How can two atomic orbitals with the same n value be orthogonal?

[fsci]

I am wondering how two orbitals of same n values can be orthogonal, for example how are a 2s and 3s orbital orthogonal?
What I understand is a property of orthogonality is the product of the two wave functions integrate to zero over all space. I tried to look at this graphically and categorize overlapping regions as either positive or negative products and then cancel out positive and negative regions to yield zero, but what I am having trouble is that the 3s orbital is larger than the 2s orbital, so how can they possible integrate to zero?
If someone could also explain the significance/implications of all orbitals being orthogonal that would be helpful too! I do not understand the importance of orthogonality in orbitals!
Thank you!

 

[Quantum Defect]

I am wondering how two orbitals of same n values can be orthogonal, for example how are a 2s and 3s orbital orthogonal?
What I understand is a property of orthogonality is the product of the two wave functions integrate to zero over all space. I tried to look at this graphically and categorize overlapping regions as either positive or negative products and then cancel out positive and negative regions to yield zero, but what I am having trouble is that the 3s orbital is larger than the 2s orbital, so how can they possible integrate to zero?
If someone could also explain the significance/implications of all orbitals being orthogonal that would be helpful too! I do not understand the importance of orthogonality in orbitals!
Thank you!

There are radial nodes that you usually cannot see in the higher n (n>=2) representations, as typically drawn. 2s has one radial node, 3s has two radial nodes. On either side of the radial node there is a change in sign for the wave function.

When you do the integral for any of the products you will get positive regions cancelling the negative regions.

 

[fsci]

There are radial nodes that you usually cannot see in the higher n (n>=2) representations, as typically drawn. 2s has one radial node, 3s has two radial nodes. On either side of the radial node there is a change in sign for the wave function.

When you do the integral for any of the products you will get positive regions cancelling the negative regions.

Ok thank you. I may be heading in the wrong direction... But how do the product of a 1s x 3s and then product of a 2s x 3s both integrate to zero when 1s and 2s orbitals are not equal to each other?

 

[Quantum Defect]

Ok thank you. I may be heading in the wrong direction... But how do the product of a 1s x 3s and then product of a 2s x 3s both integrate to zero when 1s and 2s orbitals are not equal to each other?

You might look into some Linear Algebra, Differential Equations textbooks. For the H-atom wave functions, you will have an infinite set of wavefunctions, that are solutions to the Schroedinger Eq. Each l=0 wavefunction will be "orthogonal" to the other, as long as n_1<> n_2.

This is kind of like the way that the unit vectors in 3D space are all orthogonal to one another. x_hat dot y_hat = 0, x_hat dot z_hat = 0, y_hat dot z_hat =0.

The solutions to the Schoredinger Eq for H are like these basis vectors in 3D space, except the space is infinite.