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Unformatted text preview: Theoretical Physics KTH December 11, 2007 Quantum Mechanics Formula Collection 1 Vectors and operators To a ket vector  V ) corresponds a dual bra vector ( V  To an operator corresponds an adjoint operator :  V ) ( V  Hermitean operator: = . Unitary operator: = 1 , = 1 Completeness relations: i  i )( i  = I , integraltext  )(  d = I Fourier transforms: f ( k ) = 1 (2 ) 1 / 2 integraldisplay e ikx f ( x ) dx, f ( x ) = 1 (2 ) 1 / 2 integraldisplay e ikx f ( k ) dk Dirac delta function: integraltext f ( x ) ( x x ) dx = f ( x ), ( x x ) = ( x x ), ( ax ) = ( x ) /  a  , d dx ( x x ) = d dx ( x x ) = ( x x ) d dx ( x x ) = d dx ( x x ), step function: ( x ) = 0 , x < 0; ( x ) = 1 , x > 1 2 integraltext e ik ( x x ) dk = ( x x ) 2 Classical mechanics and electromagnetism Lagranges equations: L q i d dt L d q i = 0 Lagrangian: L = T V , Action: S = integraltext t 1 t Ldt Canonical momentum conjugate to q i : p i = L q i Generalized force conjugate to q i : F i = L q i Hamiltonian: H = i p i q i L = T + V = p 2 2 m + V ( r ) Hamiltons equations: q i = H p i , p i = H q i Force on charge q : F = q ( E + v c B ) (no factor c in SI units) Electric field: E = 1 c A t . Magnetic field B = A Electromagnetic Lagrangian: L = 1 2 m v 2 q + q c v A Electromagnetic Hamiltonian: H =  p q A /c  2 2 m + q 3 Basic Quantum Mechanics Schrodinger equation: i planckover2pi1  ( t ) ) t = H  ( t ) ) 1 Expansion in discrete eigenfunction basis:  ) = i  i )( i  ) . Expectation value: ( ) = i i ( i  ) 2 . Probability for a system in state  ) to be in state  i ) : P ( i ) = ( i  ) 2 Expansion in continuous eigenfunction basis:  ) = integraltext d  )(  ) . Expectation value: ( ) = integraltext d (  ) 2 . Probability for a system in state  ) to have a value for between and + d is P ( ) d , where P ( ) = (  ) 2 Canonical commutation relation: [ x, p ] = i planckover2pi1 Position basis: ( x  ) = ( x ) , ( x  x  ) = x ( x ) , ( x  p  ) = i planckover2pi1 d ( x ) dx Momentum basis: ( p  ) = ( p ) , ( p  p  ) = p ( p ) , ( p  x  ) = i planckover2pi1 d ( p ) dp Ehrenfests theorem: i planckover2pi1 d dt ( ) = ( [ , H ] ) Probability conservation: P t = j Probability current: j = planckover2pi1 2 mi ( ) Density matrix: = i p i  i )( i  Ensemble average: ( ) = i p i ( i ...
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This note was uploaded on 04/09/2008 for the course PHYS 502 taught by Professor Hummin during the Fall '08 term at University of Cincinnati.
 Fall '08
 hummin
 mechanics, Theoretical Physics

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