Chapter7_1 - ij = 0 if i j and ij = 1 if i = j, p 1 = p x ,...

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PHY3063 R. D. Field Department of Physics Chapter7_1.doc University of Florida Quantum Mechanics in Three Dimensions Schrödinger Equation: In three dimensional space we have dt t r d i t r H op ) , ( ) , ( r h r Ψ = Ψ where ) ( ) ) ( ) ( ) (( 2 1 ) ( 2 1 2 2 2 2 r V p p p m r V p m H op z op y op x op op r r + + + = + = and x i p op x = h ) ( y i p op y = h ) ( z i p op z = h ) ( op op i p = r h r with z z y y x x op ˆ ˆ ˆ + + = r . Schrödinger’s equation becomes dt t r d i t r r V t r m op ) , ( ) , ( ) ( ) , ( 2 2 2 r h r r r h Ψ = Ψ + Ψ where 2 op is the Laplacian operator 2 2 2 2 2 2 2 z y x op op op + + = = r r and 2 2 2 op op p = h . Stationary State Solutions: The stationary state solutions are of the form h r r / ) ( ) , ( iEt e r t r = Ψ ψ and the time independent equation is ) ( ) ( ) ( ) ( 2 2 2 r E r r V r m op r r r r h = + . Canonical Commutation Relations: It is easy to see that ij op j op i i x p δ h = ] ) ( , ) [( 0 ] ) ( , ) [( = op j op i p p 0 ] ) ( , ) [( = op j op i x x where δ ij is the Kroenecker delta function (i = 1,2,3 j = 1,2,3
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Unformatted text preview: ij = 0 if i j and ij = 1 if i = j, p 1 = p x , p 2 = p y , p 3 = p z , x 1 = x, x 2 = y, x 3 = z) and i op i x i p = h ) ( . Probability Density: The probability density is 2 | ) , ( | ) , ( t r t r r v = and the probability of finding the particle between r r and r d r r r + at time t is r d t r 3 ) , ( v with 1 ) , ( 3 = allspace r d t r v , where dxdydx r d = 3 in Cartesian coordinates and d drd r r d sin 2 3 = in Spherical coordinates....
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