Griffiths problem 2.2 states that $E$ must exceed the minimum value of $V(x)$ for every normalizable solution to the time-independent Schrodinger equation. Then, it asks for a proof and what the classical analog of this statement is. I understand the proof for this, but I am very confused about the classical analog.
I was initially thinking that this statement means that the energy must be greater than the potential energy at this $x$ value. I assumed $E$ was the total energy, so this would mean that there must be kinetic energy always. However, I remain confused about many things:
What is kinetic energy in quantum mechanics?
Is it proper to say that $E$ here is total energy?
Is this the proper classical analog? I don't need an answer, but a hint would be nice. An answer is okay as well but I'm not sure it's allowed.
Why can we use the concepts potential and potential energy interchangeably? Here, we are saying that potential has the same units as energy, which means it's potential energy I gathered?
Update: I have also thought about the classical analog being underdamped motion, but I am still curious about the above questions and interpretation.