# Ground state

1. Feb 21, 2005

### Sterj

If we calculate the ground state energy of a system we receive E=w*h(bar).
This Energy can be derived with the harmonic oscillator.

Is this Energy the Energy of a free photon or a free electron in (for ecample) the vacuum state or is this describing a particle between (potential,...) walls?

If yes, why can we describe particles as oscillators (always thought that's to do with waves)?

2. Feb 21, 2005

### dextercioby

What do you mean...?Which system...?

I frankly doubt it.Please show us....

If you refer to $E=\hbar\omega$,yes,it's for a free photon.

You might make a connection between an infinite set of quantum oscilators and the free scalar boson.

Daniel.

3. Feb 21, 2005

### Sterj

What's the ground state energy of a free electron for example? Can we also say, that a free electron is a harmonic oscillator? Why is E=h(bar)*w the energy of a photon, the derivation of this energy isn't explicit for the photon.

And another problem is: Energy of photon: E=h*f but the energy of a photon is also E=h(bar)*w*(n+1/2). That would say, that a photon can't have all frequences f.

thanks dexter

4. Feb 21, 2005

### ZapperZ

Staff Emeritus
ER.. no. You cannot mix things with different potentials and think that it fits everything. The system that produces the energy eigenstates of E = hbar*w is ONLY for the harmonic oscillator. It is NOT for electron in an atom. The potential profile for an electron in a central coulombic force is different than a harmonic oscillator.

Furthermore, harmonic oscillators are usually not associated with photon emission/absorption UNLESS these are electric dipole or higher order mode vibrations such as those found in the optical phonon modes of solids.

Zz.

5. Feb 21, 2005

### Sterj

thanks, now i got it.

6. Feb 21, 2005

### Staff: Mentor

A bound system (such as an electron in an atom, or a simple harmonic oscillator) has a ground state. An unbound system (such as a free particle) does not. So it makes no sense to talk about the ground state energy of a free particle.

No, see above.

This is a fundamental assumption, not a derived result.

No, the second formula gives the allowed energies of a simple harmonic oscillator:

$$E=\frac{1}{2}\hbar\omega, \frac{3}{2}\hbar\omega, ...$$

7. Feb 21, 2005

### Sterj

@jtbell: You said: "No, the second formula gives the allowed energies of a simple harmonic oscillator" (E=h(bar)w)

If that's the energy of a simple harmonic oscillator, what is E=h*f? The energy of an electromagnetic wave?
Whats the differrent of these two things?

And another question:
Cause of the heisenberg's uncertainty principle there is a virtual pair (photons) createn with energy E1. While its life time it gets more energy and before annihilating it has energy E2. E1<E2. Now, what happens with the energy that doesn't go back to "vacuum"?

thanks for all answers.

8. Feb 21, 2005

### masudr

No. $E=\hbar\omega$ is for the photon, not the harmonic oscillator. This equation is identical to $E=hf$, since $\hbar=\frac{h}{2\pi}$ and $\omega = 2\pi f$, we get

$$E=hf\Rightarrow E=\frac{h}{2\pi}2\pi f\Rightarrow E=\hbar\omega$$

You seemed to have confused the energy eigenvalues of the simple harmonic oscillator and the photon. We calculate the energy eigenvalues of the oscillator given the potential function. We assume that all photons have energy E given by $\hbar\omega=hf$.

The two things are complete different systems. E=hf is the energy of a photon, $E=\hbar\omega(n+\frac{1}{2})$ give the eigenvalues of a harmonic oscillator. A harmonic oscillator and a photon are two different systems!

9. Feb 22, 2005

### Sterj

If a harmonic osscillator and a photon are two different things, what's the different between this two things? But the eigenvalues of a harmonic oscillator are the energy states of a photon, right? And E=hf describes a classical electro magnetic wave.

10. Feb 22, 2005

### dextercioby

Take any decent SM book.I'm sure they discuss photonic & phononic ideal gases...

No,they are the eergy states of the LHO...

No,it describes the QUANTUM particle called photon.

Daniel.

11. Feb 22, 2005

### Sterj

Aaa, A photon is a "light particle" and a phonon is createn if an electron,... absorbates a photon. I got it, so we can describe the phonon as a harmonic oscillator and the photon as an electro magnetic wave/"light particle".

thanks, now I think I got it.

But in ground state, we can describe phonons like photons (because h(bar)*w=h*f)?
I read that the vacuum state is the state with no photons, why (I think at protons, ...)?

12. Feb 22, 2005

### dextercioby

A quanta of the EM field...

No.A phonon is a quanta (particle) which describes the quantized modes of the vibration of a crystaline structure/lattice.

Sort of...

Not really.Phonons have nonzero zero-point energy,which photons have zero zero-point energy.

The theory says so.Take the $\hat{N}$ and aply it on the vacuum state $|0\rangle$...I'm sure the result will be zero...It's in the definition of the vacuum state for any free field.

Daniel.

