Here is a video on the 1st excited states of hydrogen. It includes a HW problem at the end.
HW6:
1a) calculate the expectation value of x for the state discussed in this video, and
b) calculate the expectation values of x for the true sp2-1 state, the one with 1/sqrt(3) and sqrt(2/3) coefficients, as we discussed in class today. (How do they compare. Discuss here in the comments which one is larger as soon as you get results.)
Also, would someone please post those sp2 states, as linear combinations of our s, x, y, z basis states, here soon so everyone can see? Sooner the better. Much appreciated!
2. a) Write the 3 in-plane sp2 states in terms of \(\psi_{2s}, \: \psi_{2px}, \: \psi_{2py}\).
b) Figure out the expectation values of x and y for each state. (Many of the integrals in this problem are, i imagine, equivalent. You can use that. Keeping track of the cross-term coefficients is important too.
c) On an x-y plot, show the location of the expectation value of the vector r for each of the 3 sp2 states.
"Where does those states come from?"
Originally, from solving the wave equation. In that way we find that there are four 1st excited states. Let say they are the four \(\psi_{2,l,m}\) states. Then we use those to construct the 2s, 2px, 2py, 2pz states, which I think provide a better basis. Then we are seeking to make states that are at 120 angles to each other and in the x-y plane, and we do that via linear combinations of the 2s, 2px, 2py, 2pz states. Does that make sense?
No. I think they are all identical. You can test that by integrating each one squared and see if it integrates to 1.
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ReplyDelete#1 part a) answer is larger by a factor of 2?
ReplyDeleteI think I got something a little bit different for the 2nd one. It makes a difference in interpretation. Does your math intuition tell you anything regarding which one should be larger.
DeleteHmm. Well, you may be right, but that sounds a bit too complicated to me. The sp2 states, see below, are all normalized to start with.
DeleteI will post the 3 in-plane sp states here in 3 comments. Here is one:
ReplyDelete\(\psi_{sp2-1} = \sqrt{1/3}\:\psi_{2s} + \sqrt{2/3}\:\psi_{2px}\)
\( \psi_{sp2-2} = \sqrt{1/3}\:\psi_{2s} - \sqrt{1/6}\:\psi_{2px} + \sqrt{1/2}\:\psi_{2py} \)
ReplyDeleteCan someone else post the third one?
ReplyDeleteSame thing as Psi(sp2-2) but the y term is negative
ReplyDelete"Where does those states come from?" Do you mean the x, y z and s states, or the sp2 states? That is an important question. I will post an answer into this post.
ReplyDelete