I The energy term in Schroedinger equation

[blue_leaf77]

In the case here we are talking about free particle

I thought in post#25 you were considering a potential step, which is not normally called as free space.

why the potential of the particle is changing here?

You ask why the potential changes? In post #25, you invited us to consider a step potential. So.. isn't it you who defined the potential to change such that the overall profile looks like a step? Otherwise, do I understand you wording correctly?

 

[amjad-sh]

I thought in post#25 you were considering a potential step, which is not normally called as free space.

yes, you are right.

So.. isn't it you who defined the potential to change such that the overall profile looks like a step? Otherwise, do I understand you wording correctly?

yes but I meant just the case where x<0(before the particle reaches x>0 region), and I wanted to know if the particle while moving to x>0 if its (K.E or potential or the total energy(K.E+its potential )) changes or not.( I think they will not change)

 

[vanhees71]

For a free particle the energy is the kinetic energy, and it is conserved.

 

[Mark Harder]

the energy is actually obtained as the expectation value of the Hamiltonian, ⟨E⟩=⟨ψ|^H|ψ⟩⟨E⟩=⟨ψ|H^|ψ⟩\langle E \rangle = \langle \psi | \hat{H} | \psi \rangle.

The whole thread confuses me and I suspect others. Please forgive me, DrClaude, if I pick on you . In this case, the confusion is a failure to qualify what things the energies belong to: the particle, the barrier, or the sum of the 2. E<V when E, the particle's energy (E= U + K, to use "U" for potential of the particle and "K" for K.E. of the particle) is less than the potential energy required to surmount the barrier, which is the quantity V.

what is meant by "E < V" is that the total energy is lower than the barrier height V(x)=0 almost everywhere, except for V(x) = V₀ for a<x<b .

Also, I don't understand the geometry. From the above, it sounds like V(x) is a "bump" in potential over a finite interval, and zero potential everywhere else. Or is U(x) = U(a < x <b) <0, in other words the potential of the particle is negative while it is between the walls of zero potential? I think what is intended is a 1-D particle-in-a-box model, in which U₀ < V(barrier). In the PIB model, a particle can tunnel through a barrier of finite width. But as I (we?) defined it, the barrier is infinite in extent outside of (a,b), so that tunneling into, but not through, the barrier can occur. Is that clear, or am I completely lost?

 

[DrClaude]

I think what is intended is a 1-D particle-in-a-box model, in which U₀ < V(barrier).

The OP clearly mentioned a potential step, and in my post I discussed a potential barrier, which would give the same conceptual problem.

n this case, the confusion is a failure to qualify what things the energies belong to: the particle, the barrier, or the sum of the 2. E<V when E, the particle's energy (E= U + K, to use "U" for potential of the particle and "K" for K.E. of the particle) is less than the potential energy required to surmount the barrier, which is the quantity V.

Yes, that was the basic confusion. I and others have couched it in QM terms, discussing the difference between operators and expectation values. By the way, "the sum of the 2" doesn't make sense in this context.

 

[Mark Harder]

The OP clearly mentioned a potential step, and in my post I discussed a potential barrier, which would give the same conceptual problem.Yes, that was the basic confusion. I and others have couched it in QM terms, discussing the difference between operators and expectation values. By the way, "the sum of the 2" doesn't make sense in this context.

Yes, you're right. I seem to have fallen into the same trap, conflating the two meanings of 'energy' in this context.