13. Feb 22, 2005

### Sterj

Do you mean, that the zero point energy of a phonon is E=h(bar)w/2 and the zero point energy of a photon is E=0.

And if we have the casimir effect, the photons between the plates behave like harmonic oscillators (is written in wikipedia).

Ok, now, if I'm right, I'm understanding it:
If a photon is createn in vacuum it has energy E=hf=h(bar)w and so the virtual photon has an energy of: E=2hf=2h(bar)w.

If we are in a vacuum and there is a cristall structure the energy of one phonon in this cristall is: E=h(bar)w/2.

But something is confusing me now, see attachment (from a pdf that derives casimir force).
Why can the Hamiltonian of an electro magnetic field be written by (a part of) a harmonic oscillator? You said that phonons can be written by harmonic oscillators and not photons.

#### Attached Files:

• ###### casimir.GIF
File size:
4.1 KB
Views:
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14. Feb 22, 2005

### dextercioby

One question:what happened to normal ordering...?You know,the trick that allows zero vaccum energy.

Daniel.

15. Feb 22, 2005

### Sterj

yep: heisenberg's uncertainty principle.

16. Feb 22, 2005

### dextercioby

What does it have to do with normal ordering...?

Daniel.

17. Feb 22, 2005

### ZapperZ

Staff Emeritus
I need to ask you one thing. You were having problems finding the connection between "f" and "omega" a few postings ago (which is a rather elementary problem if you have had any basic college intro physics). So now I do not understand how you can "read" a Hamiltonian and understand what it is, especially the one you cited that contains creation/destruction operators (i.e. 2nd Quantization formalism).

I will right off the bat tell you that you cannot learn, much less understand, physics by learning things in bits and pieces like this. It appears that you are picking out stuff that maybe you read or came across, but you have no clue where they came from. This is a dangerous way to learn this subject because you will lose the coherence and interconnectivity of physics. It is one thing to want to know it generally or via hand-waving arguments. But if you start citing specific equations or expressions, then this is no longer an informal description and it is imperative that things are done carefully. You cannot mix "handwaving" with "exactness". If you want the latter, then you need to study the material from the very beginning.

Zz.

18. Feb 22, 2005

### Sterj

I started with integral and differential calculus. After that I went offer to Einsteins theory of relativity. Then I wanted to do something in quantum mechanics, so I learned to calculate with a very important equation (schrödinger equation <--- Hamiltonian). After that I took a watch at Hilbertspace (only a little) and learned to calculate and to derive the uncertainty principle. So, I've to be more then 1.5 years in school until I can study physics. In school we're discussing electricity (easy physics).
Now I want to learn some thing about ZPE and vacuum fluctuation.

I know what f is and I now what omega is, I know what a oscillation and a wave is. But a pdf document was confusing me.

@dexter:
"What does it have to do with normal ordering"
Sorry I'm not from UK, what's a "normal ordering". Don't let me think of anything in physics.

I think I understand it now really and finally, let me try:
Einstein's equation for the energy of a photon is: E=hf=h(bar)w/2. Here we describe a photon. Now, in the ground state an electro magnetic field hasn't any energy and so far no photons. Cause of the heisenberg's uncertainty principle d(E)d(t)=h/(4pi) a system can't have an energy of zero. So in vacuum the above equation wouldn't be sadisfied if we work in classical theories. In quantum mechanics this problem is solved by letting create virtual particles and letting them annihilate after a short time. The energy of such a particle pair (if electro magnetic) is: E=2hf.

If we go over to a cristall latter (my English ) that is cooled down to 0 K we see that this cristall latter oscillates. The reasen is that the lowest allowed energy of a harmonic oscillator is E=1/2h(bar)w and not zero. E=1/2h(bar)w is the energy of a phonon, a quantum we use to describe vibration mods in a cristall latter.

I saw, that my posted forumla in tha attachment is the formula to "count" the whole energy of a system of all free bosons. Can somebody give me a web page where this formula is derived (can't find one).

Thanks for everything.

19. Feb 22, 2005

### ZapperZ

Staff Emeritus
Do you know what raising/lowering/creation/destruction operators are? What about the normal modes of oscillation? Or even Hermite polynomials? Do you think not knowing how the energy eigenvalues were obtained for each of the different situations, as illustrated by your confusion in the beginning of this thread, presents no hindrance to your understanding of the material? Do you think "phonons" is simply described via the eigen energies of a harmonic oscillator? [Look up a Solid State Physics text]

The last thing I want to do is damper your enthusiasm to study the material. But you must also be realistic on your part. There are NO shortcuts to understanding physics. You snip bits and pieces of information and try to put them together, which is why you end up using principles in the wrong situations. You can put better use of your effort and time if you start from the ground up.

Zz.

20. Feb 23, 2005

### dextercioby

Davydov wrote two fabulous books,it's true,they're a bit old,but nevertheless valuable:"Quantum Mechanics" and "The Theory of Solids"...

Maybe u may take a look.

Daniel.

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