2_Formalism - II The Formalism of Quantum Mechanics Grifths...

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II. The Formalism of Quantum Mechanics Griffiths chapter 3 1
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• Less writing. • Makes mathematical manipulations easier. • Independent of basis. ψ 1 | ψ 2 d r ψ 1 ( r ) ψ 2 ( r ) (single particle case) (momentum space) Bars denote Fourier transforms of ψ . 2 A. Dirac Notation ψ 1 | ψ 2 d p ¯ ψ 1 ( p ) ¯ ψ 2 ( p )
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• | ψ 2 > : “ket”. It is a vector. (More later.) • < ψ 1 | : “bra”. It is a linear function of vectors. For example: ψ 1 | ψ 2 d r ψ 1 ( r ) ψ 2 ( r ) ψ 1 | = d r ψ 1 ( r )[ · · · ] • It will inevitably operate on a ket to give a complex number (inner product). • Thus Dirac notation is sometimes called “bra-ket” notation. 3 A. Dirac Notation
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Some properties: ψ 1 | ψ 2 = ψ 2 | ψ 1 Let c be a complex number. ψ 1 | c ψ 2 = c ψ 1 | ψ 2 c ψ 1 | ψ 2 = c ψ 1 | ψ 2 ψ 3 | ψ 1 + ψ 2 = ψ 3 | ψ 1 + ψ 3 | ψ 2 ψ 1 and ψ 2 are orthogonal if ψ 1 | ψ 2 = 0 The normalization condition is d r | ψ ( r ) | 2 = 1 ψ | ψ = 1 d r ψ ( r ) ψ ( r ) = 1 If the functions are both orthogonal and normalized, then they are orthonormal . distributive commutative associative 4 A. Dirac Notation
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B. Hilbert Space v • A vector exists geometrically, independent of a coordinate system. (Griffiths 3.1) 5 v • We can represent it in terms of a basis set, ( ê x , ê y , ê z ), for example, where ( ê x , ê y , ê z ) are unit vectors that span the space. x z y ˆ e x ˆ e y ˆ e z v = v x ˆ e x + v y ˆ e y + v z ˆ e z x’ z’ y’ ˆ e x ˆ e y ˆ e z v v = v x ˆ e x + v y ˆ e y + v z ˆ e z v
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• The inner product of two vectors is a scalar. 6 v · u = v x u x + v y u y + v z u z v · u = v x u x + v y u y + v z u z • The length of the vector is ( v v ). • A Hilbert space is the same type of space except it is for functions instead of discrete element (traditional) vectors. x f(x) x 1 x 2 x 3 x n Δ f(x 1 ) f(x 2 ) f(x 3 ) f(x n ) f = [ f ( x 1 ) , f ( x 2 ) , · · · f ( x n )] ( x 1 , x 2 , · · · , x n ) with respect to the basis v = ( v x , v y , v z ) e x , ˆ e y , ˆ e z ) with respect to the basis Compare with B. Hilbert Space
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7 • The limit n while the spacing Δ 0 gives us the state function (wave function). • These are sometime called state vectors . • Functions are vectors in Hilbert space (a vector space in infinite dimensions). f = [ f ( x 1 ) , f ( x 2 ) , · · · f ( x n )] ( x 1 , x 2 , · · · , x n ) with respect to the basis v = ( v x , v y , v z ) e x , ˆ e y , ˆ e z ) with respect to the basis Compare with x f(x) x 1 x 2 x 3 x n Δ f(x 1 ) f(x 2 ) f(x 3 ) f(x n ) B. Hilbert Space
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8 Some properties: 1. The space is linear. - If ψ space and c is a constant then c ψ space. - If ψ 1 , ψ 2 space then ψ 1 + ψ 2 space. ψ 1 | ψ 2 d r ψ 1 ( r ) ψ 2 ( r ) exists. 3. The length of a vector is related to the inner product < ψ | ψ > . This is why we always say that wave functions have to be “square integrable”.
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