# What is Ground State of SHO?

1. Jul 10, 2010

### referframe

What does the Ground State of a quantum simple harmonic oscillator represent physically?

2. Jul 10, 2010

### ZapperZ

Staff Emeritus
This is a rather vague question. For example, is there a reason why you're asking ONLY for the ground state of a SHO? Does this mean that you have no issues with the physical meaning of, say, the ground state of a hydrogenic atom, or a square-well potential? If this is true, then it would be informative to know what you mean in those cases as "represent physically", so that we know what you're looking for.

Zz.

3. Jul 10, 2010

### referframe

From what I have read, the Ground State of a quantum SHO is a gaussian and that state, as an oscillator, has "null vibrations". It is referred to as the "zero-point" energy level and is fundamentally the same as the energy associated with empty space. The ground state of an SHO, because it is a gaussian, minimizes the position-momentum uncertainty.

That is what I have read. I guess I'm trying to visualize what "null vibrations" means.

4. Jul 10, 2010

### nateHI

I haven't heard the term null vibration before but I think you are talking about quantum fluctuations. Quantum fluctuations are due to the fact that you can never have no energy. Even in the lowest energy state of a QHO

|0>

you still have hbar*omega/2 energy.

5. Jul 11, 2010

### ZapperZ

Staff Emeritus
This is getting to be even more puzzling. Gaussian? "null vibrations"?

The SHO wave functions are described via the Hermite polynomials! Maybe you should tell us what you think the ground state of a SHO is before we answer the question. It certainly would help if you cite a source that gave you such confusing description.

Zz.

6. Jul 11, 2010

### vanesch

Staff Emeritus
... times a gaussian. That's maybe where the poster's expression came from. As H0 is a constant, the wavefunction is a gaussian for the ground state of the SHO.

7. Jul 11, 2010

### sweet springs

Hi, referframe.
Wave functions both in coordinate space and in momentum space are Gaussian as you are well aware. Isn't it enough? I am not sure QM allows further visualization.
Regards.

Last edited: Jul 11, 2010
8. Jul 11, 2010

### referframe

Exactly.

Here is just one source: http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator . Do find for "null oscillations".

9. Jul 11, 2010

### nateHI

As a sidenote. There is also such a thing as a 'null space'. But that is more math than physics. It's what you get when you apply the lowering operator to the ground state of the quantum harmonic oscillator.

10. Jul 11, 2010

### Staff: Mentor

I can safely say that I personally have never seen the term "null oscillations" applied to a QM system before seeing this thread. My first guess at interpreting it was to suppose that it means "no oscillations" in some sense. However, I couldn't fit that to the actual behavior of SHO energy eigenstates.

1. All energy eigenstates (ground state and otherwise) have wave functions of the form

$$\Psi(x,t) = \psi(x) \exp \left( -i \frac{E}{\hbar} t \right)$$

that is, they all have this oscillating complex exponential factor.

2. However, the probability distribution $\Psi^*\Psi$ of any energy eigenstate does not oscillate in time; it is "stationary" because the time-dependence disappears when you calculate $\Psi^*\Psi$.

There's no difference between the ground state and other energy eigenstates in these respects.

I did a Google search for the phrase "null oscillations", using quotes to keep the words together. The vast majority of the hits are regurgitations or quotations of the two Wikipedia pages that use the phrase. Several hits have to do with neutrino oscillations, where the phrase apparently has a specialized use. I found two hits (via Google Books) to books that were apparently written by Russians, in which case it may simply be a too-literal translation of a Russian phrase that means "ground state". (What is the Russian term for "ground state"?)

Another possibility: the Wikipedia article on zero-point energy (one of the two that mention "null oscillations") says:

So maybe "null oscillations" comes from German in the same way, in which case it really should read "zero-point oscillations".