QMformulas

QMformulas - Theoretical Physics KTH Quantum Mechanics...

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Unformatted text preview: Theoretical Physics KTH June 12, 2007 Quantum Mechanics Formula Collection 1 Vectors and operators • To a ket vector α | V ) corresponds a dual bra vector ( V | α ∗ To an operator Ω corresponds an adjoint operator Ω † : Ω | V ) ↔ ( V | Ω † Hermitean operator: Ω † = Ω. Unitary operator: Ω † Ω = 1 , ΩΩ † = 1 • Completeness relations: ∑ i | i )( i | = I , integraltext | ω )( ω | dω = I • Fourier transforms: f ( k ) = 1 (2 π ) 1 / 2 integraldisplay ∞ −∞ e − ikx f ( x ) dx, f ( x ) = 1 (2 π ) 1 / 2 integraldisplay ∞ −∞ e ikx f ( k ) dk • Dirac delta function: integraltext ∞ −∞ f ( x ′ ) δ ( x − x ′ ) dx ′ = f ( x ), δ ( x − x ′ ) = δ ( x ′ − x ), δ ( ax ) = δ ( x ) / | a | , d dx δ ( x − x ′ ) = − d dx ′ δ ( x − x ′ ) = δ ( x − x ′ ) d dx ′ δ ( x − x ′ ) = d dx θ ( x − x ′ ), step function: θ ( x ) = 0 ,x < 0; θ ( x ) = 1 ,x > 1 2 π integraltext ∞ −∞ e ik ( x ′ − x ) dk = δ ( x ′ − x ) 2 Classical mechanics and electromagnetism • Lagrange’s 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 ) Hamilton’s 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 = −∇ φ − 1 c ∂ A ∂t . Magnetic field B = ∇ × A Electromagnetic Lagrangian: L = 1 2 m v 2 − qφ + q c v · A Electromagnetic Hamiltonian: H = | p − q A /c | 2 2 m + qφ 3 Basic Quantum Mechanics • Schr¨odinger equation: i planckover2pi1 ∂ | ψ ( t ) ) ∂t = H | ψ ( t ) ) 1 • Expansion in discrete eigenfunction basis: | ψ ) = ∑ i | ω i )( ω i | ψ ) . Expectation value: ( Ω ) = ∑ i ω i |( ω i | ψ )| 2 . Probability for a system in state | ψ ) to be in state | ω i ) : P ( ω i ) = |( ω i | ψ )| 2 • Expansion in continuous eigenfunction basis: | ψ ) = integraltext dω | ω )( ω | ψ ) . Expectation value: ( Ω ) = integraltext dωω |( ω | ψ )| 2 . Probability for a system in state | ψ ) to have a value for Ω between ω and ω + dω is P ( ω ) dω , where P ( ω ) = |( ω | ψ )| 2 • Canonical commutation relation: [ x,p ] = i planckover2pi1 • Position basis: ( x | ψ ) = ψ ( x ) , ( x | x | ψ ) = xψ ( x ) , ( x | p | ψ ) = − i planckover2pi1 dψ ( x ) dx • Momentum basis: ( p | ψ ) = ψ ( p ) , ( p | p | ψ ) = pψ ( p ) , ( p | x | ψ ) = i planckover2pi1 dψ ( p ) dp • Ehrenfest’s theorem: i planckover2pi1 d dt ( Ω ) = ( [Ω ,H ] ) • Probability conservation: ∂P ∂t = −∇ · j Probability current: j = planckover2pi1 2 mi ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) • Density matrix: ρ = ∑ i p i | i )( i | Ensemble average: ( Ω ) = ∑ i p i ( i | Ω | i )...
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This note was uploaded on 01/19/2012 for the course PHYSICS 101 taught by Professor Vanchu during the Spring '11 term at BC.

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QMformulas - Theoretical Physics KTH Quantum Mechanics...

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