Hydrogen energy levels question

[Robsta]

Homework Statement

Draw an energy level diagram for hydrogen (use the vertical direction for energy and separate the states horizontally by angular momentum l)

Homework Equations

I've got some fundamental misunderstandings with this one. I thought the energy levels of hydrogen were given by E_n = -R/n² where R is the Rydberg constant.

So that makes me think that E is not a function of its quantum number l

But then I read that l is "the amount of orbital angular momentum of the electron". How can the electron have more or less angular momentum and this not affect the energy of the hydrogen system?

I'd appreciate any insight.

 

[andrevdh]

Search the Hyperphysics site for the Bohr model...

 

[Robsta]

I've done that, and it doesn't really say how you can increase the angular momentum of a system without changing its energy.

I understand that you get different l values from the quantisation of AM and that the wavefunction kind of needs to be continuous around the circle.

http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html#c2

 

[Robsta]

The actual page itself on l doesn't shed any light on it either. http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydcol.html#c2

 

[andrevdh]

Each orbit of the electron has a certain amount energy and a certain amount of angular momentum.
They require you to draw such a graph or diagram.
Which means that the n values map to the 1/n² values

 

[Robsta]

I'm not really looking to answer the question so much as understand the concept. So these are points in n,l space? I don't think n and l are correlated, but I'm aware of the constraint n > l > 0 and all are integers.

 

[Robsta]

I'm really just asking by what method you could change the electron's angular momentum without changing its energy.

 

[andrevdh]

Both formulas involve n.

 

[Robsta]

You're not really helping me here at all, sorry. I can draw the energy levels of Hydrogen, but I don't understand what on Earth that has to do with l numbers.

 

[andrevdh]

The energy of an electron has two components - kinetic and potential.
The kinetic component can be split up into a radial and an orbital component.
So the angular momentum can change, that is the orbital energy can change (l-value) at the expense of the radial energy.
This way we get the s, p, d, f ... sub states.
l = 0 gives us the s state of the electron, l = 1 gives us the p state... up l = n-1 for each energy level.

 

[ehild]

The Bohr model assumes circular orbits, and the quantized energy levels result from the assumption that the angular momentum on the orbit is integer multiple of ħ: mrv=nħ. So the quantum number n determines the angular momentum on a circular orbit which radius r is also determined by n, and so is the energy.

Quantum Mechanics is an entirely different model of the atoms. It does not work with well defined orbits, rather with "shells" of different shapes.
The energy is defined by the principal quantum number n. To a certain n, different shapes belong, called s, p, d shells. http://i.ytimg.com/vi/bq8ZLECxKhc/hqdefault.jpg Only the s shells are spherical.
It should not be strange, that the energy depends only on the principal quantum number n. In case of motion in gravitational field, the energy of a planet orbiting about the Sun depends only on the mayor axis, although the orbits are more or less elongated ellipses .

 

[Robsta]

Okay great. So the Bohr model by my understanding is used to give the correct energy values for the hydrogen atom through quantisation of angular momentum, but isn't that helpful apart from that. Could you maybe elaborate on how the different shape shells are associated with different angular momentum values? Or is this not the case?

 

[Robsta]

I'd be keen to see any kind of diagram that can link n and l like in the question.

 

[Robsta]

Hang on, so the n value from the Bohr model is the l value from the quantum model? This is a major breakthrough for me if so

 

[Robsta]

So tell me if this is wrong. The n value in the bohr model corresponds to the l value in the quantum one. And the quantum one also has an m value for the component of l in the z direction. The quantum one has a different n value to the bohr one, one which is total energy. I'm now just not quite sure how it needs all three since surely you can work n out from knowing the angular momentum l?

 

[ehild]

Forget the Bohr model. The three quantum numbers, principal, angular momentum and magnetic quantum numbers appear in Quantum Mechanical model of the hydrogen atom. They are attributes of the wave functions. Have you studied Quantum Mechanics? What have you learned about the wavefunctions and quantum numbers of the Hydrogen atom?
You will find the answer for the problem at many places, here, for example. http://en.wikipedia.org/wiki/Quantum_number

 

[andrevdh]

This might be a bit more than what you currently want:
http://ocw.mit.edu/courses/physics/
Scroll down on the page to see the courses.

You get the angular momentum of the electron in the different shells by setting the orbital
quantum number l to 0,1,2,... (n-1). This gives you the angular momentum of the electron
in the s, p, d ... shells.
$$L = \sqrt{ l (l + 1)} h/2π$$
The kinetic energy of the electron do change if its angular momentum is altered,
but as I previously mentioned its total energy is constant.