Prove zero point energy without calculation?

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Discussion Overview

The discussion revolves around the concept of proving that quantum mechanical systems with bounded particles possess non-zero ground state energy, specifically focusing on methods to demonstrate this without performing explicit calculations. The scope includes theoretical reasoning and conceptual understanding within quantum mechanics.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants inquire about elegant ways to prove non-zero ground state energy without calculations, suggesting a desire for conceptual approaches.
  • One participant questions how to establish the non-zero nature of energy results without calculations, indicating a search for non-mathematical reasoning.
  • Another participant suggests using the Heisenberg uncertainty principle as a basis for reasoning, proposing that localizing a free particle leads to uncertainty in momentum and thus a non-zero kinetic energy expectation value.
  • A different viewpoint mentions the idea of using a Taylor series expansion around the global minimum of a potential, though this is acknowledged as not truly avoiding calculations.
  • One participant offers a conceptual explanation involving a harmonic oscillator, arguing that the lowest energy state cannot be zero due to the uncertainty principle, implying that the lowest vibrational state must have energy on the order of ~\hbar.

Areas of Agreement / Disagreement

Participants express various approaches and reasoning methods, but there is no consensus on a definitive non-calculation proof. Multiple competing views and uncertainties remain regarding the best way to conceptualize the problem.

Contextual Notes

Some limitations include the dependence on interpretations of the uncertainty principle and the ambiguity in what constitutes a proof "without calculation." The discussion also reflects varying levels of rigor in the proposed methods.

Gerenuk
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What is an elegant way to prove that quantum mechanical system (with bounded particles?) have some non-zero ground state energy?
 
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How does one prove that the result of a calculation is non-zero (or non-any value) without doing a calculation?
 
Without the full calculation of energy. Basically the easiest way possible, whatever that is.
 
Gerenuk said:
Without the full calculation of energy. Basically the easiest way possible, whatever that is.

Well, should you take Heisenberg principle as true, there is no much caliculation involved : as an example, take a free particle and localize it ... SNIP ...such uncertainty in momentum, it's easy to derive p2 expectation value thus kinetic energy.

erorr : I didn't read the question as careful as I thought I did =) Having general QM problem in mind, I am no longer sure ... maybe taking Taylor series around global minimum and recalling oscillator, but that is hardly "without of caliculation".
 
Last edited:
A simple hand-waving conceptual answer would be something like: Consider a simple vibrating system, e.g. a harmonic oscillator. The lowest possible energy (classically) would be for the thing to simply not vibrate. Quantum mechanically, it can't do that since it'd mean having a well-defined position and momentum, in violation of the uncertainty principle.

So the lowest vibrational state must have some energy, on the order of ~[tex]\hbar[/tex].
 

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