  We consider the ground state of simple quantum systems coupled to an
environment. In general the system is entangled with its environment. As a
consequence, even at zero temperature, the energy of the system is not sharp: a
projective measurement can find the system in an excited state. We show that
energy fluctuation measurements at zero temperature provide entanglement
information. For two-state systems which exhibit a persistent current in the
ground state, energy fluctuations and persistent current fluctuations are
closely related. The harmonic oscillator serves to illustrate energy
fluctuations in a system with an infinite number of states. In addition to the
energy distribution we discuss the energy-energy time-correlation function in
the zero-temperature limit.
