How Can You Have Negative Energy in a Positive Potential?

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Discussion Overview

The discussion revolves around the concept of energy states in quantum mechanics, particularly in relation to the delta potential and the definitions of scattering and bound states. Participants explore the implications of having positive potentials and the conditions under which negative energy states can exist.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants question how negative energy can exist in a positive potential, referencing Griffiths' definitions of scattering (E>0) and bound states (E<0).
  • One participant suggests that the definition of zero energy is arbitrary, emphasizing that only the differences between energy levels hold physical significance.
  • Another participant argues that positive potential corresponds to repulsion, implying that bound states (negative energy) cannot occur in such potentials, contrasting with attractive potentials like those in hydrogen atoms.
  • A participant proposes an alternative definition of bound states, suggesting they are states where the particle's energy is classically unable to exceed the potential barrier, using the example of a positive step potential.
  • There is mention of a convention where a constant term is added to potentials to set the value at infinity to zero, which allows for the classification of bound states as negative energy and scattering states as positive energy.
  • Another participant reinforces the idea that the relevant condition for determining state types is the relationship between energy and the maximum value of the potential (E - V_max).

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of energy states in relation to potentials. There is no consensus on the definitions or the conditions under which negative energy states can occur in positive potentials.

Contextual Notes

Participants highlight the dependence on the chosen reference point for energy and the conventions used in quantum mechanics, which may affect the interpretation of energy states.

CPL.Luke
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so I am studying the delta potential now and I notice that griffiths defines scattering and bound states as cases where E>0 and E<0 respectively. but I have to ask if you have a positive potential, then how an you have negative energy?
 
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Isn't this another of the many cases in which it is completely arbitrary where you define the energy to be zero, since only the difference between successive energy levels has any physical meaning?
 
CPL.Luke said:
so I am studying the delta potential now and I notice that griffiths defines scattering and bound states as cases where E>0 and E<0 respectively. but I have to ask if you have a positive potential, then how an you have negative energy?

Positive potential corresponds to repulsion, so it cannot have bound (negative energy) states. All attractive potentials (e.g., in the hydrogen atom) are negative.

Eugene.
 
this is where I'm wondering wether or not he was just demonstrating the concept of a bound state by using a negative energy value (the potential where he did this most blatantly was the positive dirac delta potential) however its a common theme in the succesive sections for him to say that a scattering state is a state with positive energy, and a bound state is one with negative energy.

as I didn't like this definition I've been using one where a bound state is any state where the energy of the particle would be classically unable to exceed the potential barrier.

for instance in the positive step potential if the the wavicle has energy less than v then it will exponentially decay after the step, whic would be the "bound state" whereas the scattering state would be the one with energy greater than v.
 
There is a convention that a (inconsequential) constant term is aded to any potential, so as to make sure that the value of the potential at infinity is zero. Only with this condition it is correct to say that bound states have negative energy and scattering states have positive energy.

Eugene.
 
meopemuk said:
There is a convention that a (inconsequential) constant term is aded to any potential, so as to make sure that the value of the potential at infinity is zero. Only with this condition it is correct to say that bound states have negative energy and scattering states have positive energy.

Quite so. More generally, and without caring about where the zero of your energy axis is, it is [itex]E-V_{\rm max}[/itex] that must be positive (free state) or negative (bound state), where [itex]V_{\rm max}[/itex] is the maximum value of the potential.
 

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