From Heisenberg to Superposition states

In summary: 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 (Schrodinger 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)5) The Heisenberg uncertainty principle is just a special case for conjugate observables with [A,B] = iIn nonlinear quantum mechanics you can have Heisenberg Uncertainty relations and no superposition principle stricto sensu (time evolution is non-linear). See e
  • #1
skynelson
58
4
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 (Schrodinger 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?
 
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  • #2
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
 
  • #3
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" [Broken] and references therein.
 
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  • #4
What about current status and broad acceptance of nonlinear quantum mechanics?
 
  • #5
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.
 
  • #6
tom.stoer said:
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.
 
  • #7
skynelson said:
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.
 

1. What is the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle, also known as the uncertainty principle, states that it is impossible to know the exact position and momentum of a particle at the same time. This is due to the fundamental nature of quantum mechanics and the limitations of our measurement tools.

2. How does superposition states relate to the Heisenberg uncertainty principle?

Superposition states refer to the quantum state of a particle being in multiple states at the same time. This concept is closely related to the Heisenberg uncertainty principle because it shows that the precise state of a particle cannot be determined, but rather exists as a combination of all possible states.

3. What is the significance of superposition states in quantum computing?

In quantum computing, superposition states allow for the potential of performing multiple calculations simultaneously, leading to faster and more efficient computing. This is because instead of processing information one bit at a time, quantum computers can process information in parallel through the use of superposition states.

4. Can superposition states be observed in everyday life?

No, superposition states are only observable at the quantum level. In our everyday world, objects have definite positions and cannot exist in multiple states at the same time.

5. How do scientists manipulate superposition states in experiments?

Scientists use various methods such as quantum entanglement and quantum gates to manipulate and control superposition states in experiments. These techniques involve precise control and measurement of quantum systems, allowing for the observation and manipulation of superposition states.

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