What Quantum Numbers Define a Hydrogen Atom's State with No Angular Dependence?

[unscientific]

Homework Statement

What quantum numbers are used to define state of hydrogen? The wavefunction has no angular dependence. Find the values of all the angular momentum quantum numbers for the electron.

Homework Equations

The Attempt at a Solution

The numbers are n, l and m.

n: Energy level
l(l+1): Eigenvalues of total orbital angular momentum
m: z component of orbital angular momentum

The complete wavefunction is given by: ##\psi = u_n^l Y_l^m##.

Thus the only spherical harmonic that doesn't have angular dependence is ##Y_0^0 = \sqrt{\frac{1}{4\pi}}##.

Thus the wavefunctions are ##\sqrt{\frac{1}{4\pi}}u_n^0##.

Thus n = any integer, l = 0, m = 0.

I'm slightly bothered by the term 'spatial part' of the wavefunction.

 

[BvU]

"Spatial part" as opposed to "spin part". The exercise probably doesn't want you to worry about spin (my guess -- change that if you just finished a chapter on spin...)

 

[unscientific]

"Spatial part" as opposed to "spin part". The exercise probably doesn't want you to worry about spin (my guess -- change that if you just finished a chapter on spin...)

We learn about the gross structure of Hydrogen, which ignores spin as the Hamiltonian is the KE of the nucleus and electron, and the potential energy.

Are my answers right then?

 

[BvU]

I would say yes. A nitpicker would argue n isn't a quantum number for angular momentum. In that case the answer is: l = 0 and m = 0

 

[paranormal]

In general the wavefunction is a function of space and time, but I assume you mean the part that depends only on the spatial coordinates. In that case, the wavefunction for the hydrogen atom is a product of the radial wavefunction and the spherical harmonic, which represents the angular part. The radial wavefunction is dependent on the principal quantum number, n, while the spherical harmonic is dependent on the angular momentum quantum numbers, l and m.

To find the values of all the angular momentum quantum numbers for the electron, we can use the following rules:

1. The principal quantum number, n, can take on any positive integer value.

2. The angular momentum quantum number, l, can take on values from 0 to n-1.

3. The magnetic quantum number, m, can take on values from -l to l.

Therefore, for the electron in a hydrogen atom, the possible values for l are 0, 1, 2, ..., n-1, and the possible values for m are -l, -l+1, -l+2, ..., l-2, l-1, l.

In the case of the wavefunction having no angular dependence, as stated in the problem, it means that l = 0 and m = 0. This corresponds to the s orbital, which is spherical in shape and has no angular dependence. Therefore, the only possible angular momentum quantum numbers for the electron in this case are l = 0 and m = 0.