it seems that when we look at the x,y,x behavior of a state, we only look at the r, thetta, and exp(i*phi) parts. why dont we use the exp(-r/a_0) part??? that decaying exponential in 'r' contains all 3 components x,y,z, but in my discussions with peers and in these solutions, it isnt talked about when we consider x,y,z dependence.
i understand the logic of mixing states to get x-spice, radial spice, z-spice etc., but dont see why that decaying exponential is ignored when looking at the x,y,z dependence. is it that it just cuts-off the wave-function at large r (or large x,y,z)?
I think we keep the exponential in terms of r because it has radial symmetry plus we gain nothing from writing it in cartesian coordinates. In fact, it makes the equation look more complicated, whereas rewriting the r, theta and phi in terms of x, y, and z relate the state function's asymmetry to a coordinate system that we are VERY familiar with.
I forgot if anything having to do with crystals was going to be on the final. Does anyone remember the consensus on Sunday? Geno makes a good point. To elaborate, we're talking about mixed states here where everything has that factor of e^-r/na. They're not interesting because they don't say anything interesting about oscillation or rotation or overall trajectory when we make the transformation with the L_z operator. What're the most interesting parts, are the combinations of x's and y's and the differences in energy between each state. These tell you the nature of rotation, and the frequency of that rotation.
These are really helpful.
ReplyDeleteit seems that when we look at the x,y,x behavior of a state, we only look at the r, thetta, and exp(i*phi) parts. why dont we use the exp(-r/a_0) part??? that decaying exponential in 'r' contains all 3 components x,y,z, but in my discussions with peers and in these solutions, it isnt talked about when we consider x,y,z dependence.
ReplyDeletei understand the logic of mixing states to get x-spice, radial spice, z-spice etc., but dont see why that decaying exponential is ignored when looking at the x,y,z dependence. is it that it just cuts-off the wave-function at large r (or large x,y,z)?
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I think we keep the exponential in terms of r because it has radial symmetry plus we gain nothing from writing it in cartesian coordinates. In fact, it makes the equation look more complicated, whereas rewriting the r, theta and phi in terms of x, y, and z relate the state function's asymmetry to a coordinate system that we are VERY familiar with.
ReplyDeleteTo summarize, e^-r is not exciting.
I forgot if anything having to do with crystals was going to be on the final. Does anyone remember the consensus on Sunday?
ReplyDeleteGeno makes a good point.
To elaborate, we're talking about mixed states here where everything has that factor of e^-r/na. They're not interesting because they don't say anything interesting about oscillation or rotation or overall trajectory when we make the transformation with the L_z operator. What're the most interesting parts, are the combinations of x's and y's and the differences in energy between each state. These tell you the nature of rotation, and the frequency of that rotation.