How Can You Have Negative Energy in a Positive Potential?

[CPL.Luke]

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?

 

[cesiumfrog]

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?

 

[meopemuk]

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.

 

[CPL.Luke]

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.

 

[meopemuk]

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.

 

[blechman]

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 $E-V_{\rm max}$ that must be positive (free state) or negative (bound state), where $V_{\rm max}$ is the maximum value of the potential.