Total energy of Coulomb model of Hydrogen atom

In summary, the conversation discusses a Coulomb model of a hydrogen atom and specifies the Hamiltonian operator, energy eigenfunction, and quantum numbers associated with the system. The formulas for the magnitude and z-component of orbital angular momentum are provided, but the total energy cannot be evaluated without more information. The conversation also mentions a relation between principal quantum number and energy, but it is unclear if it applies to the hydrogen atom.
  • #1
Roodles01
128
0

Homework Statement


Hi, my question is regardng a Coulomb model of an H atom specified with Hamiltonian operator, Hhat, by spherical coordinates of energy eigenfunction
ψ2,1,-1 (r,θ, ∅) =(1/ 64∏a02)1/2 r/a0 e-r/2a0 sinθ e-iθ

Principal quantum numer n = 2
orbital an mom l = 1
magnetic quantum number m = -1

I must specify magnitude of orbital ang momentum, Lhat2, and z-component of orbital ang momentum as well as total energy.


Homework Equations


so;
Lhat2 = l (l + 1) hbar2
Lz = m hbar (1/ √2∏) eim∅



The Attempt at a Solution


Lhat2 = 1 (1 + 1)hbar2
Lhat2 = 2 hbar2

Lz = m hbar (1/ √2∏) eim∅
Lz = -1 hbar / √2∏ ei-1∅
= -1 hbar

but what about total energy?
I'm lost in the forest again and can't see the wood for the trees.
 
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  • #2
There is a relation between principal quantum number and energy that you should know.
 
  • #3
Hmmm!

Well, En = (n + 1/2) hbar ω0

and

En = n22 hbar2 / 2mL2
where principal quantum number n = 0, 1, 2 . . .

but I can't evaluate ω0 and I don't have the mass for the other one . . do I?
 
  • #4
Ah!
n = principal quantum number
m = mass of electron
L2 = square of magnitude of orbital ang momentum.

so . . . .

E = 22 ∏2 hbar2 / 2(9.1x10-31) 2 hbar2
E = -2∏hbar/m

er! surely too messy and probably wrong.
 
  • #5
Roodles01 said:
but I can't evaluate ω0 and I don't have the mass for the other one . . do I?
Why? Your formula (edit: wait, those are the wrong formulas) assumes an infinite central mass, which is usually a reasonable approximation in a hydrogen atom.

The total energy of the system does not matter unless you want to consider special relativity.
 
Last edited:
  • #6
Roodles01 said:
Hmmm!

Well, En = (n + 1/2) hbar ω0

and

En = n22 hbar2 / 2mL2
where principal quantum number n = 0, 1, 2 . . .

Does either of these formulas apply to the hydrogen atom?
 

1. What is the Coulomb model of the hydrogen atom?

The Coulomb model of the hydrogen atom is a theoretical model that describes the behavior of the electron and proton in a hydrogen atom using classical mechanics and the laws of electrostatics, specifically Coulomb's law.

2. How is the total energy of the hydrogen atom calculated in the Coulomb model?

In the Coulomb model, the total energy of the hydrogen atom is calculated by adding together the potential energy, which is determined by the distance between the electron and proton, and the kinetic energy, which is determined by the electron's velocity.

3. What is the significance of the total energy in the Coulomb model of the hydrogen atom?

The total energy in the Coulomb model represents the stability of the hydrogen atom. If the total energy is negative, the electron and proton are attracted to each other and the atom is stable. If the total energy is positive, the electron and proton will repel each other and the atom is not stable.

4. How does the total energy of the hydrogen atom change as the electron moves closer or further from the proton?

In the Coulomb model, as the electron moves closer to the proton, the potential energy decreases and the kinetic energy increases, resulting in a decrease in the total energy. As the electron moves further from the proton, the potential energy increases and the kinetic energy decreases, resulting in an increase in the total energy.

5. Can the total energy of the hydrogen atom be negative?

Yes, in the Coulomb model, the total energy of the hydrogen atom can be negative if the electron and proton are close enough that the attractive force between them is greater than the repulsive force. This indicates a stable and bound state of the hydrogen atom.

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