# Zero-point energy for Zeroes

1. Jan 8, 2009

### NJV

I'm fed up. Quantum mechanics keeps confusing me. Is there anyone who can explain in as straightforward a way as possible what exactly zero-point energy is? In particular, there are two things I find quite confusing:

1) Why could zero-point energy be infinite?

2) It is said that the zero point energy of a quantum harmonic oscillator (which I assume includes is equal to one half h-bar times the angular velocity, and that this is the lowest energy it can achieve. Here, the angular velocity in the equation is a variable, and therefore so is the zero point energy. How can there be a lower limit to the angular velocity of the system, and therefore to energy the system can achieve?

2. Jan 8, 2009

### malawi_glenn

The zero point energy you can think of that since you can't determine position and momentum simultaneuosly, the particle/state must have momentum (hence energy) to be localized before a measurment.

Think of this: A particle subject to the harmonic oscillator potential would have zero energy and hence zero momentum (recall we are in non relativistic quantum mechanics where E = p^2/2m). Now that particle/state would be totally smeared in space:
$$\Delta p \Delta x > \hbar$$
So if momentum, p, is known for sure (it is equal to zero) then Delta x must "be infinity" i.e not certain / localized at all!

Classically, you can think of the quantum harmonic oscillator to vibrate even at 0K !

3. Jan 8, 2009

### NJV

Thank you for your reply, Malawi. I must say I'm still in the dark, however.

Heisenberg is about measurement. Does this really have anything to do with how the particle actually is? Why can't one just assume a particle, hypothetically, to have an energy below the zero-point energy? Also, your deduction does not explain why there should be a lower limit at exactly h-bar/(2*omega).

For the record, I also still don't understand why omega would have a lower limit.

4. Jan 8, 2009

### malawi_glenn

you don't know what the particle have for energy until you measure it so the thing is that you will never be able to measure the energy = 0 for a particle.

No, no particle can have energy below energy below 0-point energy, no particle can have energy between the energy eigenvalues:

E_n = (1/2 + n)*hbar*omega

Omega has first of all discrete values and of them are the lowest one (which is non zero) Mathematically, this is just consequence of theory of differential equations: http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator
That is the REAL DEAL why the states have the energies they have.

So if you can accept that omega is DISCRETE, you should be able to accept that it can (and does have) a smallest value.

5. Jan 8, 2009

### NJV

A-ha! Right, I forgot — the energy is quantized. Quantum mechanics. Duh.

That positively answers my question. Thank you.

6. Jan 9, 2009

### granpa

an infinite number of quantum oscillators (if you believe that) equals an infinite amount of energy.

what would happen if the quantum oscillators had more energy than 0-point energy. would it radiate away as photons?

7. Jan 9, 2009

### malawi_glenn

it depends on what transition matrix element you have between the states