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# Ch2 - SJP QM 3220 Ch 2 part 1 Once again the Schrdinger...

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SJP QM 3220 Ch. 2, part 1 Page 1 Once again, the Schrödinger equation: ) , ( ) , ( ) , ( t x V x t x m t t x i ψ ψ ψ + - = 2 2 2 2 (which can also be written Ĥ ψ(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 Ψ (x,t) = u(x) φ(t). Griffiths calls u(x) ≡ ψ(x), but I can't distinguish a "small ψ" from the "capital Ψ" so easily in my handwriting. You’ll find different authors use both of these notations…) So 2 2 2 2 dx x u d t x dt d x u t ) ( ) ( ) ( φ φ = Ψ = Ψ Note full derivative on right hand side! So Schrödinger equation reads (with df dt f , and du dx u ' ) ( 29 ) ( ' ' φ φ φ + - = u V u m u i 2 2 Now divide both sides by Ψ = u • φ. 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

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SJP QM 3220 Ch. 2, part 1 Page 2 So (1) ) ( ) ( t E t i φ φ = (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 Schrödinger equation". You can also write this as which is an "eigenvalue equation". Ĥ = "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 - = 0 φ φ / ) ( ) ( ) ( ) , ( iEt e x u t x u t x - = = Ψ φ 2 2 dx x u d ) ( ) ( )] ( [ ˆ x u E x u H = [ ] ) ( x V dx d m + - 2 2 2 2 ) ( )] ( [ ˆ 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 Ĥ". Why? Let’s see… Notice that Ψ n (x,t) corresponding to u n is ◄══ go back a page! So ◄══ no time dependence (for the probability density ). It's not evolving in time; it's "stationary". (Because ) Convince yourself! (If you think back to de Broglie's free particle with E = , it looks like we had stationary states Ψ n (x,t) with this (same) simple time dependence, e iωt , with . This will turn out to be quite general) If you compute Q (the expectation value of any operator “Q” for a stationary state) the e - iE n t / h in Ψ multiplies the e iE n t / h in Ψ *

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

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