Solving Spherical Symmetry in Hydrogen Atom

In summary: so \frac{3}{4\pi}((\cos\theta)^2 + (\sin\theta)^2((\cos\delta)^2 - (\sin\delta)^2))=\frac{3}{4\pi}\left(\frac{1}{2}\right)^2+\frac{1}{4\pi}\left(\frac{-1}{2}\right)^2+(\frac{1}{4\pi}\left(\frac{1}{2}\right)^2-\frac{1}{4\pi}\left(\frac{-1}{2}\right)^2\)
  • #1
rubertoda
33
0
I have a problem; I am trying to show the spherical symmetry in a hydrogen atom, for a sum over the l=1 shell i.e the sum over the quadratics over three angular wave equations in l=1,

|Y10|^2 + |Y11|^2 + |Y1-1.|^2 .

This should equal up to a constant or a zero to yield no angular dependence. m goes from -l to l.
The problem is that when trying to do this, i get all kinds o different sin, cos formulas, which i cannot reduce to zero or a constant to make the total equation only radial dependent - or am i doing this the wrong way?Anyone who did this problem?
 
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  • #2
rubertoda said:
I have a problem; I am trying to show the spherical symmetry in a hydrogen atom, for a sum over the l=1 shell i.e the sum over the quadratics over three angular wave equations in l=1,

|Y10|^2 + |Y11|^2 + |Y1-1.|^2 .

This should equal up to a constant or a zero to yield no angular dependence. m goes from -l to l.



The problem is that when trying to do this, i get all kinds o different sin, cos formulas, which i cannot reduce to zero or a constant to make the total equation only radial dependent - or am i doing this the wrong way?


Anyone who did this problem?

It sounds like you have the correct approach. Why don't you show us your calculation so we can tell you where you are going wrong.
 
  • #3
gabbagabbahey said:
It sounds like you have the correct approach. Why don't you show us your calculation so we can tell you where you are going wrong.

I have: [tex]|\frac{1}{2}\sqrt{3/2\pi}\sin\theta e^{-i\delta}|^2+|\frac{1}{2}\sqrt{3/2\pi}\sin\theta e^{i\delta}|^2 + |\frac{1}{2}\sqrt{3/\pi}\cos\theta|^2[/tex]

which i finally got to: [tex]\frac{3}{4\pi}((\cos\theta)^2 + (\sin\theta)^2((\cos\delta)^2 - (\sin\delta)^2)[/tex]

which i obviously cannot get to a constant or zero.
 
  • #4
rubertoda said:
I have: [tex]|\frac{1}{2}\sqrt{3/2\pi}\sin\theta e^{-i\delta}|^2+|\frac{1}{2}\sqrt{3/2\pi}\sin\theta e^{i\delta}|^2 + |\frac{1}{2}\sqrt{3/\pi}\cos\theta|^2[/tex]

which i finally got to: [tex]\frac{3}{4\pi}((\cos\theta)^2 + (\sin\theta)^2((\cos\delta)^2 - (\sin\delta)^2)[/tex]

which i obviously cannot get to a constant or zero.

The deltas should disappear when you take the modulus of each term. For a complex number, the modulus is defined as [itex]|z|=\sqrt{zz^*}[/itex], you don't just square the number, you multiply by its complex-conjugate.
 
  • #5


I understand the importance of solving the spherical symmetry in the hydrogen atom. This is a fundamental concept in quantum mechanics and plays a crucial role in understanding the behavior of atoms and molecules.

Firstly, it is important to note that the angular wave equations in l=1 represent the p-orbitals, which have a duality in terms of spherical symmetry. This means that the wave functions are symmetric under inversion through the origin, but not under rotation about the axis passing through the origin.

To show the spherical symmetry in the hydrogen atom, we need to solve for the wave functions that are independent of the angular coordinates (theta and phi). In other words, we need to eliminate the angular dependence in the wave functions. This can be done by using the spherical harmonics, represented by the Ylm term in the wave function, which are eigenfunctions of the angular momentum operator.

In the case of the l=1 shell, the spherical harmonics are Y10, Y11, and Y1-1. As you correctly pointed out, these terms should sum up to a constant or zero to eliminate the angular dependence. However, this is not always the case. The coefficients of the spherical harmonics depend on the orientation of the p-orbital, which can be different for different atoms or molecules. Therefore, the sum may not always reduce to a constant or zero, but it will still be independent of theta and phi.

It is also important to note that the m quantum number represents the orientation of the orbital in space. So, for the l=1 shell, m can take values from -1 to 1, representing the three possible orientations of the p-orbital. This means that the sum over the spherical harmonics should include all three terms, as you have correctly done in your equation.

In conclusion, it is possible that the sum over the l=1 shell may not reduce to a constant or zero, but it will still satisfy the condition of spherical symmetry by being independent of the angular coordinates. I suggest double-checking your calculations and making sure that you are using the correct coefficients for the spherical harmonics. Additionally, consulting with a colleague or a textbook may also help in tackling this problem. I hope this helps in solving the spherical symmetry in the hydrogen atom.
 

1. What is spherical symmetry in the hydrogen atom?

Spherical symmetry in the hydrogen atom refers to the equal distribution of electric charge and energy around the nucleus in all directions. This means that the electron cloud surrounding the nucleus is symmetrical, resembling a sphere.

2. Why is solving spherical symmetry in the hydrogen atom important?

Solving the spherical symmetry in the hydrogen atom is important because it allows us to accurately describe and understand the behavior and properties of the atom. It also serves as a basis for understanding the electronic structure of more complex atoms.

3. How is spherical symmetry in the hydrogen atom solved?

Spherical symmetry in the hydrogen atom is solved using mathematical equations and principles, such as the Schrödinger equation and the quantum mechanical model. These equations take into account the principles of quantum mechanics and the interactions between the electron and the nucleus.

4. What are the implications of solving spherical symmetry in the hydrogen atom?

By solving spherical symmetry in the hydrogen atom, we gain a better understanding of the behavior of electrons in atoms. This knowledge has many practical applications, such as in the development of new materials and technologies.

5. Are there any limitations to solving spherical symmetry in the hydrogen atom?

While solving spherical symmetry in the hydrogen atom is a major breakthrough in our understanding of atomic structure, it is not a complete description of all atoms. Other factors, such as the presence of multiple electrons and the effects of relativity, must be taken into account for a more accurate understanding of atomic behavior.

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