Quantum Theory of the Hydrogen Atom: Separation of Variables in Schrödinger's Equation, Slides of Physics

Download Quantum Theory of the Hydrogen Atom: Separation of Variables in Schrödinger's Equation and more Slides Physics in PDF only on Docsity! Review of Modern Physics Zeroeth Semester Lecture 20: k Docsity.com Quantum Theory of Hydrogen shrödinger's equation for hydrogen separation of variables Docsity.com The potential looks quite simple, but it is a function of r, not x or (xyz). What can we do about that? Lots of luck solving that!  2 2 2 2 2 2 2x + y + z = r ⇒ r = x + y + z   22 2 2 2 2 2 0 2m e + E - - = 0 x 4 ε x +y +z                 (To say nothing of the fact that we shouldn’t be using the 1D Schrödinger equation for a 3D problem.) We need to let the symmetry of the problem dictate our mathematical approach. Docsity.com The spherically symmetric potential ―tells‖ us to use spherical polar coordinates! http://hyperphysics.phy-astr.gsu.edu/hbase/sphc.html Docsity.com In spherical polar coordinates, r is the length of the radius vector from the origin to a point (xyz)  2 2 2r = x +y +z ,  is the angle between the radius vector and the +z axis   -1 2 2 2 z θ = cos , x + y +z           and  is the angle between the projection of the radius vector onto the xy plane and the +x axis -1 y = tan . x        point (xyz) Docsity.com When we solved Schrödinger's equation in one dimension, we found that one quantum number was necessary to describe our systems. At the end of this section Beiser tells what the quantum numbers for the hydrogen atom are, and gives their possible values. I’ll skip that for now, because until we see where they come from and what they mean, they aren't of much use to us. For example, in the Bohr atom, the electron moves in an orbit, but we need only one parameter to specify its position in the fixed orbit, so we only need one quantum number. Here, in three dimensions and with three boundary conditions, we will find that we need three quantum numbers to describe our electron. Docsity.com Another comment: we are really solving Schrödinger's equation for the electron in a hydrogen atom, aren't we. Nevertheless, we talk about solving the "hydrogen atom," because our solution will provide us with much of what we need to know about hydrogen. 6.2 Separation of Variables We now ―solve‖ the hydrogen atom. Here are some math activities,  solving linear algebraic equations  solving coupled algebraic equations (e.g. xy together)  solving linear differential equations  solving coupled differential equations (e.g. derivatives mixed together) Docsity.com We have a coupled linear differential equation to solve. Maybe if we are clever, like we were with the tunneling calculation, we can make the problem easier. A big improvement would be to uncouple the variables. Stated more mathematically, when we have an equation like the one above, we like to see if we can "separate" the variables; i.e., "split" the equation into different parts, with only one variable in each part. Our problem will be much simplified IF we can write (r,θ, ) = R(r) (θ) ( ) = R       I’ve never known you (as a class) to shy away from leaps of logic, so how about if we assume that and see where it leads us? Docsity.com                      2 2 2 2 2 2 2 0 sin θ d dR sinθ d d r + sinθ R dr dr dθ dθ 2mr sin θ e 1 d + + E = - . 4 ε r d This equation has the form f is a function of r and  only, and g is a function of  only. f(r,θ) = g( ) ―How can this be? The RHS has only  in it (but no r and ), and the LHS has only r and  in it (but no ).‖ And LHS=RHS? ―And yet you’re telling me LHS = RHS. I repeat, how can this be?‖ Yes, you heard right! Docsity.com Only one way!   f(r,θ) = a constant, independent or r,θ, and = g( )  ―Are you telling me everything is just a constant?‖ Absolutely not! It’s just that the particular combination of terms on the LHS happens to add up to a constant, which is the same as the constant given by the particular combination of terms on the RHS.                      2 2 2 2 2 2 2 0 sin θ d dR sinθ d d r + sinθ R dr dr dθ dθ 2mr sin θ e 1 d + + E = constant = - . 4 ε r d This is really good. We've taken the one nasty equation in r, and separated it into two equations, one in r, and the other in  only. Do you think maybe we can separate the r part… Docsity.com It turns out (although we won't do the math in this course) that the ―constant‖ must be the square of an integer. If not, our differential equations have no solution. Thus, we can write the RHS of this equation as    2 21 d = m . d Where did this mℓ come from? It’s an integer. We just ―happened‖ to give it that ―name.‖ Kind of hard to see, but that’s a lowercase script ℓ                      2 2 2 2 2 2 2 0 sin θ d dR sinθ d d r + sinθ R dr dr dθ dθ 2mr sin θ e 1 d + + E = constant = - . 4 ε r d Docsity.com