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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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