class29 - Today everything you ever wanted to know about...

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Unformatted text preview: Today everything you ever wanted to know about the Schrdinger equation but never dared to ask. From last class: t t x i t x t x V x t x m = + - ) , ( ) , ( ) , ( ) , ( 2 2 2 2 h h Time dependent Schrdinger equation in 1D: t t z y x i t z y x z y x ke z m t z y x y m t z y x x m t z y x = + +- + + + - ) , , , ( ) , , , ( ] ) /( [ 2 ) , , , ( 2 ) , , , ( 2 ) , , , ( 2 / 1 2 2 2 2 2 2 2 2 2 2 2 2 2 h h h h Time dependent Schrdinger equation in 3D (Cartesian): Simplification #1 when V(x) only (or V(x,y,z) in 3D): (x,t) separates into position part dependent part (x) and time dependent part (t) =exp(-iEt/ ). (x,t)= (x) (t) Plug in, get equation for (x) Most physical situations (e.g. H atom) no time dependence in V! t t x i t x t x V x t x m = + - ) , ( ) , ( ) , ( ) , ( 2 2 2 2 h h Plug in, get equation for (x) Seems like a good exercise for HW ;) Same as in the book with V(x) for U(x) Time independent Schrdinger equation ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V x x m = + - h (Will use this eqn. in all Schrdinger Eqn problems in this class!) Time Independent Schrodinger Equation ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V x x m = + - h (Basically just conservation of Energy) KE + PE = E (Basically just conservation of Energy) If (x)=Ae i(kx- t) (plane wave) d (x)/dx = ikAe i(kx- t) d 2 (x)/dx 2 = -k 2 Ae i(kx- t) = -k 2 (x) Example: How to use the Schrdinger equation Example: How to use the Schrdinger equation Say you wanted to calculate electron waves in a thin wire (1-D). Whats the first step youd take? a. figure out how many electrons will be interacting t t x i t x t x V x t x m = + - ) , ( ) , ( ) , ( ) , ( 2 2 2 2 h h Q1 a. figure out how many electrons will be interacting b. figure out what general solutions will be by b....
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class29 - Today everything you ever wanted to know about...

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