# What does a negative energy eigenvalue mean?

If I have a Hamiltonian $$H$$ that only has kinetic energy and no potential energy, do the energy eigenvalues have to be non-negative? Could the ground state of Hamiltonian have negative eigenenergy? If so, is the negative value results from potential? What's the physical meaning of negative eigen-energy?

• What do you think negative energy means in classical physics, and why do you think quantum physics would be any different? Sep 24, 2020 at 15:55
• Which Hamiltonian are you considering? If it's a relativistic Hamiltonian, then negative energy states are an important issue. Sep 24, 2020 at 18:08
• @JoshuaTS I'm considering the Hamiltonian in some scenarios like quantum oscillator. Why are the negative energy states important in relativistic Hamiltonian? Thanks!
– ZR-
Sep 24, 2020 at 18:32

For example in classical physics when talking about gravitational potential energy, one usually defines $$E=0$$ to be at the ground, and then when I rise up a height, $$h$$, my energy becomes $$E=mgh$$. If I wanted to then convert this energy to kinetic energy (for example by letting myself fall) the resultant kinetic energy when I reach the ground would be the (positive) difference between these two energies, and then $$\frac{1}{2}mv^2=mgh\;\Rightarrow\;v=\sqrt{2gh}$$ etc.
But there is no reason we have to set $$E=0$$ at the ground. I can define the zero of energy to be at height $$h$$, and then my potential energy when I'm on the ground would be $$-mgh$$. Still, if I wanted to work out what my kinetic energy would be when falling from $$h$$ to the ground, I would still get $$mgh$$, which is the difference between these energies. In this way, energy differences are more important than absolute energies.
Moving to quantum mechanics, exactly the same arguments apply. I can define my "zero of energy" to be whatever I like, often something that is convenient to me. For example if we were just considering a particle moving with no potential, I would define my $$E=0$$ to be when its kinetic energy is 0, for convenience.