Ch2 - SJP QM 3220 Ch. 2, part 1 Page 1 Once again, the...

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Unformatted text preview: SJP QM 3220 Ch. 2, part 1 Page 1 Once again, the Schrdinger equation: ) , ( ) , ( ) , ( t x V x t x m t t x i + - = 2 2 2 2 (which can also be written (x,t) if you like.) And once again, assume V = V(x) (no t in there!) We can start to solve the PDE by SEPARATION OF VARIABLES . Assume (hope? wonder if?) we might find a solution of form (x,t) = u(x) (t). Griffiths calls u(x) (x), but I can't distinguish a "small " from the "capital " so easily in my handwriting. Youll find different authors use both of these notations) So 2 2 2 2 dx x u d t x dt d x u t ) ( ) ( ) ( = = Note full derivative on right hand side! So Schrdinger equation reads (with df dt f , and du dx u ' ) ( 29 ) ( ' ' +- = u V u m u i 2 2 Now divide both sides by = u . i h f ( t ) f ( t ) function of time only 1 2 3 = - h 2 2 m u ''( x ) u ( x ) + V ( x ) function of space only 1 2 4 4 4 3 4 4 4 This is not possible unless both sides are constants . Convince yourself; that is the key to the "method of separation of variables". Let's name this constant "E". [Note units of E are time or dist m 2 2 2 ) ( or simply V(x), either way, check, it's Energy !] (S. Pollock, taken from M. Dubson) with thanks to J. Anderson for typesetting. Fall 2008 SJP QM 3220 Ch. 2, part 1 Page 2 So (1) ) ( ) ( t E t i = (2) ) ( ) ( ) ( ) ( ' ' x u E x u x V x u m = +- 2 2 These are ordinary O.D.E.'s Equation (1) is about as simple as ODE's get! Check any constant, it's a linear ODE. (1 st order linear ODE is supposed to give one undetermined constant, right?) This is "universal", no matter what V (x) is, once we find a u(x), we'll have a corresponding But be careful, that u(x) depends on E, (2) ) ( ) ( ) ( ) ( ' ' x u E x u x V x u m = +- 2 2 . this is This is the "time independent Schrdinger equation". You can also write this as which is an "eigenvalue equation". = "Hamiltonian" operator = In general, has many possible solutions. eigenfunctions eigenvalues u 1 (x), u 2 (x), u n (x) may all work, each corresponding to some particular eigenvalue E 1 , E 2 , , E n . (What we will find is not any old E is possible if you want u(x) to be normalizable and well behaved. Only certain E's, the E n 's, will be ok.) (S. Pollock, taken from M. Dubson) with thanks to J. Anderson for typesetting. Fall 2008 / ) ( iEt e t- = / ) ( ) ( ) ( ) , ( iEt e x u t x u t x- = = 2 2 dx x u d ) ( ) ( )] ( [ x u E x u H = [ ] ) ( x V dx d m +- 2 2 2 2 ) ( )] ( [ x u E x u H = SJP QM 3220 Ch. 2, part 1 Page 3 Such a u n (x) is called a "stationary state of ". Why? Lets see Notice that n (x,t) corresponding to u n is go back a page!...
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This note was uploaded on 02/27/2012 for the course PHYSICS 3220 taught by Professor Stevepollock during the Fall '08 term at Colorado.

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Ch2 - SJP QM 3220 Ch. 2, part 1 Page 1 Once again, the...

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