Lect12_1DMotion

# Lect12_1DMotion - Simple Problems in 1D To Describe the...

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Simple Problems in 1D To Describe the motion of a particle in 1D, we need the following four QM elements: Putting them together yields the Schrödinger wave equation: ) ( ) ( t H t dt d i ! ! = h ) ( 2 2 X V m P H + = ) , ( ) ( t x t x ! ! = ) , ( ) ( t x x i t P x ! ! " " # = h ) , ( ) ( ) , ( 2 ) , ( 2 2 2 t x x V t x x m t x dt d i ! ! ! + " " # = h h Coordinate Representation In coordinate representation, we can make the following substitutions: Example: This is just a shortcut, whose validity is readily verified via Dirac notation ) , ( ) ( t x t ! ! " x X ! x i P ! ! " h ! " " # = \$ x x x dx i P ) ( ) ( % % h ) ( ) , ( ) ( ) , ( x i x f x dx P X f x ! ! " # # \$ % & h

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Wavevector basis Instead of momentum, it is often convenient to use wavevector states: h P K = k k k K = ) ( k k k k ! " = ! # ! 2 x k i e k x = k p h = 1 = ! + " " # k k dk [ ] i K X = , [ ] 0 , = K P k k k k k k p p p p ! = ! " = ! " = ! " = ! h h h h 1 ) ( 1 ) ( ) ( # # # " k = h p p = h k Bound States and/or Scattering States Problems dealing with motion in 1D fall into one of two categories 1. Bound-state problems: V ( x ) < E over finite region only Energy levels are discrete Typical problem: Find Energy eigenvalues: { E n }; n =1,2,3,…
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