# In what conditions we can take ground state energy $E_0$ equal to zero?

Like in 3D harmonic oscillator , $$E_m = {h} \omega {(m_x +m_y + m_z + 3/2)}$$, At ground state energy $$m=0$$ and $$E_m = h \omega(3/2)$$. While discussing Bose-Einstein condensation, for calculations we considered ground state energy $$E_0= 0$$. In what conditions In what conditions we can take ground state energy $$E_0=0$$ equal to zero ?

You can only measure differences between the ground state energy and excitations (setting aside the cosmological constant problem). That means you can safely "reset" ground state energy to zero. In fact, in quantum field theory, since each point in space(time) is assigned an oscillator of matter/radiation fields, if you don't subtract the ground states of oscillators, your total ground state energy would be infinite, since you integrate over continuous volume of space.

As the other answers have said it is almost always safe to subtract off a constant from your energy. One other case when it may not be safe is if your Hamiltonian can be varied by an external parameter. For example if you have a harmonic oscillator with a controlable frequency $$H = \frac{p^2}{2m} + \frac{1}{2}m\, \omega(s)^2 x^2\;.$$ You could then ask questions such as

If I have an oscillator in its ground state with frequency $$\omega_1$$ and adibatically vary the frequency to $$\omega_2$$, how much work must I do on the system?

In this case it is crusial that include the zero point energy as it depends on $$\omega$$ and so will vary as $$\omega$$ is varied. If you neglect the zero point energy the result will be 0, but the correct result is $$\frac{1}{2}\hbar (\omega_2-\omega_1)$$

The key point here is that while I can select an arbitrary point for my 0 of energy, once I have done so I must use that zero constently. I cannot vary my zero as I change my Hamiltonian.

Energy is always defined within an additive constant. Therefore, a shift of energy is always possible (in statistical mechanics one has to remember that a shift of energy scale has to be accompanied by an equal shift of the chemical potential). Moreover, in thermodynamics the shift is necessary to allow a simple expression of the the consequence of extensiveness.

The only case where the shift is questionable is the case of hamiltonians which are not limited below or if they are limited below but the ground state energy of an $$N-$$particle system ($$E_G$$) is not limited by $$E_G\geq-B N$$ with $$B$$ positive constant independent on $$N$$. However, in such pathological cases, the real problem is not the shift of the energy scale, but the impossibility to get a well behaved thermodynamic.