Prove zero point energy without calculation?

[Gerenuk]

What is an elegant way to prove that quantum mechanical system (with bounded particles?) have some non-zero ground state energy?

 

[Vanadium 50]

How does one prove that the result of a calculation is non-zero (or non-any value) without doing a calculation?

 

[Gerenuk]

Without the full calculation of energy. Basically the easiest way possible, whatever that is.

 

[xlines]

Without the full calculation of energy. Basically the easiest way possible, whatever that is.

Well, should you take Heisenberg principle as true, there is no much caliculation involved : as an example, take a free particle and localize it ... SNIP ...such uncertainty in momentum, it's easy to derive p² expectation value thus kinetic energy.

erorr : I didn't read the question as careful as I thought I did =) Having general QM problem in mind, I am no longer sure ... maybe taking Taylor series around global minimum and recalling oscillator, but that is hardly "without of caliculation".

 

[alxm]

A simple hand-waving conceptual answer would be something like: Consider a simple vibrating system, e.g. a harmonic oscillator. The lowest possible energy (classically) would be for the thing to simply not vibrate. Quantum mechanically, it can't do that since it'd mean having a well-defined position and momentum, in violation of the uncertainty principle.

So the lowest vibrational state must have some energy, on the order of ~$$\hbar$$.