# Homework Help: Hydrogen energy levels (quantum theory)

1. Jul 7, 2007

### physicshelp1

Hello,

I really need help with this question.

Continuum is when the proton/electrons exist in the continuous band of energy levels above the highest bound energy level of the atom.

What can be said about the combined energies of the proton and electron is the hydrogen atom is in continuum.

Thanks! Any ideas will be a help.

2. Jul 7, 2007

### jambaugh

When you define the energy of a pair of interacting particles it decouples into the kinetic energy of the whole system's center of mass (COM) in motion plus the kinetic and potential energies of their relative motion about the COM. This second part can be treated as a single particle system with combined mass (called the reduced mass) orbiting about the COM. For bound system e.g. hydrogen you have a total energy which is the sum of the first component which is continuous and the second which is discrete.

In the case of the hydrogen atom, since the proton mass is so much greater than that of the electron, then the COM is almost located at the proton anyway so usually it is treated as a "free proton system" with an electron bound to it. This then, relative to your question, means for all practical purposes the proton has a continuous energy spectrum and the electron when bound has a discrete one.

3. Jul 7, 2007

### physicshelp1

Thanks so much, James, for your help.

I don't quite understand what you are saying about the combined energies of the proton and electron.

I have to state the answer in one sentence.

What should I say?

4. Jul 7, 2007

### olgranpappy

You should say, "The value of the expectation value of the kinetic energy plus the value of the expectation value of the Hamiltonian should equal my grade on this question."

5. Jul 7, 2007

6. Jul 7, 2007

### jambaugh

Pardon, I didn't realize that you were asking a specific homework question. (And if I did I would have been a bit less direct.) I can't just give you a "one sentence answer". You need to understand the issues well enough to phrase your own.

What more I can say is this.
The electron has kinetic energy.
The proton has kinetic energy.
The hydrogen atom (electron and proton) have total kinetic energy equal to the sums of that of each part.
It also has a potential energy due to the electrical attraction between proton and electron.

When you consider the hydrogen atom (by definition the electron is bounded) where the center of mass is stationary then this whole system will have discrete total energy eigen-states in its quantum description.

The hydrogen atom as a whole is (we suppose from this problem) moving freely and so has continuous total energy.

(If on the other hand you consider say a hydrogen atom in orbit around a planet or confined to a box then the energy spectrum is again discrete.)

There are some other nit-picking issues. Specifically if you are considering the center of mass frame for the whole hydrogen atom then you are supposing you are measuring this variable which does not commute with the total momentum or total energy of the hydrogen atom. It is sufficient however to consider a hydrogen atom whose COM you don't know but whose total momentum is zero and thus speak of its discretized energy in this context.

7. Jul 7, 2007

### Astronuc

Staff Emeritus
When protons and electrons are 'unbound', that is a plasma, and the energy levels depend on the temperature which may be decribed by a Maxwellian distribution. But reality (Nature) is not so simple, and in plasmas there is a constant churning of ionziation and recombination.

Hydrogen atoms do have a continuum of kinetic (translational) energy, but then atoms may be ionized by radiation (both EM and particles).

The unbound proton and electron do not an atom make. The proton and electron are in a continuum of energy, and the atom as whole has a contiuum of energy, but within the atom, the electron occupies more or less a discrete energy level.

Last edited: Jul 7, 2007
8. Jul 7, 2007

### physicshelp1

So, the proton and electron have equal energies when the atom is in continuum?

9. Jul 7, 2007

### physicshelp1

I think the answer is much more simple than this, but just don't know. It's a state answer, ie. the combined energy is greater than 13.60eV? That's what it would take to go to continuum? Is that right?

10. Jul 7, 2007

### jambaugh

Actually ignoring rest energy the total threshold energy will be zero. Bound particles in a potential well have a negative total energy (kinetic plus potential) and discrete energy spectrum, whereas free particles have positive and continuous energy spectrum.

The number 13.60eV is the energy difference. Put another way, -13.60eV is the negative energy (disregarding rest mass) of the electron in the hydrogen ground state.

But I'm just being technical... you can if you like reset the potential energy to any value you like, e.g. set the ground energy to zero. This is called a choice of gauge. But the convention is to set the zero energy as that for electron and proton stationary and separated by infinite distance.

Getting even more technical this choice does make a difference in the relativistic treatment where the choice of potential energy affects the rest energy (mass) of the hydrogen atom (it is less than the sum of the rest energies of a free proton and free electron). So there is a good reason to stick to negative energy (apart from component masses) for bound particle systems.