Can I Tell if a State is Ground State?

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Discussion Overview

The discussion revolves around the question of whether one can determine if a given quantum state is the ground state based on knowledge of the potential, energy, and a specific eigenstate, without knowing the general forms of eigenstates and eigenvalues. The scope includes conceptual and theoretical aspects of quantum mechanics.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether it is possible to identify a state as the ground state given certain parameters, specifically the potential, energy, and an eigenstate.
  • Another participant suggests that a ground state wave function typically has no nodes, implying a characteristic feature that could aid in identification.
  • A different participant proposes that the ground state wave function minimizes the expectation value of the Hamiltonian, indicating a potential method for proving the ground state status.
  • One participant references the Variational Principle, stating that the ground state energy is less than or equal to the expectation value of the Hamiltonian for any state.
  • Another participant shares an observation from quantum mechanics problems, noting that for a specific potential, the product of the standard deviations of position and momentum equals hbar/2 in the ground state, suggesting a unique property of the ground state that may not hold for other potentials.

Areas of Agreement / Disagreement

Participants express differing views on the ability to determine if a state is the ground state, with some proposing characteristics that may help in identification while others remain skeptical about the feasibility of such determination without additional information.

Contextual Notes

Some assumptions about the nature of potentials and eigenstates are not explicitly stated, and the discussion does not resolve the applicability of the proposed characteristics across different potentials.

KFC
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If I know the explicit form of potential, the energy and a specific eigenstate, but I don't know the general form of eigenstates and eigenvalues, can I tell if the state is ground state or not?
 
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No i don't think so, why are you asking?
 
KFC said:
If I know the explicit form of potential, the energy and a specific eigenstate, but I don't know the general form of eigenstates and eigenvalues, can I tell if the state is ground state or not?

A ground state wave function usually has no nodes.
 
clem said:
A ground state wave function usually has no nodes.
I think you can prove this statement by noting that the ground state wave function minimizes
[tex]\left < \psi \right| H \left | \psi \right>[/tex]
 
The Variational Principle guarantees that

[tex] E_g \leq \langle \psi|H|\psi \rangle \equiv \langle H \rangle[/tex]
 
Last edited:
In doing some QM problems a few weeks ago I noticed that, for the PARTICULAR potential I was working with, the product of the standard deviation of the position (sx) and the standard deviation of the momentum (sp) was exactly hbar/2 in the ground state (sx*sp=hbar/2), and the product grew larger for higher energy states. So the uncertaintly principle was closest to getting violated in the lowest energy state. I assume that's NOT true for other potentials, but maybe something you could look into depending upon what exactly you are trying to do.
 

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