Schrödinger Eq: Neglect Uncertainty Principle? Intensity Probability?

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Discussion Overview

The discussion revolves around the relationship between the Schrödinger equation and the uncertainty principle, exploring whether the equation neglects the uncertainty principle and how this affects the probability distribution of intensity. Participants examine theoretical implications, interpretations of inputs to the equation, and the nature of observables in quantum mechanics.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants question whether the Schrödinger equation completely neglects the uncertainty principle, suggesting that this could imply a distinct probability distribution for intensity.
  • Others argue that the Schrödinger equation predicts the wave function with certainty, from which uncertainties of observables can be derived.
  • It is noted that the momentum-space wave function is related to the position-space wave function through Fourier transforms, which inherently involve the uncertainty principle.
  • One participant emphasizes that the inputs to the Schrödinger equation are not energy or mass but rather the wave function, which evolves over time.
  • Another participant challenges the notion that no information is input into the time-independent Schrödinger equation, questioning how one determines the particle being described.
  • Concerns are raised about how uncertainties in direct observables might complicate the intensity distribution, particularly in relation to Fourier transforms.
  • A later reply clarifies that while the Hamiltonian is expressed as if inputs are known, it does not imply that exact positions of particles are known, aligning with the uncertainty principle.
  • One participant expresses satisfaction with the clarification regarding the interpretation of the Hamiltonian and its relation to the uncertainty principle.
  • Another participant references a source that discusses the relationship between the wave function and the uncertainty principle through Fourier integrals.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the Schrödinger equation regarding the uncertainty principle. There is no consensus on whether the equation neglects the uncertainty principle or how it affects the probability distribution of intensity.

Contextual Notes

Discussions involve assumptions about the inputs to the Schrödinger equation and the nature of observables, which remain unresolved. The relationship between the wave function and the uncertainty principle is also explored but not definitively concluded.

Mr-T
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Does the Schrödinger equation completely neglect the uncertainty principle? If so, wouldn't this imply that our intensity distribution has its own probability distribution?
 
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The Schrödinger equation predicts the wave function with certainty; but from this wave function the uncertainties of observables can be derived exactly
 
The momentum-space wave function ##\Phi(p,t)## is basically the Fourier transform of the position-space wave function ##\Psi(x,t)##. The uncertainty principle comes from the properties of Fourier transforms. Any pair of functions that are related by Fourier transforms has a similar uncertainty principle.
 
I understand what both of you are saying and I appreciate the replies.

In the Schrödinger equation we input values for energy/mass assuming we know with 100% certainty what these values for energy/mass are. Due to the input of these values is where my question holds its regards.
 
Mr-T said:
In the Schrödinger equation we input values for energy/mass assuming we know with 100% certainty what these values for energy/mass are.
No

The input is a wave function, the output is a wave function at a later time. This predicts with certainty that a system will be in a state A' at time t' > t provided that it was in state A at time t; A is specified by a wave function or a state vector |A>.

In case of the time-indep. SE the input is not energy, the input is nothing! The outputs are a) the allowed energy eigenvalues and b) the corresponding eigenfunctions. The SE does not tell you in which state the system is, in only tells you what the allowed state are
 
tom.stoer said:
In case of the time-indep. SE the input is not energy, the input is nothing!

If you are not inputting any information into the T-I SE then how do you know what particle it is talking about?!
 
Mr-T said:
If you are not inputting any information into the T-I SE then how do you know what particle it is talking about?!
Do you mean you specify a potential, then solve the SE equation for a given potential? Or you plug in the values of the eigenvalues?
 
Mr-T said:
If you are not inputting any information into the T-I SE then how do you know what particle it is talking about?!

The remark by Tom is an overstatement, an exaggeration. The input is the specific form of the Hamiltonian in terms of fundamental observables such as position, momentum, spin.
 
If all direct observables have some uncertainty, won't this mess up our intensity distribution even more than the fouriers already do?
 
  • #10
Mr-T said:
If all direct observables have some uncertainty, won't this mess up our intensity distribution even more than the fouriers already do?

OK, I think I see where you were going with your original question...

In the time-dependent Schrödinger equation [itex]H\Psi=E\Psi[/itex] the Hamiltonian is written as if all of its inputs were exactly known. For example, if we're dealing with two charged particles, there will be a [itex]\frac{1}{r1-r2}[/itex] term somewhere in it, where r1 and r2 are the positions of the two particles. You should read that as saying not that the two particles are at those exact positions, but rather that if they were in those positions that would be the exact distance between them. The uncertainty principle doesn't stop us from talking about how things would be if we knew exactly where a particle was, it just forbids us from knowing exactly where it is.

Once I have the Hamiltonian written down, I solve Schrödinger's equation; and as tom.stoer said in #2, the uncertainty principle is inherent in the ψ that comes out.
 
  • #11
Ahh yes, talking in this fashion resolves my concerns.

Thank you nug
 

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