Energy of a QM system, stationary states

[yeahhyeahyeah]

I'm pretty confused by the rules regarding the total energy, the kinetic energy, and potential of a QM system.

Does the total energy have to be positive or greater than zero? And if so, why not? I don't really understand what it means to have a negative total energy of a system I guess. I know that we treat a lot of potentials as negative, like gravitational potential, but I guess I have never found complaint with it until now.

As an example, what if you had a system described by:

V = -V₀ for x< -a
0 for |x|<a
V₀ for x>a

What ranges can its energy exist in? And in what range does its energy have to be for the stationary states to be physically possible (this only happens when it is a bound state?)


Lastly, this is a kind of unrelated question, but say you have a wave packet whose initial state is such that $$\Psi$$(-x) = $$\Psi$$ (x)

In other words, an odd function. Now, because of its oddness, it can only be comprised of the odd (sin functions) stationary states. If I add time evolution, since the coefficients of any even states is 0, there will NEVER in time EVER be a contribution from any even states.

So basically, if the system starts out odd/even it will remain like so forever?

 

[Simon Bridge]

The zero for potential energy is arbitrary ... for instance, we often take objects on the ground as having zero gravitational potential energy, they have positive mgh as they go higher and negative for lower than the ground. But we could equally put the zero at the top of a building or at the bottom of the sea - or at the center of the Earth. In order to get general equations, we have to do better than that and what we usually do is set the zero at the edge of the Universe ... eg. the potential energy is how much work you have to do to remove the object to infinity.

This is why you will find atomic potentials expressed as negative numbers.

In your example, you have defined the zero potential in a particular region of space.

A particle can have a negative kinetic energy if it is at position x < -a - and may tunnel slightly into the barrier at x=a. The rest of the wave reflects to give standing wave solutions.

You should be able to complete the picture for higher energies.