possible energy level)

And there we have arrived at superposition states. Can someone revise this if it is incorrect?

 

[tom.stoer]

I would state it differently

1) QM uses a Hilbert space plus linear operators as building blocks
2) Linear operators automatically guarantuee the superposition principle (it is already used in the definition of "linear")
3) Canonically conjugate variables x and p are translated to operators satisifying [x,p] = i
4) For all non-commuting hermitean operators A, B a generalized uncertainty relation can be derived from the commutator [A,B]
5) The Heisenberg uncertainty principle is just a special case for conjugate observables with [A,B] = i

 

[arkajad]

In nonlinear quantum mechanics you can have Heisenberg Uncertainty relations and no superposition principle stricte senso (time evolution is non-linear). See e.g. http://www.slac.stanford.edu/econf/C9707077/papers/art36.pdf" and references therein.

 

[tom.stoer]

What about current status and broad acceptance of nonlinear quantum mechanics?

 

[arkajad]

Current status and acceptance of anything beyond "mainstream" is "not the mainstream physics" - thus risky and shaky. You better do not spend too much time with that, as you can easily loose your position. I mean unless you are Haag or Weinberg.

 

[yoda jedi]

What about current status and broad acceptance of nonlinear quantum mechanics?

and maybe with superposition (plus heinserberg principle).

On the nonlinear extension of quantum superposition and uncertainty principles.
Renzo Cirellia, Mauro Gattib, Alessandro Manià.

i think, maybe going beyond (alternative or unlike) Hilbert space, we can gain new insight on physics.

 

[dextercioby]

Hi All,
I am trying to remember the logical argument that leads from Heisenberg's uncertainty principle to the existence of quantum superposition states.

Here's my sketchy version:

1) postulate of Quantization leads to non-commuting operators
[...]. Can someone revise this if it is incorrect?

Operators on what ? That's right, linear / vector spaces. Linearity is thus assumed in your first assertion. It cannot be logically derived from it without circular agruments.