Solving Spherical Symmetry in Hydrogen Atom

[rubertoda]

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?

 

[gabbagabbahey]

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.

 

[rubertoda]

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: $$|\frac{1}{2}\sqrt{3/2\pi}\sin\theta e^{-i\delta}|^2+|\frac{1}{2}\sqrt{3/2\pi}\sin\theta e^{i\delta}|^2 + |\frac{1}{2}\sqrt{3/\pi}\cos\theta|^2$$

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

which i obviously cannot get to a constant or zero.

 

[gabbagabbahey]

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

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

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 $|z|=\sqrt{zz^*}$, you don't just square the number, you multiply by its complex-conjugate.