Subject: higher mass -> more zero-point energy? or zero-point energy is constant??

Assuming "mass on a spring" * can be used for explaining quantum jitters, as the
mass (of a particle etc.) increases does the zero-point energy increases? or
stays the same? For example, does a heavier atom (say, a gold atom) have higher
zero-point energy than a lighter atom (say, a hydrogen)? I guess if the
zero-point energy does not change from field to field or from particle to
particle regardless of mass, the frequency must decrease as the mass of the
particle increases. but if the fields is empty of real particle, do all fields
(photon, electron, gluon, Higgs field etc.) have the same zero-point energy or
the same energy for the square or absolute value of zero-point energy for each
field? or zero-point energy differs from field to field? (if so, what factors
(=mass, field type (=fermionic, bosonic, etc.), vacuum expectation value of
field? etc.) increase / decrease, determine zero-point energy of each field, and
why?) if so or not so, how / why can you say your explanations/answers are
valid, correct, and consistent?


*
http://van.physics.illinois.edu/qa/listing.php?id=1256

Re: higher mass -> more zero-point energy? or zero-point energy is constant??