lecture19_umn - Lecture 19 Variational Calculations on the...

Info icon This preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 19 October 26, 2009 Variational Calculations on the H Atom Gaussian Functions and their linear combinations
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Variational Calculations on the H Atom Consider a system where the answer is known exactly: H atom. Try different variational protocols . Replace the exact 1s wave function (19-1) with a different functional dependence on r , namely (19-2) N normalization constant and α variational constant . " 100 r , # , $ ( ) = Z 3 / 2 % e & Zr " 1 s r , # , $ ; % ( ) = Ne & % r 2
Image of page 2
Normalization Constant We can find N (19-3) (19-4) " 1 s r , # , $ ; % ( ) " 1 s r , # , $ ; % ( ) = 1 = N 2 e & 2 % r 2 r 2 dr sin # d # d $ 0 2 ( 0 ( 0 ) ( = 4 N 2 r 2 e & 2 % r 2 dr 0 ) ( = 4 N 2 1 8 % 2 % * + , - . / 1/ 2 0 1 2 2 3 4 5 5 N = 2 ! " # $ % & ( 3/ 4
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Variational Condition Variational condition for a normalized wave function is (19-5) Evaluate the expectation values of the kinetic and potential energy operators . 0 = d d ! " 1 s r , # , $ ; ! ( ) H " 1 s r , # , $ ; ! ( ) [ ] = d d ! " 1 s r , # , $ ; ! ( ) % 1 2 & 2 % 1 r " 1 s r , # , $ ; ! ( ) ( ) ) * + , ,
Image of page 4
Kinetic Energy Operator In spherical polar coordinates (and a.u.) (19-6) Angular momentum of an s wave function is 0 , we need only consider the r component: (19-7) Take the constants out front, and integrate over θ , φ to get 4 π . " 2 r , # , $ ( ) = % 1 r 2 & & r r 2 & & r % L 2 ( ) * + , ! 1 s r , " , # ; $ ( ) % 1 2 & 2 ! 1 s r , " , # ; $ ( ) = % 1 2 2 $ ( ) * + , - 3 /2 4 ( ) e %$ r 2 1 r 2 d dr r 2 d dr ( ) * + , - 0 . / e %$ r 2 r 2 dr
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Evaluating the Integral The d/dr operators only operate on the wave function. First differentiate the exponential function multiply by r 2 ; differentiate the product once again ; divide by r 2 ; then finally include the volume element (19-8) = 2 " # $ % & ( ) 3/ 2 2 # ( ) 6 " r 2 e * 2 " r 2 dr 0 + , * 4 " 2 r 4 e * 2 " r 2 dr 0 + , ( ) " 1 2 2 # $ % & ( ) * 3/ 2 4 $ ( ) e " # r 2 1 r 2 d dr r 2 d dr % & ( ) * 0 + , e " # r 2 r 2 dr =
Image of page 6
Kinetic Energy Integral In an integral table (19-9) This gives (19-10) r 2 n e " ar
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern